Vol. 19, No. 1, March 2009 CODEN: CHAOEH ISSN: 1054-1500
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Cover image from Carsten Allefeld, Harald Atmanspacher, and Jiří Wackermann, Chaos 19, 015102 (2009).
GENERAL EDITORIAL POLICIES (2 pages). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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INFORMATION FOR CONTRIBUTORS (4 pages). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ANNOUNCEMENTS Announcement: Chinese, Japanese, and Korean characters available for author names (1 page) Mark M. Cassar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Announcement: Reduced fees for color in print (1 page) Mark M. Cassar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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REGULAR ARTICLES On Thermostats: Isokinetic or Hamiltonian? Finite or infinite? (7 pages) Giovanni Gallavotti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013101
Generalized projective synchronization in time-delayed systems: Nonlinear observer approach (9 pages) Dibakar Ghosh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013102
Transition to complete synchronization in phase-coupled oscillators with nearest neighbor coupling (5 pages) Hassan F. El-Nashar, Paulsamy Muruganandam, Fernando F. Ferreira, and Hilda A. Cerdeira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013103
Anticipating synchronization of chaotic systems with time delay and parameter mismatch (10 pages) Qi Han, Chuandong Li, and Junjian Huang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013104
Enhanced synchronizability in scale-free networks (5 pages) Maoyin Chen, Yun Shang, Changsong Zhou, Ye Wu, and Jürgen Kurths . . . . . . . . . . . . .
013105
Outer synchronization of coupled discrete-time networks (7 pages) Changpin Li, Congxiang Xu, Weigang Sun, Jian Xu, and Jürgen Kurths . . . . . . . . . . . . . .
013106
Yet another 3D quadratic autonomous system generating three-wing and four-wing chaotic attractors (6 pages) L. Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013107
State and parameter estimation of spatiotemporally chaotic systems illustrated by an application to Rayleigh–Bénard convection (10 pages) Matthew Cornick, Brian Hunt, Edward Ott, Huseyin Kurtuldu, and Michael F. Schatz . . . .
013108
Generalized outer synchronization between complex dynamical networks (9 pages) Xiaoqun Wu, Wei Xing Zheng, and Jin Zhou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013109
Regular and chaotic dynamics of magnetization precession in ferrite–garnet films (14 pages) Anatoliy M. Shuty and Dmitriy I. Sementsov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013110
Distinguished trajectories in time dependent vector fields (18 pages) J. A. Jiménez Madrid and A. M. Mancho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013111
Pinning control of fractional-order weighted complex networks (9 pages) Yang Tang, Zidong Wang, and Jian-an Fang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013112
Chimera states in heterogeneous networks (8 pages) Carlo R. Laing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013113
Plykin-type attractor in nonautonomous coupled oscillators (10 pages) Sergey P. Kuznetsov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013114
(Continued)
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© 2009 American Institute of Physics
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Chaos, Vol. 19, No. 1, 2009
Coherence resonance induced by rewiring in complex networks (4 pages) Mi Jiang and Ping Ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013115
Experimental verification of rank 1 chaos in switch-controlled Chua circuit (9 pages) Ali Oksasoglu, Serdar Ozoguz, Ahmet S. Demirkol, Tayfun Akgul, and Qiudong Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013116
Effect of chemical synapse on vibrational resonance in coupled neurons (6 pages) Bin Deng, Jiang Wang, and Xile Wei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013117
Generalized synchronization of chaotic systems: An auxiliary system approach via matrix measure (10 pages) Wangli He and Jinde Cao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013118
Power grid vulnerability: A complex network approach (6 pages) S. Arianos, E. Bompard, A. Carbone, and F. Xue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013119
Pinning synchronization of delayed dynamical networks via periodically intermittent control (8 pages) Weiguo Xia and Jinde Cao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles? (10 pages) Carlo Bianca and Lamberto Rondoni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013121
Node-to-node pinning control of complex networks (11 pages) Maurizio Porfiri and Francesca Fiorilli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013122
Influence of noise on the sample entropy algorithm (7 pages) Sofiane Ramdani, Frédéric Bouchara, and Julien Lagarde . . . . . . . . . . . . . . . . . . . . . . . . .
013123
Simple driven chaotic oscillators with complex variables (7 pages) Delmar Marshall and J. C. Sprott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Adaptive gain fuzzy sliding mode control for the synchronization of nonlinear chaotic gyros (9 pages) Mehdi Roopaei, Mansoor Zolghadri Jahromi, and Shahram Jafari . . . . . . . . . . . . . . . . . . .
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Self-organization of a neural network with heterogeneous neurons enhances coherence and stochastic resonance (6 pages) Xiumin Li, Jie Zhang, and Michael Small . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013126
Dynamical mechanism of intrinsic localized modes in microelectromechanical oscillator arrays (9 pages) Qingfei Chen, Liang Huang, Ying-Cheng Lai, and David Dietz . . . . . . . . . . . . . . . . . . . . . .
013127
Four twins for a paradox: On “sensitive” twins and the biological counterpart of the “twin paradox” (4 pages) Fortunato A. Ascioti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013128
Low dimensional description of pedestrian-induced oscillation of the Millennium Bridge (5 pages) Mahmoud M. Abdulrehem and Edward Ott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013129
The development of generalized synchronization on complex networks (9 pages) Shuguang Guan, Xingang Wang, Xiaofeng Gong, Kun Li, and C.-H. Lai . . . . . . . . . . . . . .
013130
The effect of noise on the complete synchronization of two bidirectionally coupled piecewise linear chaotic systems (10 pages) Yuzhu Xiao, Wei Xu, Xiuchun Li, and Sufang Tang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013131
Invariant submanifold for series arrays of Josephson junctions (9 pages) Seth A. Marvel and Steven H. Strogatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013132
„ , p , q…-vulnerabilities: A unified approach to network robustness (11 pages) Regino Criado, Javier Pello, Miguel Romance, and María Vela-Pérez . . . . . . . . . . . . . . . .
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Onset of synchronization in weighted scale-free networks (8 pages) Wen-Xu Wang, Liang Huang, Ying-Cheng Lai, and Guanrong Chen . . . . . . . . . . . . . . . . .
013134
Arm splitting and backfiring of spiral waves in media displaying local mixed-mode oscillations (6 pages) Qingyu Gao, Lu Zhang, Qun Wang, and I. R. Epstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013135
A dynamical systems proof of Kraft–McMillan inequality and its converse for prefix-free codes (5 pages) Nithin Nagaraj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013136
(Continued)
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Chaos, Vol. 19, No. 1, 2009
A new criterion to distinguish stochastic and deterministic time series with the Poincaré section and fractal dimension (13 pages) Abbas Golestani, M. R. Jahed Motlagh, K. Ahmadian, Amir H. Omidvarnia and Nasser Mozayani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
013137
Capture and release of traveling intrinsic localized mode in coupled cantilever array (7 pages) Masayuki Kimura and Takashi Hikihara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Cantori of the dissipative sawtooth map (6 pages) Alessandra Celletti and Massimiliano Guzzo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chaos of elementary cellular automata rule 42 of Wolfram’s class II (6 pages) Fang-Yue Chen, Wei-Feng Jin, Guan-Rong Chen, Fang-Fang Chen, and Lin Chen . . . . .
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FOCUS ISSUE: NONLINEAR DYNAMICS IN COGNITIVE AND NEURAL SYSTEMS Introduction to Focus Issue: Nonlinear Dynamics in Cognitive and Neural Systems (3 pages) F. Tito Arecchi and Jürgen Kurths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Mental states as macrostates emerging from brain electrical dynamics (12 pages) Carsten Allefeld, Harald Atmanspacher, and Jiří Wackermann . . . . . . . . . . . . . . . . . . . . . .
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Inverse problems in dynamic cognitive modeling (21 pages) Peter beim Graben and Roland Potthast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
015103
Control of transient synchronization with external stimuli (5 pages) Marzena Ciszak, Alberto Montina, and F. Tito Arecchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
015104
A combined method to estimate parameters of neuron from a heavily noise-corrupted time series of active potential (9 pages) Bin Deng, Jiang Wang, and Yenqiu Che . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
015105
Automated synchrogram analysis applied to heartbeat and reconstructed respiration (8 pages) Claudia Hamann, Ronny P. Bartsch, Aicko Y. Schumann, Thomas Penzel, Shlomo Havlin, and Jan W. Kantelhardt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
015106
Numerical studies of slow rhythms emergence in neural microcircuits: Bifurcations and stability (8 pages) M. A. Komarov, G. V. Osipov, J. A. K. Suykens, and M. I. Rabinovich . . . . . . . . . . . . . . . .
015107
Hypothesis test for synchronization: Twin surrogates revisited (14 pages) M. Carmen Romano, Marco Thiel, Jürgen Kurths, Konstantin Mergenthaler, and Ralf Engbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
015108
Multistability, local pattern formation, and global collective firing in a small-world network of nonleaky integrate-and-fire neurons (8 pages) Alexander Rothkegel and Klaus Lehnertz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
015109
Controlling the onset of traveling pulses in excitable media by nonlocal spatial coupling and time-delayed feedback (14 pages) Felix M. Schneider, Eckehard Schöll, and Markus A. Dahlem . . . . . . . . . . . . . . . . . . . . . . .
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Nonlinear analysis and modeling of cortical activation and deactivation patterns in the immature fetal electrocorticogram (8 pages) Karin Schwab, Tobias Groh, Matthias Schwab, and Herbert Witte . . . . . . . . . . . . . . . . . . .
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Noise-enhanced target discrimination under the influence of fixational eye movements and external noise (7 pages) Christian Starzynski and Ralf Engbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
015112
Hypotheses on the functional roles of chaotic transitory dynamics (10 pages) Ichiro Tsuda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
015113
Detecting nonlinear oscillations in broadband signals (7 pages) Martin Vejmelka and Milan Paluš . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
015114
From working memory to epilepsy: Dynamics of facilitation and inhibition in a cortical network (17 pages) Sergio Verduzco-Flores, Bard Ermentrout, and Mark Bodner . . . . . . . . . . . . . . . . . . . . . . .
015115
(Continued)
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Chaos, Vol. 19, No. 1, 2009
Generalized memory associativity in a network model for the neuroses (11 pages) Roseli S. Wedemann, Raul Donangelo, and Luís A. V. de Carvalho . . . . . . . . . . . . . . . . . .
015116
Graph analysis of cortical networks reveals complex anatomical communication substrate (7 pages) Gorka Zamora-López, Changsong Zhou, and Jürgen Kurths . . . . . . . . . . . . . . . . . . . . . . .
015117
ERRATA Erratum: “Robust Hⴥ synchronization of chaotic Lur’e systems” †Chaos 18, 033113 „2008…‡ (1 page) He Huang and Gang Feng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Erratum: “Synchronization in monkey visual cortex analyzed with an information-theoretic measure” †Chaos 18, 037130 „2008…‡ (1 page) Nikolay V. Manyakov and Marc M. Van Hulle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CUMULATIVE AUTHOR INDEX (1 page) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CHAOS 19, 010201 共2009兲
Announcement: Chinese, Japanese, and Korean characters available for author names Mark M. Cassar Publisher, Journals & Technical Publications, American Institute of Physics, Suite 1NO1, 2 Huntington Quadrangle, Melville, New York 11747-4502, USA
共Received 19 December 2008; published online 20 January 2009兲 关DOI: 10.1063/1.3078244兴 Beginning on 1 January 2009, authors with Chinese, Japanese, or Korean names may choose to have their names published in their own language, alongside the English versions of their names, in the author list of their publications.1 For Chinese, authors may use either Simplified or Traditional characters. In order to participate in this free service, interested authors must include Chinese, Japanese, or Korean characters within the author list of their manuscripts when submitting or resubmitting. The publisher does not offer a translation service and will not insert names; this is the author’s responsibility. To participate, authors must prepare their manuscripts using Microsoft Word or using the CJK LaTeX package. Specific guidelines for each authoring
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Announcement: Reduced fees for color in print Mark M. Cassar Publisher, Journals & Technical Publications, American Institute of Physics, Suite 1NO1, 2 Huntington Quadrangle, Melville, New York 11747-4502, USA
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CHAOS 19, 013101 共2009兲
On Thermostats: Isokinetic or Hamiltonian? Finite or infinite? Giovanni Gallavotti Dept. Fisica-INFN, Universita di Roma 1, P. le Moro 5, 00185 Roma, Italy and IHP, Rue P&M Curie, 75005 Paris, France
共Received 9 October 2008; accepted 2 December 2008; published online 7 January 2009兲 The relation between finite isokinetic thermostats and infinite Hamiltonian thermostats is studied and their equivalence in the thermodynamic limit is heuristically discussed. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3054710兴 Studies on nonequilibrium statistical mechanics progressed considerably after the introduction of artificial forces supposed to simulate the interaction of a “test system” with “heat reservoirs,” also called “thermostats.” Simulations could be developed eliminating the need of very large systems to model the action of heat reservoirs. The drawback is that the equations of motion are no longer Hamiltonian. The simulations led to developments and to many new insights into nonequilibrium, particularly with regard to the theory of large fluctuations (fluctuation theorems, work relations, and attempted applications to problems ranging from biophysics to fluid turbulence). An ongoing question has been, therefore, whether such thermostat models are just devices to generate simulations that may have little to do with physical reality and, therefore, in the end not really relevant for physics. There are, however, conjectures of equivalence between various kinds of thermostats and the often preferred “infinite thermostats” which, being Hamiltonian, are considered more fundamental (in spite of being infinite in size) or the stochastic thermostats. Here we try to substantiate, via a heuristic analysis, the equivalence conjecture between “Hamiltonian” and “isokinetic” thermostats by discussing it in precise terms. “Isokinetic” will mean that artificial forces are introduced whose role is to turn into an exact constant of motion the total kinetic energy of the particles identified as particles of any of the thermostats interacting with the test system. The novelty here is that a careful distinction is made between the test system particles and the particles of the thermostats in contact with it but physically located in containers outside the system (as in most real thermal baths). The artificial forces only act on the latter: this is a substantial difference from most cases considered in the literature in which the artificial forces act also on the test system particles (technically called “bulk thermostats”). It is convenient to call the thermostats considered here “peripheral thermostats.” The test system will be kept fixed but the thermostats will be allowed to be of arbitrary size, and their behavior as the size becomes infinite is what will interest us. The conclusion is that, under a suitable assumption, a peripheral isokinetic thermostat becomes in the thermodynamic limit, when its container becomes infinite, completely equivalent to a Hamiltonian infinite thermostat: in the sense that the time evolution of the configurations (i.e., of the phase space point representing 1054-1500/2009/19共1兲/013101/7/$25.00
test and interaction systems) is, with probability 1, the same as that obtained by letting the isokinetic containers become infinite. In bulk thermostats there cannot be such strict equivalence because motion remains nonHamiltonian even in the limit of infinite systems. The analysis reinforces, as a by-product, the identification (modulo an additive total time derivative) between phase space contraction and entropy production. I. THERMOSTATS
A classical model for nonequilibrium, for instance, in Ref. 1, is a test system in a container ⍀0, for instance, a sphere of radius R0 centered at the origin O, and several Interaction systems containing the thermostats: we denote their containers ⍀ j and they can be thought of 共to fix ideas兲 as the sets ⍀ j consisting of disjoint sectors ⍀ j = 兵 苸 R3 , 兩兩 ⬎ R0 , · k j ⬍ 兩兩 j其, j = 1 , . . . , n, k j distinct unit vectors, realized, for instance, as disjoint sectors in R3 共see Fig. 1兲; i.e., as cones in R3 with vertex at the origin deprived of the points inside the sphere containing the test system. For precision of language, we shall call such containers “spherically truncated cones,” but the actual shape could be rather arbitrarily changed, as it will appear. The terms “test” and “interaction” systems were introduced in Ref. 1. The contact between test system and thermostats occurs only through the common boundaries 共located on the boundary of the ball ⍀0兲 of the test system. No scaling, of time or space, will be considered here. In the quoted reference, as well as in later related works,2–4 the particles contained in ⍀0 , . . . , ⍀n were quantum particles and the interaction systems were infinitely extended 共and obeying a linear Schrödinger equation兲 and each was initially in a Gibbs state at respective temperatures T1 , . . . , Tn. Here the particles will be classical, with unit mass, elastically confined in ⍀0 , ⍀1 艚 ⌳r , . . . , ⍀n 艚 ⌳r with ⌳r a finite ball, centered at O, of radius r ⬎ R0. The temperatures in the interaction systems, here called thermostats, will be defined by the total kinetic energies in each of them, and will be kept a constant of motion by adding a phenomenological “thermostatting force.” Hence, the qualification of isokinetic that will be given to such thermostats. More appropriately, one should call such thermostats “peripherally isokinetic” because most often in the literature the term “isokinetic,” instead refers to systems in which the total kinetic
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© 2009 American Institute of Physics
013101-2
Chaos 19, 013101 共2009兲
Giovanni Gallavotti
U共x兲 = 兺q* ,q
⬘ ⬙苸X艚⌳r
共q⬘ − q⬙兲,
K共x兲 = 兺qi苸X艚⌳rq˙2i /2,
where the ⴱ means that the sum is restricted to the pairs q⬘ , q⬙, which are either in the same ⍀ j or consist of two elements q⬘ , q⬙, of which one is in ⍀0; this means that particles in ⍀0 interact with all the others, but the particles in ⍀ j interact only with the ones in ⍀ j 艛 ⍀0. The ’s will be, for simplicity, the same for all pairs. The system in ⍀0 interacts with the thermostats but the thermostats interact only with the system 共see Fig. 1兲: FIG. 1. The 1 + n boxes ⍀ j 艚 ⌳r, j = 0 , . . . , n, are marked C0 , C1 , . . . , Cn and contain N0 , N1 , . . . , Nn particles, mass m = 1, with positions and velocities ˙ ,X ˙ , ... ,X ˙ , respectively. The E are external, denoted X0 , X1 , . . . , Xn, and X 0 1 n positional, nonconservative, forces; the multipliers ␣ j are so defined that the 1 ˙2 kinetic energies K j = 2 X j are exact constants of motion.
˙ ,X ,X ˙ , . . . ,X ,X ˙ 兲. x = 共X0,X 0 1 1 n n Hence, if x 苸 H共⌳r兲, the energy U共x兲 can be written as n
U共x兲 = U0共X0兲 + 兺 共U j共X j兲 + U0,j共X0,X j兲兲
共1.2兲
j=1
energy of all particles 共in the test and interaction systems兲 is maintained constant. The latter are called bulk thermostats; our models will correspond to a system in which no internal microscopic friction occurs and which exchanges energy with external systems kept at constant temperature. The properties that we discuss cannot hold for bulk thermostats. However, we shall call our thermostats simply “isokinetic” except in the last section. The arrangement is illustrated in Fig. 1. Remark: Peripherally isokinetic thermostats have been considered in the literature in simulations,5 and their physically correct behavior was immediately noted, sparking investigations about the equivalence problem 共see also Refs. 6 and 7兲. Recently, a case of a model in which only hard core interactions between particles were present, and the test system was thermostatted peripherally, has been studied in Ref. 8, showing the thermostat action being efficient and measurable even in such an extreme situation. Phase space: Phase space H is the collection of locally finite particle configurations x = 共. . . , qi , q˙i , . . . 兲⬁i=1: ˙ ,X ,X ˙ , . . . ,X ,X ˙ 兲 = 共X,X ˙兲 x = 共X0,X 0 1 1 n n
共1.1兲
with X j 傺 ⍀ j; hence, X 傺 ⍀ = 艛nj=0⍀ j, and q˙i 苸 R3. In every ball B共r , O兲, of radius r and center at the origin O, fall a finite number of points of X. The space H共⌳r兲 will be the space of the finite configurations with X 傺 ⌳r. It will be convenient to imagine a con˙ 兲 figuration x as consisting of a configuration 共X0 , X 0 ˙ 兲 苸 H共B共R0 , O兲兲 and by n configurations 共X j , X j 苸 共H共⍀ j 艚 R3 / B共R0 , O兲兲兲, j = 1 , . . . , n. Interaction: The interparticle interaction will be a pair potential with finite range r and superstable in the sense that is non-negative, decreasing in its range (i.e., “repulsive”), smooth and positive at the origin. Remark: Singularities such as hard core could be also considered 共at the heuristic level of this paper兲 but are left out for brevity. For more general cases, like Lennard-Jones interparticle potentials or for modeling by external potentials the containers walls, see Ref. 9. The potential and kinetic energies of the configuration x 苸 H共⌳r兲 are
˙ 兲= 1X ˙2 and the kinetic energies will be K j共X j 2 j . The equations of motion will be 共see Fig. 1兲 ¨ = − U 共X 兲 − X 0i i 0 0
兺 iU0,j共X0,X j兲 + Ei共X0兲,
j⬎0
共1.3兲 ˙ ¨ = − U 共X 兲 − U 共X ,X 兲 − ␣ X X ji i j j i 0,j 0 j j ji , where the first label, j = 0 , . . . , n, denotes the thermostat 共or system兲 and the second the derivatives with respect to the coordinates of the points in the corresponding thermostat 关hence the labels i in the subscripts 共j , i兲 have 3N j values兴. The multipliers ␣ j are, for j = 1 , . . . , n,
␣j ⬅
˙ Qj − U j , 2K j
˙ · U 共X ,X 兲, Qj ⬅ − X j j 0,j 0 j
共1.4兲
and the “walls” 共i.e., the boundaries ⍀i , ⌳r兲 delimiting the different containers will be supposed elastic. A more general model to which the analysis that follows also applies is in Ref. 10. It is also possible to imagine thermostats acting in the ˙ ; this bulk of the test system by adding a further force −␣0X 0 is, for instance, of interest in electric conduction models,11 where the dissipation is due to energy exchanges with oscillations 共“phonons”兲 of an underlying lattice of obstacles. Such bulk thermostatted systems will not be discussed because, for physical reasons, their dynamics cannot be expected to be equivalent to the Hamiltonian one in the strong sense that will be considered here. The thermostat forces would introduce an effective friction on the system motion not disappearing as the size of the systems grows, as is always the case in bulk thermostatted systems. Other thermostats considered in the literature,12,13 could be studied and be subject to a similar analysis, which would be interesting; e.g., the Nosé–Hoover or the isoenergetic thermostats. Note that even the isoenergetic thermostat does not conserve Gibbs states 共in the presence of a test system兲. The equations of motion will be called isokinetically thermostatted because the multipliers ␣ j are so defined to as keep the K j exactly constant for j ⬎ 0. The forces Ei共X0兲 are positional nonconservative, smooth, forces. The numbers N j of particles in the initial data may be random but will be
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picked with a distribution giving them average values of within positive and asymptotically N j / 兩⍀ j 艚 ⌳r兩 ⌳r-independent bounds as r → ⬁. Initial data: The probability distribution 0 for the ran˙ / N !, dom choice of initial data will be, if dx ⬅ 兿nj=0dX jdX j j the limit as ⌳r → ⬁ of
0,⌳r共dx兲 = const e−H0共x兲dx
共1.5兲
˙ 兲 − N + U 共x兲兲 and  ⬅ 1 / k T , j with H0共x兲 = 兺nj=0 j共K j共X j j j j j B j ⬎ 0 and 0 ⬎ 0 arbitrary. Here, = 共0 , 1 , . . . , n兲 and T = 共T0 , T1 , . . . , Tn兲 are, respectively, fixed chemical potentials and temperatures 共kB being Boltzmann’s constant兲. The limit 0 as ⌳r → ⬁ of the distribution in Eq. 共1.5兲 makes sense 共with particles allowed to be located in the infinite containers ⍀ j, j ⬎ 0兲 provided it is interpreted as a Gibbs distribution 0 obtained by taking the “thermodynamic limit” ⌳r → ⬁, supposing for simplicity that the parameters j , T j, j ⬎ 0 do not correspond to phase transition points 共which would require care to consider boundary conditions which generate pure phases14兲. It will be convenient to think always the initial data chosen with respect to the latter distribution: if ⌳r ⬍ ⬁, the particles positions and velocities outside ⌳r will, however, be imagined fixed in time 共“frozen,” see Refs. 15 and 16兲. Therefore, defining ⬁
Z j共, 兲 =
兺 N=0
冕
e−共K j+U j−N j兲
共⌳r艚⍀ j兲⫻R3
˙ dXdX N!
共1.6兲
and  p j共 , 兲 = lim⌳r→⬁共1 / 兩⌳r 艚 ⍀ j兩兲 log10 Z j共 , 兲, the thermostat’s density and average potential energy will be
␦j =
j p j ,  j
uj = −
 j p j 3 − k BT j ␦ j − j ␦ j 2  j
共1.7兲
and ␦ j , u j , 23 kBT j will be supposed, respectively, to be the average density, average potential energy density, and average kinetic energy per particle in the initial configurations: without loss of generality because this holds with 0-probability 1 共by the no-phase-transitions assumption兲. II. DYNAMICS
In general, time evolution with the thermostatted dynamics changes the measure of a volume element in phase space by an amount related to 共but different from兲 the variation of the Liouville volume. Minus the change per unit time of a volume element measured via Eq. 共1.5兲 is, in the sectors of phase space containing N j ⬎ 0 particles inside ⌳r 艚 ⍀ j, j = 0 , 1 , . . . , n, with kinetic energy K j,⌳r共x兲,
共x兲 =
Q
j ˙ +U ˙ 兲, 共1 − 共3N j兲−1兲 + 0共K 兺 0 0 k T 共x兲 j⬎0 B j
共2.1兲
where kBT j共x兲 = 32 K j,⌳r / N j, and kBT j共x兲 → −1 j for ⌳r → ⬁, at least for the initial data, with 0-probability 1.
Remarks: 共1兲 The dynamics given by the equations of motion Eq. 共1.5兲 or by the same equations with ␣ j ⬅ 0 are, of course, different. We want to study their difference. 共2兲 The choice of the initial data with the distribution 0 regarded as obtained by a thermodynamic limit of Eq. 共1.5兲 rather than 共more naturally兲 with 0,⌳ ⬘ 共dx兲, r
⬘ r共dx兲 0,⌳ dx
n
冉
= const e−H0共x兲 兿 ␦ K j − j=1
3N jkBT j 2
冊
共2.2兲
with N0 , N1 , . . . , Nn fixed, N j / 兩⍀ j 艚 ⌳r兩 = ␦ j, j ⬎ 0, and no particles outside ⌳r is done to refer, in the following, to Refs. 15 and 16. A heuristic analysis would be possible also with this, and other, alternative choices. 共3兲 Equation 共2.2兲 is natural, although less convenient notationally, because in the case n = 1, E = 0, and 0 = 1 =  with −1 = kBT1共1 − 1 / 3N1兲−1, it is exactly stationary 共a minor extension of Ref. 12兲, if multiplied by the density 共x兲 = exp共−兺 j⬎0U共X0 , X j兲兲, which is the “missing” Boltzmann factor in Eq. 共1.5兲, and can therefore be called an equilibrium distribution. Choosing initial data with the distribution 0, let x r,a兲 → x共⌳r,a兲共t兲 ⬅ S共⌳ x, a = 0 , 1 be the solution of the equations t of motion with ␣ j = 0 共a = 0, “Hamiltonian thermostats”兲 or ␣ j given by Eq. 共1.3兲 共a = 1, “isokinetic thermostats”兲 and ignoring the particles initially outside ⌳r,16 and let S共0兲 t x be the dynamics lim⌳r→⬁S共⌳r,0兲x. Existence of a solution to the equations of motion is a problem only if we wish to study the ⌳r → ⬁ limit; i.e., in the case in which the thermostats are infinite 共thermodynamic limit兲. It is a very difficult problem even in the case in which ␣ = 0 and the evolution is Hamiltonian. For n = 1, ␣1 = 0,  = 0 ⬅ , and E = 0 共a case that will be called equilibrium兲, it was shown15 that a solution to the 共Hamiltonian兲 equations of motion exists for almost all initial data x chosen with a distribution obtained by multiplying 0共dx兲 by an arbitrary density function 共x兲; it is defined as the limit as ⌳r → ⬁ of r,0兲 x in ⍀ 艚 ⌳r. the finitely many particle evolutions S共⌳ t Recently, the related problem of a single finite system and no thermostat forces has been solved in Refs. 15 and 16, where it has been shown that, for a set of initial data which have probability 1 with respect to all distributions like Eq. 共1.5兲, the Hamiltonian equations make sense and admit a unique solution, but the general nonequilibrium cases remain open. Therefore, in the following, I shall suppose, heuristically, a property 共called below “locality of evolution”兲 of the equations of motion Eq. 共1.3兲 with and without the thermostatting ˙ . forces ␣ jX j The question will then be: Are the two kinds of thermostats equivalent? This is often raised because the isokinetically thermostatted dynamics is considered “unphysical” on grounds that are viewed, by some, sufficient to ban isokinetic thermostats from use in physically meaningful problems, such as their
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use to compute transport coefficients.12 The following heuristic considerations show that the latter would be too hasty a conclusion.
E共x兲 = sup
,兩兩⬎r R⬎r log共2兩兩/r兲
lim
共⌳r,1兲 ˙ ⌳r,1共t兲兲 r,1共t兲,X x⌳r,1共t兲 = 共X⌳ x, i=0,. . .,n = St i i 共⌳r,0兲 ˙ ⌳r,i共t兲兲 r,0共t兲,X x⌳r,0共t兲 = 共X⌳ x, i=0,. . .,n = St 0 i
共3.1兲
共0兲 ˙ 0共t兲兲 x0共t兲 = 共X0i 共t兲,X i=0,. . .,n = St x. i
A particle 共qi , q˙i兲 located at t = 0 in, say, the jth thermostat then evolves 关see Eqs. 共1.3兲兴 as qi共t兲 = qi +
冕
t
q˙i共t⬘兲dt⬘ ,
0
q˙i共t兲 = e
−兰t0␣ j共t⬘兲dt⬘
q˙i +
冕
共3.2兲
t
dt⬙e
t −兰t ␣ j共t⬘兲dt⬘ ⬙
Fi共t⬙兲dt⬙ ,
⌳r→⬁
i
冉
q˙2i 1 + 兺 共qi − q j兲 + Fr 2 2 j⫽i
N共j,⌳r兲 = ␦ j, 兩⌳r 艚 ⍀ j兩
lim
⌳r→⬁
U共j,⌳r兲 = uj , 兩⌳r 艚 ⍀ j兩
共3.5兲
(1) there is B共x , t兲 ⬎ 0, continuous and nondecreasing in 兩t兩, such that E共x共⌳r,a兲共t兲兲 艋 B共x , t兲 , a = 0 , 1; (2) the limits x共a兲共t兲 = lim⌳r→⬁x共⌳r,a兲共t兲 exist and are in H0 for all t, with E共x共0兲共t兲兲 艋 B共x , t兲; (3) x共0兲共t兲 solve the Hamiltonian equations and the latter admit a unique solution in H0. Remarks:
where Fi共t兲 = −qi共U j共X j共t兲兲 + U j,0共X0共t兲 , X j共t兲兲兲. The above relations hold up to the first collision of the i-th particle with the containers walls, afterwards they continue until the next collision with the new initial condition given by the elastic collision rule; they hold for the three dynamics considered in Eqs. 共3.1兲, provided ␣ j = 0 in the second and third cases and ⌳r is finite in the first and second cases. The first difficulty with infinite dynamics is to show that the speeds and the number of particles in a finite region of diameter r ⬎ R0 remain finite and bounded in terms of the region diameter 共and the initial data兲 for all times or, at least, for any prefixed time interval. Therefore, we shall suppose that the configurations evolve in time keeping the “same general statistical properties” that certainly occur with probability 1 with respect to the equilibrium distributions or the distributions like 0 in Eq. 共1.5兲, i.e., density and velocity that grow at most logarithmically with the size of the region in which they are observed,15,16 and average kinetic energy, average potential energy, average density having, asymptotically as ⌳r → ⬁, values 23 kBT j , ␦ j , u j depending only on the thermostat’s parameters 共 j , T j , j ⬎ 0兲 关see Eq. 共1.7兲兴. More precisely, let the local energy in ⍀ 艚 B共 , R兲, 苸 R3, R ⬎ R0 + r be
兺 q 苸X艚B共,R兲
共3.4兲
with ␦ j ⬎ 0 , u j given by Eq. 共1.7兲, if N共j , ⌳r兲 and U共j , ⌳r兲 denote, respectively, the number of particles and their internal potential energy in ⍀ j 艚 ⌳r兲. The set of configurations x 苸 H0 has 0,⌳r-probability 1.16 The discussion in this paper relies on the assumptions 1–3 below, motivated by the partial results in Refs. 15 and 16, as it will appear shortly. It is to be expected that the probability distributions 共⌳r,a,t兲 , 共0,t兲 obtained by the evolut 共0兲 r,a兲 tion of 0 with S共⌳ , S 共a = 0 , 1兲, and all configurations t t in x 苸 H0 share the following properties. Local dynamics assumption: With 0-probability 1 for x 苸 H0, the number of collisions 共i , t , ⌳r , a兲 that the i-th particle of x共⌳r,a兲共t⬘兲 has with the containers walls, for 0 ⱕ t⬘ ⱕ t, is bounded uniformly in ⌳r , a, and
0
W共x; ,R兲 =
W共x; ,R兲 R3
and call H0 the configurations in H with E共x兲 ⬍ ⬁ and
III. HEURISTIC DISCUSSION AND EQUIVALENCE ISOKINETIC VERSUS HAMILTONIAN
The first paper dealing with equivalence issues is Ref. 6; its ideas are taken up here, somewhat modified, and extended. A detailed comparison with Ref. 6 is in the last section. In the Hamiltonian approach, the thermostats are infinite systems with no thermostatting forces 共␣ j ⬅ 0兲, the initial data are still chosen with the distribution 0 discussed above. Let
sup
冊
with F = max兩q兩, and its “logarithmic scale” average
共3.3兲
共a兲
共b兲 共c兲 共d兲 共e兲
共f兲
The limits of x共⌳r,a兲共t兲, as ⌳r → ⬁, are understood in the 共t兲 , p共0,a兲 共t兲兲 of sense that for each i, the limits 共q共0,a兲 i i 共⌳r,a兲 共⌳r,a兲 共t兲 , pi 共t兲兲 exist together with their first two 共qi 共t兲 , p共0,a兲 共t兲兲 are twice continuderivatives; and 共q共0,a兲 i i ously differentiable in t for each i. It can be shown that, in the Hamiltonian case a = 0, the uniform bounds in 共2兲 imply the existence of the limits. However, they do not imply that x共0兲共t兲 苸 H0; i.e., they do not imply the second of Eq. 共3.5兲. The number of points of x共⌳r,a兲共t兲, a = 0 , 1, in a ball B共R , 兲 is bounded by B共x , t兲R3, for all R , with R ⬎ r log10 2兩兩 / r and 兩t⬘兩 ⬍ t. The speed of a particle located in q 苸 R3 is bounded by B共x , t兲共2 log10 2兩q兩 / r兲3 for 兩t⬘兩 艋 t. Comments 共b兲 and 共c兲 say that locally, the particles keep a finite density and reasonable energies and momentum distributions. An implication is that Eqs. 共3.2兲 have a meaning with probability 1 on the choice of the initial data x. It is very important that the assumption that dynamics develops within H0 implies that at all times Eq. 共3.5兲 will hold with ␦ j , u j time independent, physically reflecting the infinite sizes of the thermostats whose density and energy cannot change in any finite time. The analysis of the nonequilibrium cases can be partially performed in similar Hamiltonian cases as done in the detailed and constructive analysis in Ref. 16, but dropping the requirement in Eq. 共3.5兲.
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It seems reasonable that by the method in Ref. 16, the restriction of satisfying Eq. 共3.5兲 can be removed in the Hamiltonian model. New ideas seem needed to obtain the local dynamics property in the case of the thermostatted dynamics. The multipliers ␣ j are sums of two terms. The first is ˙ · U 共X ,X 兲兩 兩X j j 0,j 0 j 2 ˙ X
共3.6兲
j
关see Eq. 共1.4兲兴 and the short range of the potential implies that the force − jU0,j共X0 , X j兲 is a sum of contributions bounded by F ⬅ max兩共q兲兩 times the number of pairs of particles in the band of width r around the boundary of the container ⍀0 关because, by Eq. 共3.5兲, E共x兲 ⬍ + ⬁兴; this is of order O共共R20rF␦兲2兲 if ␦ is an upper bound on the densities near ⍀0. Note that such a bound exists and is time independent, by the local evolution hypothesis 共above兲, but of course it is not uniform in the choice of the initial data x. Applying Schwartz’s inequality, B1 ⬎ 0 exists with ˙ · U 共X ,X 兲兩 兩X R20rF␦ j j 0,j 0 j 艋 B1 冑3N jkBT j␦⬘ ˙2 X
共3.7兲
j
for ⌳r large and ␦⬘ = min j⬎0␦ j, having used the first of Eq. 共3.5兲. ˙ = U共j , ⌳ 艚 ⍀ 兲, contribThe second term in ␣ j, with U j r j utes to the integrals in the exponentials of Eqs. 共3.2兲 as
冕
˙ u j共t兲 − u j共t⬘兲 U j dt⬙ ⯝ , 3kBT j t⬘ 2K j t
共3.8兲
where u j共t兲 is the specific energy at time t and the ⯝ reflects the use of the second equation in Eq. 共3.5兲 to estimate U j / 2K j as U j / 关kBT jN共j , ⌳r 艚 ⍀ j兲兴; it means equality up to quantities tending to 0 as r → ⬁. By the above hypothesis, the right-hand side of Eq. 共3.8兲 tends to 0 as ⌳r → ⬁ because the configurations 共initial and after evolution兲 are in H0, and, hence, have the same specific potential energies u j 关by Eq. 共3.5兲; see also comment 共e兲 above兴, while the contribution to the argument of the same exponentials from Eq. 共3.6兲 also tends to 1 by Eq. 共3.7兲. Taking the limit of Eq. 共3.2兲 at fixed i, this means that, for initial data in H0, hence with 0-probability 1, the limit motion as ⌳r → ⬁ 共with  j , j , j ⬎ 0, constant兲 satisfies the Hamiltonian equations qi共t兲 = qi +
冕
t
0
q˙i共t⬘兲dt⬘,
q˙i共t兲 = q˙i +
冕
t
Fi共t⬙兲dt⬙ ,
共3.9兲
0
and the solution to such equations is unique with probability 1. The conclusion is that in the thermodynamic limit, the thermostatted evolution becomes identical, in any prefixed time interval, to the Hamiltonian evolution on a set of configurations which have probability 1 with respect to the initial distribution 0, in spite of the nonstationarity of the latter.
In other words, suppose that the initial data are sampled with the Gibbs distributions of the thermostats particles 共with given chemical potentials and temperatures兲 and with an arbitrary distribution for the finite system in ⍀0 关with density with respect to the Liouville volume, for instance, with a Gibbs distribution at temperature T0 and chemical potential 0, as in Eq. 共1.5兲兴. Then, in the thermodynamic limit ⌳r → ⬁, the time evolution is the same that would be obtained, in the same limit, via a isokinetic thermostat acting in each container ⍀ j 艚 ⌳r to keep the total kinetic energy constant and equal to 23 N jkBT j. IV. ENTROPY PRODUCTION
It is important to stress that while, in the thermodynamic limit, the dynamics becomes the same for isokinetic and Hamiltonian thermostats, because the thermostat force on each particle tends to 0, the phase space contraction in the isokinetic dynamics does not go to zero, by Eqs. 共3.7兲 and 共3.8兲. Instead it becomes, up to an additive time derivative 关see Eq. 共2.1兲兴, = 兺 j⬎0Q j / kBT j. This is possible because is a sum of many quantities 共the ␣ j’s兲, each of which tends to 0 in the thermodynamic limit while their sum does not. The interest of the remark is that, while 兺 j⬎0Q j / kBT j is the natural definition of entropy production in both cases, in the literature it is often stated 共correctly so in the contexts兲 that entropy production is the phase space contraction, raising eyebrows because the latter vanishes in Hamiltonian models. However, in finite thermostat models, the phase space contraction rate depends on the metric used to measure volume in phase space; it has been stressed that the ambiguity affects the phase space contraction only by an additive quantity which is a time derivative of some function on phase space. Such ambiguity will not affect the fluctuations of the long time averages of the phase space contraction, which, therefore, has an intrinsic physical meaning for this purpose.17 In both the isokinetic and Hamiltonian cases the above 共which is, physically, the physical entropy production兲 dif˙ , from fers, by a time derivative H tot ¯ =
兺 j⬎0
冉
冊
˙ E共X0兲 · X Qj Qj 0 − − . k BT j k BT 0 k BT 0
共4.1兲
The time derivative in question here is the derivative of the total energy Htot = 0共兺 j艌0共K j共x兲 + U j共x兲兲 + 兺 j⬎0U0,j共x兲兲.17 Furthermore, ¯ generates the matter and heat currents.17 For this reason, the equivalence conjectures, of which the isokinetic-Hamiltonian is a prominent example 共see Ref. 6; Ref. 18, Sec. 8; Ref. 19; Ref. 11, Sec. 6; Ref. 20; Ref. 14, Sec. 9.11兲, to quote a few, are relevant for the theory of transport and establish a connection between the fluctuation dissipation theorem and the fluctuation theorem.17,21 References 15 and 16 bring the present analysis closer to a mathematical proof for repulsive interaction, and I hope to show in a future work that they actually lead to a full proof of the locality of the dynamics, at least in dimension d = 1 , 2, for other thermostat models.
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Giovanni Gallavotti
V. COMPARISON WITH REF. 6 AND COMMENTS
共1兲 Equivalence between different thermostats is widely studied in the literature and it is surprising that there are so many questions still raised about the very foundations, while little attention is devoted to trying to expand the analysis of the early works. A clear understanding of the problem was already set up in comparing isokinetic, isoenergetic, and Nosé–Hoover bulk thermostats in Ref. 6, where a history of the earlier results is presented as well 共see also Ref. 22兲. 共2兲 Finite thermostats acting on the boundary were studied already in Ref. 5 in special cases, and were recognized to be equivalent to thermostats acting on the bulk of the test system. More recently,23 isokinetic versus isoenergetic thermostats equivalence has been analyzed and the splitting of the phase space contraction into an entropy part and an “irrelevant” additive time derivative has been first stressed 共see also the later Refs. 24 and 10兲 and related to the interpretation and prediction of numerical simulations. 共3兲 The basic idea in Ref. 6 for the equivalence is that the multipliers defining the forces that remove the heat in finite thermostat models have equal average 共“equal dissipation”兲 in the thermodynamic limit, 关Ref. 6, Eq. 共15兲兴, thus making all evolutions equivalent. In Ref. 6, the expectation of observables in two thermostatted evolutions is represented via Dyson’s expansion of the respective Liouville operators starting from an equilibrium distribution: equivalence follows order by order in the expansion 共in the joint thermodynamic limit and infinite time limit兲 if a mixing property 关Ref. 6, Eq. 共23兲兴, of the evolution with respect to both the equilibrium and the stationary distributions is assumed. The method is particularly suitable for bulk thermostatted systems close to equilibrium where application of Dyson’s expansion can be justified, at least in some cases.25 共4兲 The main difference between the present work and Ref. 6 is that here, even far out of equilibrium, we discuss equivalence between the boundary thermostatted dynamics and Hamiltonian dynamics; therefore, we compare a situation in which the average value of the dissipation 关analog of Eq. 共32兲, Ref. 6兴 is⫽ 0 with one in which it is 0 exactly, at least formally. This is achieved by showing that the multipliers in the models in Fig. 1 vanish in the thermodynamic limit not only in average but also pointwise with probability 1; this is in agreement with the results in Ref. 5 and provides more theoretical grounds to explain them. It also means that in boundary thermostatted systems the analog of 关Eq. 共32兲, Ref. 6兴 does not tend to 0 when N → ⬁, although the analog of the average of the multipliers, corresponding to Eq. 共33兲, Ref. 6, does. 共5兲 In bulk thermostatted systems, there cannot be equivalence between the Hamiltonian and the isokinetic dynamics in the sense discussed in this paper; i.e., the identity of the dynamics of individual particles. However, as discussed already in Ref. 6, the expectation values of extensive observables could hold. On the other hand, the analysis in Ref. 6 should be extendible to cover also the
boundary thermostatted systems because, while the dissipation 共i.e., entropy production兲 does not vanish in the thermodynamic limit, the average of the multipliers still does 关see point 共3兲 above兴, and this is what is really needed in Ref. 6. 共6兲 Neither Dyson’s expansion convergence questions nor time-mixing properties, on which Ref. 6 is based, enter into the present analysis, but the assumptions needed on the dynamics 共local dynamics兲 are still strong and are only under partial control via the theory in Refs. 15 and 16. 共7兲 An important question is whether taking the time t → ⬁ limit after the thermodynamic limit ⌳r → ⬁ 共when, therefore, the dynamics are identical兲, the probability distribution S共0兲 t 0 tends to a limit , and still attributes probability 1 to H0; this is an apparently much harder question related to the difference between the transient results and the deeper, steady state results.26 共8兲 Finally, the choice, made here, of dimension 3 for the ambient space is not necessary for the analysis. Dimensions d = 1 , 2 , 3 would be equally suited. However, it is only if the thermostat’s container’s dimension is d = 3 that the system with infinite thermostats is expected to reach a stationary state. If d = 1 , 2, the equalization of the temperatures is expected to spread from the system to the reservoirs and to proceed indefinitely, tending to establish a constant temperature over larger and larger regions of size growing with a power of time.27
ACKNOWLEDGMENTS
I am grateful to C. Maes for bringing up the problem, and to him, E. Presutti, and F. Zamponi for criticism and hints. E. Presutti made the essential suggestion to use Ref. 16 to try to put the ideas in a precise mathematical context and to state properly the conditions on the walls collisions. I am also grateful to the referees for their remarks, which I have incorporated in the revised version. Work was partially supported by I.H.P., Paris, at the workshop Interacting Particle Systems, Statistical Mechanics and Probability Theory, 2008, through the Fondation Sciences Mathématiques de Paris. R. P. Feynman and F. L. Vernon, Ann. Phys. 24, 118 共1963兲. D. Abraham, E. Baruch, G. Gallavotti, and A. Martin-Löf, Stud. Appl. Math. 51, 211 共1972兲. 3 J. L. Lebowitz, in Proceedings of the VI IUPAP Conference on Statistical Mechanics, volume edited by S. A. Rice, K. F. Freed, J. C. Light 共University of Chicago Press, Chicago, 1971兲. 4 J. P. Eckmann, C. A. Pillet, and L. Rey Bellet, Commun. Math. Phys. 201, 657 共1999兲. 5 S. Y. Liem, D. Brown, and J. H. K. Clarke, Phys. Rev. A 45, 3706 共1992兲. 6 D. J. Evans and S. Sarman, Phys. Rev. E 48, 65 共1993兲. 7 G. Gallavotti, Eur. Phys. J. B 61, 1 共2008兲. 8 P. Garrido and G. Gallavotti, J. Stat. Phys. 126, 1201 共2007兲. 9 F. Bonetto, G. Gallavotti, A. Giuliani, and F. Zamponi, J. Stat. Phys. 123, 39 共2006兲. 10 G. Gallavotti, Chaos 16, 043114 共2006兲. 11 G. Gallavotti, J. Stat. Phys. 84, 899 共1996兲. 12 D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Fluids 共Academic, New York, 1990兲. 13 D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 共1993兲. 1 2
013101-7
G. Gallavotti, Statistical Mechanics. A Short Treatise 共Springer Verlag, Berlin, 2000兲. 15 C. Marchioro, A. Pellegrinotti, and E. Presutti, Commun. Math. Phys. 40, 175 共1975兲. 16 E. Caglioti, C. Marchioro, and M. Pulvirenti, Commun. Math. Phys. 215, 25 共2000兲. 17 G. Gallavotti, “Thermostats, chaos and Onsager reciprocity,” arXiv:0809.2165 共2008兲. 18 G. Gallavotti and E. G. D. Cohen, J. Stat. Phys. 80, 931 共1995兲. 19 G. Gallavotti, J. Stat. Phys. 78, 1571 共1995兲. 14
Chaos 19, 013101 共2009兲
Isokinetic–Hamiltonian thermostats
C. Wagner, R. Klages, and G. Nicolis, Phys. Rev. E 60, 1401 共1999兲. G. Gallavotti, Scholarpedia J. 3, 5893 共2008兲. 22 D. Ruelle, J. Stat. Phys. 100, 757 共2000兲. 23 F. Zamponi, G. Ruocco, and L. Angelani, J. Stat. Phys. 115, 1655 共2004兲. 24 A. Giuliani, F. Zamponi, and G. Gallavotti, J. Stat. Phys. 119, 909 共2005兲. 25 N. I. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sinai, Commun. Math. Phys. 154, 569 共1993兲. 26 E. G. D. Cohen and G. Gallavotti, J. Stat. Phys. 96, 1343 共1999兲. 27 D. Ruelle, J. Stat. Phys. 98, 57 共1998兲. 20 21
CHAOS 19, 013102 共2009兲
Generalized projective synchronization in time-delayed systems: Nonlinear observer approach Dibakar Ghosha兲 Department of Mathematics, Dinabandhu Andrews College, Garia, Calcutta-700 084, India
共Received 24 June 2008; accepted 2 December 2008; published online 12 January 2009兲 In this paper, we consider the projective-anticipating, projective, and projective-lag synchronization in a unified coupled time-delay system via nonlinear observer design. A new sufficient condition for generalized projective synchronization is derived analytically with the help of Krasovskii– Lyapunov theory for constant and variable time-delay systems. The analytical treatment can give stable synchronization 共anticipatory and lag兲 for a large class of time-delayed systems in which the response system’s trajectory is forced to have an amplitude proportional to the drive system. The constant of proportionality is determined by the control law, not by the initial conditions. The proposed technique has been applied to synchronize Ikeda and prototype models by numerical simulation. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3054711兴 In 1990, Pecora and Carroll1 first observed the synchronization of chaotic dynamical systems, and chaos synchronization has become a topic of great interest. Different kinds of synchronization phenomena have been discovered, such as complete synchronization, phase synchronization, lag synchronization, generalized synchronization, and so on. While most studies focused on these types of chaos synchronization in various fields, little attention has been paid to projective synchronization. In 1998, Gonzalez–Miranda2 observed that when chaotic systems exhibit invariance properties under a special type of continuous transformation, amplification and displacement of the attractor occurs. This degree of amplification or displacement obtained is smoothly dependent on the initial condition. Later, in 1999, Mainieri and Rehacek3 observed projective synchronization in partially linear three-dimensional chaotic systems. This paper proposes a new analytical treatment of stable projective synchronization (anticipating and lag) algorithm based on nonlinear observer design without the limitation of partially linearity. The scaling factor ␣ is determined by the control law, not the initial conditions. It is proved that nonlinear observer approach provides a unified treatment for synchronization in chaotic and hyperchaotic systems. Therefore, nonlinear observer approach provides a unified treatment for generalized projective synchronization of a large class of time-delayed systems with constant and variable time delay. Numerical simulations are proved to verify the effectiveness of the proposed scheme. I. INTRODUCTION
Synchronization between two dynamical system has stimulated a wide range of research activity.1 The phenomena of synchronization in coupled systems have been extensively studied in the context of laser dynamics, electronics circuits, chemical, and biological systems.4 Over the past dea兲
Electronic mail: drghoshគ
[email protected].
1054-1500/2009/19共1兲/013102/9/$25.00
cade, following complete synchronization,1,5 several new types of synchronization have been found in interesting chaotic systems, such as generalized synchronization 共GS兲,6 phase synchronization,7 lag synchronization,8 anticipatory synchronization,9 antiphase synchronization,10 etc. Among all kinds of chaos synchronization, projective synchronization, characterized by a scaling factor that two systems synchronize proportionally, is one of the most interesting problems. Gonzalez–Miranda2 observed that when chaotic systems exhibit invariance properties under a special type of continuous transformation, amplification and displacement of the attractor occurs. This degree of amplification or displacement obtained is smoothly dependent on the initial condition. By this definition, complete synchronization is not a special case of projective synchronization. Mainieri and Rehacek,3 in 1999, called this type of synchronization projective synchronization, and declared that the two identical systems could be synchronized up to a scaling factor ␣. The scaling factor is a constant transformation between the synchronized variables of the master and slave systems. In Refs. 2 and 3, projective synchronization occurred because the response system had neutral stability, i.e., the largest conditional Lyapunov exponent was zero, so it followed a trajectory that was proportional to the drive system attractor with a scaling factor ␣ which was determined by initial conditions. Projective synchronization is not in the category of GS because the slave system of projective synchronization is not asymptotically stable. The response system attractor possesses the “same topological characteristic 共such as Lyapunov exponents and fractal dimensions兲” as the slave system attractor.11 Recently, Hoang et al.12 proposed a new type of synchronization in multidelay feedback systems, which is called projective-anticipating synchronization. This synchronization is a combination of projective and anticipatory synchronization, and the states of master and slave are related by ay共t兲 = bx共t + 兲共 ⬎ 0兲, where a and b are nonzero real num-
19, 013102-1
© 2009 American Institute of Physics
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bers. Projective-lag synchronization is defined as ay共t兲 = bx共t − 兲共 ⬎ 0兲. Projective synchronization is interesting because of its proportionality between the synchronized dynamical states. In applications to secure communications, this feature can be used to M-nary digital communication for achieving fast communication. Recently, Li et al.13 proposed generalized projective synchronization between two different chaotic systems using combination of active control and backstepping method. Very recently, Grassi and Miller14 introduced projective synchronization of time-delay, continuous-time, and discrete-time systems via a linear observer. In Ref. 15, Grassi and Mascolo proposed nonlinear observer design to synchronize hyperchaotic systems. An observer15 is a dynamic system designed to be driven by the output of another dynamic system 共plant兲 and having the property that the state of the observer converges to the state of the plant. In the present paper, projective-anticipating, projective, and projective-lag synchronization in a unidirectional coupled time-delay system is investigated using nonlinear observer design. The proposed technique is based on nonlinear control theory and it can be successfully applied to a wide class of time delayed systems. Recently, chaotic time delay system has been suggested as a good candidates for secure communication.16,17 It is proved that low dimensional chaotic systems do not ensure a sufficient level of security for communications, as the associated chaotic attractors can be reconstructed with some effort and the hidden message can be retrieved by an attacker. In this regard, the timedelayed system received a lot of attention, i.e., x˙ = f共x共t兲 , x共t − 兲兲, where is a delay time. With increases in time delay , the system becomes more complex, the number of positive Lyapunov exponents increases, and the system eventually transits to hyperchaos. In a recent study, it was discussed that a time-delayed system is still vulnerable for communication because time delay can be exposed by several measures; e.g., autocorrelation,18 filling factor,19 one step prediction error,20 average mutual information,21 etc. If the delay time is known, the time-delayed system becomes quite simple and the message encoded by the chaotic signal can be extracted by the common attack methods.22 From the above point of view, we can see that the study of projectiveanticipating, projective, or projective-lag synchronization in a variable time-delay system is of high practical importance. The organization of the remaining part is as follows: In Sec. II, some definitions and remarks used in this paper are presented. A general approach for projective synchronization for constant and variable time-delay systems via nonlinear observer are presented by Propositions 1 and 2. In Sec. III, two well-known systems 共Ikeda and prototype兲 are considered to verify the effectiveness of the proposed scheme. Finally, conclusions are made in the last section. II. NONLINEAR OBSERVER-BASED PROJECTIVE SYNCHRONIZATION SCHEMES
Consider the coupled time-delay system as x˙ = f共x,x1兲,
共1兲
y˙ = f共y,y 1兲 + u共x,y兲,
共2兲
where x , y 苸 Rn, f : Rn → Rn is a nonlinear vector field, u共x , y兲 is the control term, and x1 = x共t − 1兲. A definition of projective synchronization is presented below. Definition 1: Let there be two chaotic time-delayed systems with initial conditions x共0兲 and y共0兲, whose trajectories are described by the time-delayed systems 共1兲 and 共2兲. We can state that systems 共1兲 and 共2兲 are synchronized in the sense of projective-anticipating, projective, or projective-lag according as ⬍ 0, = 0 or ⬎ 0, respectively, if the dynamical behavior in which the amplitude of the masters state variable and that of the slave’s synchronizes up to a constant scaling factor ␣, i.e., e共t兲 = 共y共t兲 − ␣x共t − 兲兲 → 0
as t → ⬁,
共3兲
where e represents the synchronization error. Definition 2: A projective observer is a dynamic system in which the state of the master is proportional by a constant scaling factor ␣ with the state of the slave’s. Let the output of system 共1兲 be z = s共x , x1兲. The dynamic system y˙ = f共y,y 1兲 + g共z − s共y,y 1兲兲
共4兲
is then said to be nonlinear projective 共anticipating, exact, or lag兲 observer of system 共1兲 if its state y → ␣x共t − 兲 as t → ⬁, where g : Rn → Rn is a suitably chosen nonlinear function.23 Moreover, system 共4兲 is said to be global projective 共anticipating, exact, or lag兲 observer of system 共1兲 if y → ␣x共t − 兲 as t → ⬁ for any initial condition x共0兲 and y共0兲. The synchronization manifold of systems 共1兲 and 共2兲 is y = ␣x .
共5兲
Remark 1: System 共4兲 is a global projective synchronization if the error system e˙ = y˙ − ␣x˙共t − 兲 = f共y,y 1兲 + g共z − s共y,y 1兲兲 − ␣ f共x,x+1兲 = f共e + ␣x,e1 + ␣x+1兲 + g共s共x,x1兲 − s共e + ␣x,e1 + ␣x+1兲兲 − ␣ f共x,x+1兲 = h共e,e1兲 has a 共globally兲 asymptotically stable equilibrium point for e = 0. Assumption: We consider the dynamic 共1兲 in the form x˙ = Ax + Bp共x1兲,
共6兲
where p共x兲 is a nonlinear function of x, characterizing the system, e.g., p共x兲 = sin2共x − x0兲 for sine-squared model,24 p共x兲 = ax / 共1 + xc兲 for the Mackey–Glass model,25 26 2 p共x兲 = 关b − sin 共x − x0兲兴 for the Vallee model, p共x兲 = sin x for Ikeda model,27 etc. Proposition 1: Given system 共6兲, let z = s共x , x1兲 = p共x1兲 + kx be the synchronizing signal 共k is the coupling strength兲 and let
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g共z − s共y,y 1兲兲 = B关s共␣x, ␣x+1兲 − s共y,y 1兲兴
共7兲
be a function in Eq. 共4兲. Projective synchronization then occurs if k⬎
1 关A + 兩B兩兩sup p⬘共␣x+1兲兩兴 B
共8兲
under the condition 2 = ␣1. Proof: We consider the coupled system as x˙ = Ax + 1Bp共x1兲, y˙ = Ay + 2Bp
共9兲
冉 冊 y 1
+ g共z − s共y,y 1兲兲.
␣
共10兲
tems 共9兲 and 共10兲 is k ⬎ 共1 / B兲关A + 兩B兩兩supp⬘共␣x+1兲兩兴, where sup p⬘共·兲 is the supreme limit of p⬘共·兲. This complete the proof. 䊐 Consider the feedback time delay 1 and coupling delay as a function of time instead of constant delay as
1共t兲 = 10 + a1 sin共1t兲,
= Ay + 2Bp
冉 冊 y 1
+ g关s共␣x, ␣x+1兲 − s共y,y 1兲兴
␣
− ␣Ax − ␣1Bp共x+1兲 = Ae + 2Bp
冉 冊 y 1
␣
− ␣1Bp共x+1兲 + B关p共␣x+1兲
+ ␣kx − p共y 1兲 − ky兴. According to Eq. 共5兲, the function p共y 1 / ␣兲 can be written as p共x+1兲; then e˙ = 共A − kB兲e + 2Bp共x+1兲 − ␣1Bp共x+1兲 − Bp⬘共␣x+1兲e共t − 1兲 = − 共kB − A兲e − Bp⬘共␣x+1兲e共t − 1兲, where The error dynamics can be written as e˙ = − r共t兲e + s共t兲e共t − 1兲, where r共t兲 = kB − A and s共t兲 = −Bp⬘共␣x+1兲. Consider a positive definite Lyapunov functional of the form16,28,29 1 V共t兲 = e2共t兲 + 共t兲 2
冕
k⬎
e2共t + 兲d ,
where 共t兲 ⬎ 0 for any time. The solution e = 0 is stable if the derivative of the functional V共t兲 along the error equation, i.e., V˙共t兲 = − r共t兲e2 + s共t兲ee1 + 共t兲e2 − 共t兲e2 , 1
is negative. V˙ will be negative if 4共r − 共t兲兲 ⬎ s2 and r ⬎ 共t兲 ⬎ 0. The asymptotic stability condition for e = 0 is given for r共t兲 ⬎ 兩s共t兲兩
冋
兩B兩兩sup p⬘共␣x共t兲+1共t兲兲兩 1 A+ 冑1 − 1⬘共t兲 B
with 共t兲 =
兩s共t兲兩 . 2
If this inequality holds for all t ⬎ 0, the projective synchronization manifold is asymptotically stable. Thus, the sufficient condition for projective synchronization between sys-
册
共12兲
under the condition 2 = ␣1. Proof: Let e = y − ␣x共t兲 be the synchronization error between Eqs. 共9兲 and 共10兲. The error dynamic then becomes e˙ = − r共t兲e + s共t兲e共t − 共t兲兲, where r共t兲 = kB − A, s共t兲 = −Bp⬘共␣x共t兲+1共t兲兲, and 2 = ␣1. Consider a positive definite Lyapunov functional of the form16 1 V共t兲 = e2共t兲 + 共t兲 2
冕
0
−共t兲
e2共t + 兲d .
Then, V˙共t兲 = ee˙ + ˙ 共t兲
冕
0
−共t兲
e2共t + 兲d + 共t兲关e2 − e2 + e2 ⬘兴
if ˙ 共t兲 艋 0 for arbitrary t, then V˙共t兲 艋 −e2F共X , 共t兲兲, where F共X , 共t兲兲 = r共t兲 − 共t兲 − s共t兲X − 关共t兲⬘ − 共t兲兴X2 with X = e / e. In order to show that V˙共t兲 ⬍ 0 for all e and e, i.e., for all X, it is sufficient to show that Fmin ⬎ 0. The absolute minimum of F occurs at X = s共t兲 / 2共t兲共1 − ⬘兲 with Fmin = r共t兲 − 共t兲 − s2共t兲 / 4共t兲共1 − ⬘兲. Thus, the sufficient condition for synchronization is r共t兲 ⬎ 共t兲 +
0
−1
共11兲
where 10, 0, a0,1, and 0,1 are nonzero constants. For modulated time delay, the condition for projective synchronization is summarized in the next proposition. Proposition 2: Sufficient condition for projective synchronization between systems 共9兲 and 共10兲 is
The error dynamic is, then, e˙ = y˙ − ␣x˙共t − 兲
共t兲 = 0 + a0e兩sin共0t兲兩 ,
s2共t兲 = 共共t兲兲. 4共t兲共1 − ⬘兲
Again, 共共t兲兲 is a function of 共t兲. The minimum value of 共共t兲兲 occurs at 共t兲 = 兩s共t兲兩 / 2冑1 − ⬘ with min = 兩s共t兲兩 / 冑1 − ⬘. Finally, we get the sufficient condition for the trivial solution e = 0 is r共t兲 ⬎
兩s共t兲兩
冑1 − ⬘1共t兲
with 共t兲 =
兩s共t兲兩
. 2冑1 − 1⬘共t兲
Thus, the sufficient condition for projective synchronization is k⬎
冋
兩B兩兩sup p⬘共␣x共t兲+1共t兲兲兩 1 A+ 冑1 − 1⬘共t兲 B
册
with 2 = ␣1. This complete the proof. 䊐 Remark 2: The conditions in Propositions 1 and 2 are
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Chaos 19, 013102 共2009兲
Dibakar Ghosh
冉 冊
successfully applied to a wide class of time-delay systems with constant and variable time delay.
y˙ = − ay + 2m2 sin
y 1
␣
+ m1关sin共␣x+1兲 + k␣x − sin共y 1兲 − ky兴.
共14兲
III. ILLUSTRATIVE EXAMPLES
Two examples will be used to illustrate the effectiveness of the obtained results. We consider the two well-known chaotic time-delay systems—Ikeda system and prototype delay system—and their numerical simulations are performed.
A. Example 1: Ikeda system with constant and variable time delay
To verify the above analytic calculation in numerically, we consider unidirectional coupled Ikeda system27 as x˙ = − ax + 1m1 sin共x1兲,
共13兲
Physically, x is the phase lag of the electric field across the resonator, a is the relaxation coefficient for the dynamical variable, and m1 is the laser intensity injected into the system. 1 is the round-trip time of the light in the resonator or feedback delay time in the coupled systems.30 The Ikeda model was introduced to describe the dynamics of an optical bistable resonator and is well known for delay-induced chaotic behavior.30 System 共13兲 is chaotic for the set of parameter values17 a = 1.8, m1 = 6.0, and 1 = 2.0. For = 0, 1 = 2.0, 2 = 3.0, ␣ = 1.5, and k = 2, which satisfied the necessary and sufficient condition 共8兲 for exact projective synchronization is shown in Fig. 1共a兲, and the corresponding projective synchronization manifold is in Fig. 1共b兲. The phase angle between the
8
8
(b)
(a) 6
6
4
4
y
x, y
2 2 0
0 −2 −2
−4
−4
−6
−6 130
132
134
136
138
−8 −5
140
x
t 6
0
5
1.8
(c)
(d)
1.6
4
1.4 1.2
e=y−αx
x, y
2 0 −2
1 0.8 0.6 0.4
−4
0.2 −6 −8 130
0 132
134
136
t
138
140
−0.2
0
5
t
10
15
20
FIG. 1. 共a兲 Projective synchronization of x共t兲 共solid line兲 and y共t兲 共dotted line兲 for ␣ = 1.5. 共b兲 Corresponding projective synchronization manifold. 共c兲 Antiphase projective synchronization of x共t兲 共solid line兲 and y共t兲 共dotted line兲 for ␣ = −1.5. 共d兲 Corresponding projective synchronization error.
Chaos 19, 013102 共2009兲
Projective synchronization
dx / dt
25
25
(a)
20
20
15
15
10
10
5
5
dy / dt
013102-5
0
0
−5
−5
−10
−10
−15
−15
−20
−20
−25 −10
−5
0
5
10
x(t)
(b)
−25 −10
−5
0
y(t)
5
10
FIG. 2. Chaotic attractors of 共a兲 drive system 共13兲 and 共b兲 response system 共14兲.
synchronized chaotic attractors is zero, with an identical pattern. For the opposite sign of scaling factor ␣ = −1.5, we see antiphase projective synchronization in Fig. 1共c兲. In Fig. 1共d兲, the antiphase projective synchronization error is shown. Figure 2共a兲 shows the attractor of the drive system 共13兲, whereas for ␣ = 1.5, Fig. 2共b兲 shows that the attractor of the response system 共14兲 has been scaled by 1.5. For = 3, projective-lag synchronization occurs. The time series of x共t兲 and y共t兲 are shown in Fig. 3共a兲 and the corresponding projective-lag synchronization manifold is shown in Fig. 3共b兲. In Fig. 3共a兲, it is shown that the lag time between drive and response is 3.0. The relation between the states of drive and response is y共t兲 = 1.5x共t − 3兲. At this position, for opposite sign of scaling factor 共␣ = −1.5兲, the antiphase projective-lag synchronization with antiphase pattern, and difference of phase angles of the synchronized trajectories, are shown in Fig. 3共c兲. In Fig. 3共d兲, antiphase projective-lag synchronization error is shown. At the same time, it is observed that the amplitudes of drive state x共t兲 and response state y共t兲 are related by the relation y共t兲 = −1.5x共t − 3兲, and that the slope of this line is −1.5. If = −3 ⬍ 0, one can observe the projective-anticipating synchronization for k = 2.0 and other parameters as before. In Fig. 4共a兲, we can observe that the driven system anticipates the driver. At the same time, the amplitude of x共t + 3兲 and y共t兲 are related by y共t兲 = 1.5x共t + 3兲. Figure 4共b兲 shows their corresponding manifold. It is also clear that the slope of this manifold is 1.5. For time delay modulation, we set the parameter values as 10 = 2.0, a1 = 0.5, 1 = 0.02, 0 = 3.0, a0 = 0.2, and 0 = 0.1. At this parameter values feedback delay varies from 1.5 to
2.5 and coupling delay from 3.2 to 3.54. Figure 5共a兲 shows the lag synchronization between Eqs. 共13兲 and 共14兲 for coupling k = 2.5, which satisfied the necessary and sufficient condition 共12兲. For ␣ = 1.5, the projective-lag synchronization is shown in Fig. 5共b兲. For opposite sign of 0 = −3.0, a0 = −0.2, we get anticipatory synchronization in Fig. 5共c兲. In Fig. 5共d兲, projective-anticipating synchronization is shown for ␣ = 1.5.
B. Example 2: Prototype system with constant and variable time delay
In this example, we consider the projective synchronization of two unidirectional coupled prototype systems. The delay differential equation describing the drive system is x˙ = ␦x共t − 1兲 − ⑀关x共t − 1兲兴3 ,
共15兲
where ␦ and ⑀ are positive system parameters, and 1 is the time delay. This system is used to observe self-oscillations in the shipbuilding industry. The chaotic behavior of system 共15兲 for constant time delay are studied by Ucar.30 The multistability of periodic orbits and fixed points present in this system for higher value of delay. The system is in a chaotic state for the parameter values16 ␦ = 1.0, ⑀ = 1.0, and 1 = 1.6. In Ref. 16, we replaced the time-delay parameter 1 as a function of time instead of constant delay as in Eq. 共11兲. The system 共15兲 is chaotic for the parameter values 10 = 1.6, a1 = 0.26, and 1 = 0.8. The drive and response systems are written as
013102-6
Chaos 19, 013102 共2009兲
Dibakar Ghosh 10 8
8
(a)
(b) 6
6 4 4 2
y(t)
x, y
2 0 −2
0 −2
−4 −4 −6 −8
−6
3.0
−10 130
135
140
145
−8 −5
150
0
t 8
5
x(t−τ) 1.6
(c)
(d)
1.4
6
1.2
e=y(t)−α x(t−τ)
4
x, y
2 0 −2
1 0.8 0.6 0.4 0.2
−4 0 −6
−0.2
−8 130
135
140
145
150
−0.4
0
5
10
15
20
t
t
FIG. 3. 共a兲 Projective-lag synchronization of x共t兲 共solid line兲 and y共t兲 共dotted line兲 for ␣ = 1.5. 共b兲 Corresponding projective-lag synchronization manifold. 共c兲 Antiphase projective-lag synchronization of x共t兲 共solid line兲 and y共t兲 共dotted line兲 for ␣ = −1.5. 共d兲 Corresponding projective-lag synchronization error.
x˙ = 1共␦x1 − ⑀x3 兲,
共16兲
1
y˙ = 2
冉
␦ y 1 ␣
−
⑀y 31
+ ⑀y 3 − ky. 1
␣3
冊
+ ␣␦x1+ − ␣3⑀x3 + + k␣x − ␦y 1 1
共17兲
When coupling delay = 0, projective synchronization is achieved, whereas for ␣ = 2.0, exact projective synchronization is observed, and for −2.0, antiphase projective synchronization is obtained. In Fig. 6共a兲, the chaotic attractor of system 共15兲 is reported. For ␣ = 2.0, exact projective synchronization occurs and the attractor of the response system 共17兲 has been scaled by twice than that the attractor of drive sys-
tem 共16兲 关Fig. 6共b兲兴. Antiphase projective synchronization with twice the scale for ␣ = −2.0 is shown in Fig. 6共c兲. Similar results are obtained for other values of scaling factor ␣. For 0 = 2.0, a0 = 0.5, and 0 = 0.02, i.e., 共t兲 ⬎ 0, Fig. 7共a兲 shows the time history of the exact projective-lag synchronization between the drive and response systems for ␣ = 1.0 and k = 6. For scaling factor ␣ = 2.0, we get projective-lag synchronization, which is shown in Fig. 7共b兲. We choose the parameter values 0 = −2.0 and a0 = −0.5, so that 共t兲 ⬍ 0. In Fig. 7共c兲, projective-anticipating synchronization between Eqs. 共16兲 and 共17兲 are shown. It is proved that the projective synchronization 共anticipating and lag兲 are shown for any values of ␣.
Chaos 19, 013102 共2009兲
Projective synchronization
x, y
8
8
(a)
6
6
4
4
2
2
y(t)
013102-7
0
0
−2
−2
−4
−4
−6
−6
−8
100
110
t
120
(b)
−8 −5
130
0
5
x(t+τ)
FIG. 4. 共a兲 Projective-anticipating synchronization of x共t兲 共solid line兲 and y共t兲 共dotted line兲 for ␣ = 1.5. 共b兲 Corresponding projective-anticipating synchronization manifold.
6
10
(a)
(b)
4 5
0
x, y
x, y
2
−2 −4
0
−5
−6 −8 100
6
110
t
120
−10 100
130
10
(c)
110
t
120
130
120
130
(d)
4 5
0
x, y
x, y
2
−2 −4
0
−5
−6 −8 100
110
t
120
130
−10 100
110
t
FIG. 5. For variable time delay, 共a兲 lag synchronization of x共t兲 共solid line兲 and y共t兲 共dotted line兲 for k = 2.5, 共b兲 projective-lag synchronization when ␣ = 1.5, 共c兲 anticipatory synchronization of x共t兲 共solid line兲 and y共t兲 共dotted line兲, and 共d兲 projective-anticipating synchronization between systems 共13兲 and 共14兲.
Chaos 19, 013102 共2009兲
Dibakar Ghosh
5
(a)
5
(b)
2
4
4
1.5
3
3
1
2
2
0.5
1
1
0
−0.5
dy(t)/dt
dx(t)/dt
2.5
dy(t)/dt
013102-8
0 −1
0 −1
−1
−2
−2
−1.5
−3
−3
−2
−4
−4
−2.5 −2
0
−5 −5
2
x(t)
0
5
y(t)
(c)
−5 −5
0
5
y(t)
FIG. 6. Chaotic attractors of 共a兲 drive system 共16兲 and response system 共17兲 for 共b兲 ␣ = 2.0 and 共c兲 ␣ = −2.0.
IV. CONCLUSIONS
In conclusion, we have analytically calculated the necessary and sufficient conditions for projective-anticipating, projective, and projective-lag synchronization of general class of time-delay system with constant and modulated delay time. We derived that the transition of projectiveanticipating, projective, and projective-lag synchronization manifold can be achieved by adjusting the sign of coupling delay. Compared with other works,9,10,13,15,31 our work has the following advantages. 共a兲 The early works of projective synchronization reported that the projective synchronization was usually observed only in the coupled partially linear systems. Although in previous studies9,10 projective synchronization occurred because the response system had neutral stability, i.e., the largest conditional Lyapunov exponent was zero, so it followed a trajectory that was proportional to the drive system attractor. The constant of proportionality is depend on the initial condition. We realized stable projective
(a)
3
(b)
3
2
2
0.5
1
1
0
0
0
−0.5
x, y
1
x, y
x, y
1.5
synchronization 共i.e., all the conditional Lyapunov exponents are negative兲 in time-delayed systems with constant and modulated delay times without the limitation of partial linearity. The scaling factor is determined by the control law, not the initial conditions. 共b兲 The method can be easily implementation using simple analog circuit in practical communication applications. 共c兲 This method is systematic and can be applied to a wide class of time-delayed systems with constant and modulated delay time. In application to secure communications, projective synchronization can be used to extend binary digital to M-nary digital communication for achieving fast communication, so the research on projective synchronization has important theory significance and important value. Furthermore, projective synchronization has been applied in the research of secure communication due to the unpredictability of the scaling factors. The effectiveness and feasibility of our method have been verified by computer
−1
−1
−1
−2
−2
−1.5
−3
−3
−2 100
120
t
140
−4 100
(c)
120
t
140
−4 100
120
t
140
FIG. 7. 共a兲 Exact projective synchronization of x共t兲 共solid line兲 and y共t兲 共dotted line兲 for ␣ = 2.0. 共b兲 Projective-lag synchronization for ␣ = 2.0. 共c兲 Projectiveanticipating synchronization for ␣ = 2.0.
013102-9
simulation of Ikeda and prototype systems, which can also be used for other time-delayed systems. ACKNOWLEDGMENTS
The author is very thankful to referees for making valuable comments which made this paper in its present form possible. The author is also thankful to Dr. Santo Banerjee for helpful discussion. L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 共1990兲. J. M. Gonzalez-Miranda, Phys. Rev. E 57, 7321 共1998兲. 3 R. Mainieri and J. Rehacek, Phys. Rev. Lett. 82, 3042 共1999兲. 4 Focus Issue on Chaos Synchronization, Chaos, 7共4兲 共1997兲; G. Chen and X. Dong, From Chaos to Order. Methodologies, Perspectives and Applications 共World Scientific, Singapore, 1998兲; Handbook of Chaos Control, edited by H. G. Schuster 共Wiley-VCH, Weinheim, 1999兲. 5 H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 共1983兲; D. Ghosh and S. Banerjee, Phys. Rev. E 78, 056211 共2008兲. 6 E. M. Shahverdiev and K. A. Shore, Phys. Rev. E 71, 016201 共2005兲. 7 M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 共1996兲; T. Yalcinkaya and Y. C. Lai, ibid. 79, 3885 共1997兲; D. V. Senthilkumar, M. Lakshmanan, and J. Kurths, Phys. Rev. E 74, 035205共R兲 共2006兲. 8 M. Zhan, G. W. Wei, and C. H. Lai, Phys. Rev. E 65, 036202 共2002兲. 9 H. U. Voss, Phys. Rev. Lett. 87, 014102 共2001兲. 10 L. Y. Cao and Y. C. Lai, Phys. Rev. E 58, 382 共1998兲. 11 C. Y. Chee and D. Xu, Phys. Lett. A 318, 112 共2003兲. 12 T. M. Hoang and M. Nakagawa, Phys. Lett. A 365, 407 共2007兲. 13 G. H. Li, S. P. Zhou, and K. Yang, Phys. Lett. A 355, 326 共2006兲. 1 2
Chaos 19, 013102 共2009兲
Projective synchronization 14
G. Grassi and D. A. Miller, Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 1337 共2007兲. 15 G. Grassi and S. Mascolo, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 44, 1011 共1997兲. 16 D. Ghosh, S. Banerjee, and A. Roy-Chowdhury, Europhys. Lett. 80, 30006 共2007兲. 17 S. Banerjee, D. Ghosh, A. Ray, and A. Roy-Chowdhury, Europhys. Lett. 81, 20006 共2008兲. 18 F. T. Arecchi, R. Meucci, E. Allaria, A. Di Garbo, and L. S. Tsimring, Phys. Rev. E 65, 046237 共2002兲. 19 R. Hegger, M. J. Bunner, H. Kantz, and A. Giaquinta, Phys. Rev. Lett. 81, 558 共1998兲; M. J. Bunner, Th. Meyer, A. Kittel, and J. Parisi, Phys. Rev. E 56, 5083 共1997兲. 20 V. I. Ponomarenko and M. D. Prokhorov, Phys. Rev. E 66, 026215 共2002兲; C. Zhou and C.-H. Lai, ibid. 60, 320 共1999兲. 21 D. V. Senthilkumar and M. Lakshmanan, Chaos 17, 013112 共2007兲; W. H. Kye, M. Choi, S. Rim, M. S. Kurdoglyan, C. M. Kim, and Y. J. Park, Phys. Rev. E 69, 055202共R兲 共2004兲. 22 K. M. Short and A. T. Parker, Phys. Rev. E 58, 1159 共1998兲. 23 A. Tamasevicius, A. Cenys, G. Mykolaitis, A. Namajunas, and E. Lindberg, IEEE Electron Device Lett. 33, 542 共1997兲. 24 J. P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. E 57, 2795 共1998兲. 25 M. C. Mackey and L. Glass, Science 197, 287 共1977兲. 26 R. Vallee and C. Delisle, Phys. Rev. A 34, 309 共1986兲. 27 K. Ikeda, H. Daido, and O. Akimoto, Phys. Rev. Lett. 45, 709 共1980兲; K. Ikeda, K. Kondo, and O. Akimoto, ibid. 49, 1467 共1982兲. 28 K. Pyragas, Phys. Rev. E 58, 3067 共1998兲. 29 N. N. Krasovskii, Stability of Motion 共Stanford University Press, Stanford, 1963兲. 30 A. Ucar, Chaos, Solitons Fractals 16, 187 共2003兲. 31 Z. Li and D. Xu, Phys. Lett. A 282, 175 共2001兲.
CHAOS 19, 013103 共2009兲
Transition to complete synchronization in phase-coupled oscillators with nearest neighbor coupling Hassan F. El-Nashar,1 Paulsamy Muruganandam,2 Fernando F. Ferreira,3 and Hilda A. Cerdeira4 1
Department of Physics, Faculty of Science, Ain Shams University, 11566 Cairo, Egypt and Department of Physics, Faculty of Education, King Saud University, P.O. Box 21034, 11942 Alkharj, Saudi Arabia 2 School of Physics, Bharathidasan University, Palkalaiperur, Tiruchirappalli-620024, India 3 Grupo Interdisciplinar de Física da Informação e Economi a (GRIFE), Escola de Arte, Ciências e Humanidades, Universidade de São Paulo, Av. Arlindo Bettio 1000, 03828-000 São Paulo, Brazil 4 Instituto de Física Teórica, Universidade Estadual Paulista, R. Pamplona 145, 01405-000 São Paulo, Brazil and Instituto de Física, Universidade de São Paulo, R. do Matão, Travessa R. 187, 05508-090 São Paulo, Brazil
共Received 29 August 2008; accepted 4 December 2008; published online 12 January 2009兲 We investigate synchronization in a Kuramoto-like model with nearest neighbor coupling. Upon analyzing the behavior of individual oscillators at the onset of complete synchronization, we show that the time interval between bursts in the time dependence of the frequencies of the oscillators exhibits universal scaling and blows up at the critical coupling strength. We also bring out a key mechanism that leads to phase locking. Finally, we deduce forms for the phases and frequencies at the onset of complete synchronization. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3056047兴 Weakly coupled oscillators play an important role in understanding collective behavior of large populations. They are often used to model the dynamics of a variety of systems that arise in nature, even though they are quite different. Synchronization is one of the interesting phenomena observed in these systems where the interacting oscillators under the influence of coupling would have a common frequency. Particularly, these systems show an extremely complex clustering behavior as a function of the coupling strength. In spite of their differences, the above-mentioned systems can be described using simple models of coupled phase equations such as the Kuramoto model. This paper analyzes the behavior of individual oscillators in the vicinity of the critical coupling where all the oscillators evolve in synchrony with each other. I. INTRODUCTION
Systems of coupled oscillators can describe problems in physics, chemistry, biology, neuroscience, and other disciplines. They have been widely used to model several phenomena, such as Josephson junction arrays, multimode lasers, vortex dynamics in fluids, biological information processes, and neurodynamics.1–3 These systems have been observed to synchronize themselves to a common frequency when the coupling strength between these oscillators is increased.4–13 The synchronization features of many of the above-mentioned systems, in spite of the diversity of the dynamics, might be described using simple models of weakly coupled phase oscillators such as the Kuramoto model.8,14 Finite range interactions are more realistic for the description of many physical systems, although finite range coupled systems are difficult to analyze and to solve analyti1054-1500/2009/19共1兲/013103/5/$25.00
cally. However, in order to figure out the collective phenomena when finite range interactions are considered, it is of fundamental importance to study and to understand the nearest neighbor interactions, which is the simplest form of the local interactions. In this context, a simplified version of the Kuramoto model with nearest neighbor coupling in a ring topology, which we shall refer to as the locally coupled Kuramoto model 共LCKM兲, is a good candidate to describe the dynamics of coupled systems with local interactions such as Josephson junctions, coupled lasers, neurons, chains with disorders, multicellular systems in biology, and in communication systems.6,14–16 For instance, it has been shown that the equations of the resistively shunted junction which describe a ladder array of overdamped, critical-current disordered Josephson junctions that are current biased along the rungs of the ladder can be expressed by a LCKM.17 In nearest neighbor coupled Rössler oscillators, the phase synchronization can be described by the LCKM.18 Therefore, LCKM can provide a way to understand phase synchronization in coupled systems, for example, in locally coupled lasers,19,20 where local interactions are dominant. Coupled phase oscillators described by LCKM can also be used to model the occurrence of travelling waves in neurons.6,21 In communication systems, unidirectionally coupled Kuramoto model can be used to describe an antenna array.22 Such unidirectionally coupled Kuramoto models can be considered as a special case of the LCKM and it often mimics the same behavior. One of the important features of the local model is that the properties of individual oscillators can be easily analyzed to study the collective dynamics while one has to rely on the average quantities, in a mean field approximation, or by means of an order parameter, etc., as in the case of the usual
19, 013103-1
© 2009 American Institute of Physics
Chaos 19, 013103 共2009兲
El-Nashar et al.
Kuramoto model of long range interactions. Therefore, due to the difficulty in applying standard techniques of statistical mechanics, one should look for a simple approach to understand the coupled system with local interactions by means of numerical study of a temporal behavior of the individual oscillators. Such analysis is necessary in order to obtain a close picture of the effect of the local interactions on synchronization. In this case, numerical investigations can assist one to figure out the mechanism of interactions at the stage of complete synchronization, which in turns help to obtain an analytic solution. Earlier studies on the LCKM show several interesting features including tree structures with synchronized clusters, phase slips, bursting behavior, saddle-node bifurcation, and so on.23–25 There have been studies showing that neighboring elements share dominating frequencies in their time spectra, and that this feature plays an important role in the dynamics of formation of clusters in the local model.26,27 It has been found that the order parameter, which measures the evolution of the phases of the nearest neighbor oscillators, becomes maximum at the partial synchronization points inside the tree of synchronization.28 Very recently, we developed a scheme based on the method of Lagrange multipliers to estimate the critical coupling strength for complete synchronization in the local Kuramoto model with different boundary conditions.29 In this paper we address the mechanism that leads to a complete synchronization in the Kuramoto model with local coupling. This is done by analyzing the behavior of each individual oscillators at the onset of synchronization. For this purpose we consider the equations governing the phase differences at the onset of synchronization. In particular, we identify that the cosine of only one among the phase differences becomes zero. Based on this property we derive the expression for the time interval between bursting behavior of the instantaneous frequencies of each individual oscillators in the vicinity of critical coupling strength. Our analysis shows that the transition to complete synchronization occurs due to a saddle-node bifurcation in agreement with the earlier studies. Further, we deduce the expressions for the phases and frequencies of the individual oscillators at the onset of complete synchronization. This paper is organized as follows. In Sec. II we present a brief overview on the dynamics of local Kuramoto model. We then analyze the behavior of the phase differences and the time interval between successive bursts at the transition to complete synchronization. In particular, we point out the mechanism that lead to complete phase locking at the critical coupling strength. Based on this, we deduce the forms of phases and frequencies at the onset of synchronization. Finally, in Sec. III we give a summary of the results and conclusions.
II. BEHAVIOR OF PHASES AND FREQUENCIES AT THE ONSET OF SYNCHRONIZATION
Even when there has been an extensive exploration of the dynamics of the Kuramoto model 共global coupling among all oscillators兲, the local model of nearest neighbor interactions, which can be considered as a diffusive version
3
N = 15
2
Kc (14 - 6)
˙ θ
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1 (7 - 13) 0
0
1
2
K
3
4
5
FIG. 1. 共Color online兲 Synchronization tree for a system of 15 oscillators.
of the Kuramoto model, has been receiving attention only recently. The LCKM is expressed as25–29 K ˙ i = i + 关sin共i+1 − i兲 + sin共i−1 − i兲兴, 3
共1兲
where i are the natural frequencies, K is the coupling strength, i is the instantaneous phase, ˙ i is the instantaneous frequency, and i = 1 , 2 , . . . , N. Many interesting features of the LCKM remain unknown, especially an analytic solution,14 which would be of great importance in understanding the mechanism that leads to synchronization. In order to find such an analytic solution, one should study carefully the temporal evolution of frequency and phase of each individual oscillator in the neighborhood of the critical coupling for complete synchronization. If we consider the oscillators in a ring, with periodic boundary conditions i+N = i, the nonidentical oscillators 共1兲 cluster in time averaged frequency until they completely synchronize to a common value of the average frequency 0 = 共1 / N兲兺Ni=1i, at a critical coupling Kc.24–26,28,29 At K 艌 Kc, the phases and the frequencies are time independent and all the oscillators remain synchronized. In Fig. 1, we show the synchronization tree for a periodic system with N = 15 oscillators, where the elements which compose each one of the major clusters that merge into one at Kc are indicated in each branch. In terms of phase differences i = i+1 − i, system 共1兲 can be rewritten as K ˙ i = i+1 − i + 关sin i−1 − 2 sin i + sin i+1兴, 3
共2兲
with ˙ *i = 0 at Kc for i = 1 , 2 , . . . , N. In addition, all quantities ˙ *i , i*, and ˙ i*, which become time independent25,26,28,29 at ˙* the critical coupling, remain like that for k 艌 Kc, when i = 0 and ˙ *i = 0. Earlier attempts to obtain a solution of Eq. 共2兲 show that for only two oscillators which have phase dif* , results in 兩sin *兩 = 1,23 and indeed this ference *l = *l − l−1 l is a necessary condition for Eq. 共2兲 to have a phase-locked solution. This fact has been used by Daniels et al.17 to estimate the value of critical coupling strength Kc at which the transition to complete synchronization occurs. However, the determination of which two oscillators among N oscillators that have 兩sin *l 兩 = 1, remains difficult. From the study of the temporal evolution of phases and frequencies of each indi-
013103-3
Chaos 19, 013103 共2009兲
Phase-coupled oscillators
1
sin φl = sin φ13
TABLE I. Calculated values of Kc and A for different values of N.
N = 15
sin φi
0.5 0 -0.5 -1 0
10
20 30 time ×102
40
50
N
Kc
A
3 5 10 15 20 25 50 100
0.850 412 27 3.170 827 13 3.547 010 35 3.870 238 66 4.958 300 14 3.641 060 38 9.457 200 49 12.723 208 7
1.9994 2.0001 1.9996 2.0000 2.0002 1.9989 1.9993 1.9985
FIG. 2. 共Color online兲 Values of sin i at K 艌 Kc for a system of 15 oscillators 共see Fig. 1兲.
˙ l = B共A − 2 sin l兲,
where A = 3兩共l+1 − l兲兩 / K + sin l−1 + sin l+1 and B = K / 3. Equation 共3兲 takes the form of a phase synchronization of * and * two coupled limit-cycles.30 At Kc, l* = ⫾ / 2, l−1 l+1 are constants and time independent, and A = 2. A detailed numerical study shows also that, at the onset of synchronization, A ⬇ 2 and the values of l, l−1, and l+1 remain equal to their values at Kc, for a time interval T. The values of Kc and A for different numbers of oscillators N from numerical simulations are tabulated in Table I. It is clear that in all the cases, A ⯝ 2 when K approaches Kc. The relation A ⯝ 2 is found to be valid for different choices of initial frequencies i for each N in the vicinity of Kc. Further, it should be noted that when the time interval T → ⬁, one can find that A = 2. The time interval T can be found analytically, according to Eq. 共3兲, to be T⬇
3冑2
1
共4兲
. K冑A 冑A − 2
In Fig. 3, we clearly see that T blows up as A becomes close to 2 共where K goes to Kc兲, for the case of N = 15. We find that T blows up as 共A − 2兲−0.5, which is a numerical proof that a saddle-node bifurcation occurs at Kc. Assuming that sin l−1 and sin l+1 remain constant for a time interval T in the vicinity of Kc, and are equal to their values at Kc 共which has been verified numerically兲, we find that AK / 2 ⬇ Kc. Table II shows this fact where the error is small and decreases as K approaches Kc. Therefore, Eq. 共4兲 takes the form
5 N = 15
4 log T
vidual oscillator, it has also been found numerically that, at the onset of synchronization K ⱗ Kc, the values of ˙ i共t兲 and ˙ i共t兲 remain equal to 0 and zero, respectively, for a certain time interval T. During this time T, a stable phase-locked solution exists; then they burst,18,24,25 and this stable phaselocked solution is lost. In between bursts, the phases remain fixed and then they have an abrupt change 共phase-slip behavior兲 by an amount which depends on the initial values of the frequencies i,24,25 corresponding to the burst in the frequen˙ i = 0 are always cies, while the quantities 兺Ni=1i = 0 and 兺Ni=1 preserved by the topology. Integrated with the above information, it has been shown by numerical investigation that the time interval T blows up as K becomes close to Kc and T → ⬁ at Kc. All this information leads one to conclude that there is a saddle-node bifurcation at Kc, and the synchronizationdesynchronization transition at the critical coupling can be interpreted using this knowledge. In this work, we perform numerical investigations of the temporal evolution of the phases and frequencies for the individual oscillators in order to arrive to specific conditions which will lead to criteria to obtain an analytic solution. A detailed study of all quantities sin *i at Kc for several values of N and for different sets of i, shows that there is only one value of phase difference between two neighboring oscilla* − * for which 兩sin *兩 = 1, while for all other tors l* = l+1 l l values 兩sin i*兩 ⫽ 1, i ⫽ l. In Fig. 2, we show sin *i for a case of N = 15 as time progresses at the critical coupling Kc, with the same initial frequencies of Fig. 1. We see that the value of 兩sin l*兩 = 1, is for l = 13 and that this quantity 兩sin *l 兩 = 1 holds for only one value of phase difference l = / 2, where these two oscillators l + 1 and l belong to different clusters, and these two nearest neighbors oscillators are always at the borders between the major clusters that merge at Kc, which can be seen from Fig. 1. We find the same result for different initial frequencies i and for different values of N. In addition, the sign of sin *l is negative for l ⬎ l+1 and positive for the reverse. The knowledge of the burst and phase slip 共in the vicin˙ i共t兲 and i共t兲, respectively, as ity of Kc兲 of the quantities well as the finding of 兩sin l*兩 = 1 共at Kc兲, will allow us to rewrite equation 共2兲, for the index l as
共3兲
3 2 1
slope = −0.5 ± 0.02 -8
-6
-4 log(A − 2)
-2
FIG. 3. 共Color online兲 log10 T vs log10共A − 2兲, which shows the divergence of the time interval T when A approaches 2 with a slope ⯝−0.5.
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TABLE II. Calculated values of AK / 2 for N = 15 oscillators at the vicinity of Kc = 3.870 238 658.
3.869 480 136 3.870 198 414 3.870 226 709 3.870 237 122 3.870 238 325
2
AK 2
兩Kc − K兩
2.000 700 0 2.000 035 0 2.000 010 0 2.000 001 0 2.000 000 3
3.870 834 454 3.870 266 142 3.870 246 060 3.870 238 909 3.870 238 727
5.960⫻ 10−4 2.750⫻ 10−5 7.402⫻ 10−6 2.480⫻ 10−7 6.920⫻ 10−8
φ˙
A
K
φ˙ 13 φ˙ 6
4
0 -2 -4 0
10
4
3
冑2冑Kc冑Kc − K
共5兲
.
Thus, within a good approximation, the periodic time interval T blows up as 共Kc − K兲−0.5, in good agreement with the numerical calculation by Zheng et al.,24,25 showing that a saddle-node bifurcation occurs at Kc.23 Therefore, Eq. 共3兲 can be written as
˙ l ⬇ L共Kc − K sin l兲,
共6兲
φ˙
2
T⬇
T 992
(a) 20 ˙φ13 φ˙ 6
30
40
50
0 -2
T 989
(b)
-4 0
10
20
30
time ×102
40
50
˙ 6 according to 共a兲 system FIG. 4. 共Color online兲 Time evolution of ˙ 13 and 共1兲 and 共b兲 Eqs. 共7b兲 and 共9b兲, at K = 3.870 226 709, for 15 oscillators with the same initial conditions of Fig. 1.
which can be solved analytically, and its solution reads
冋
l ⬇ 2 arctan
␣ tan共 21 ␣Lt兲 ⫾ K Kc
册
共7a兲
and
共
␣2 sec2 21 ␣Lt
L ˙ l ⬇ Kc 1+
˙ l+m ⬇
兲
再 冋 冉 冊 册冎 1 1 ␣ tan ␣Lt ⫾ K 2 Kc
,
共7b兲
2
where ␣ = 冑K2c − K2 and L = 32 . Equations 共7a兲 and 共7b兲 show ˙ * = 0. It that, at Kc, the values sin *l = ⫾ 1, which lead to l can also be seen that in the vicinity of Kc, sin l = ⫾ 1 and ˙ l = 0 for a period T. The 共⫹兲 sign in Eqs. 共7a兲 and 共7b兲 corresponds to the case l+1 ⬎ l and the 共⫺兲 sign for the reverse. In order to understand the mechanism of full synchronization which occurs at Kc, we use the fact that sin *l = ⫾ 1 and each ˙ i* = 0, where these quantities remain unchanged for T in the vicinity of Kc. Hence, from system 共1兲, we are able to obtain the following relations: N−l
* = sin l+m
3 兺 共0 − l+m兲 ⫾ sin *l , Kc m=1
* =− 3 sin l−n
共8a兲
l−1
兺 共0 − l−n−1兲 ⫾ sin *l .
Kc n=1
共8b兲
Using this fact, we write the following equations, in addition to Eqs. 共7a兲 and 共7b兲:
l−n ⬇ sin−1共an ⫾ sin l兲, ˙ l−n ⬇
˙l cos l
冑1 − 共an ⫾ sin l兲2 ,
l+m ⬇ sin−1共am ⫾ sin l兲,
共9a兲 共9b兲
˙l cos l
冑1 − 共am ⫾ sin l兲2 ,
共9c兲
共9d兲
where an = 共−3 / Kc兲兺ni=1共0 − l−i−1兲 with n = 1 , 2 , 3 , . . . , l − 1 and am = 共3 / Kc兲兺mj=1共0 − l+j兲 with m = 1 , 2 , 3 , . . . , N − l. It is clearly seen that according to the above equation, each i ˙i can be expressed in terms of l and, consequently, each ˙ can be expressed in terms of l and l. Therefore, all values of i will be shifted from each other by some constant which is determined by the location of the indexes l − n and l + m relative to oscillators with indexes l and l + 1. This is shown in Fig. 2, where sin i values are shifted from each other at ˙ l due to saddleKc. Therefore, at Kc, what occurs to l and node bifurcation diffuses through the ring via interaction between neighboring oscillators. This means that, at the vicinity of Kc, the value of l has an abrupt change after being ˙ l after constant for a time T, caused by a burst behavior of being zero for the same time interval T. The abrupt change in l produces a sudden change in the values of i of their ˙ l in turn yields neighbors, while the bursting behavior of ˙ i 共i ⫽ l兲. In order to demonstrate this fact, we plot bursts in ˙ 13 and ˙ 6, in the vicinity of Kc, the temporal evolution of according to numerical simulation of Eq. 共1兲 in Fig. 4共a兲, while we plot both quantities according to Eqs. 共7b兲 and 共9b兲 in Fig. 4共b兲. As shown in Fig. 4, the results of numerical simulation agree with that from the analytic solution. The above-mentioned behavior is reflected in the time dependence of the ˙ ⬘i s, which in turn remain equal to 0 for a time T and burst around 0 corresponding to the burst of l. ˙l Henceforward, we argue that it is the behavior of l and which drives the system to fall into full synchronization.
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Phase-coupled oscillators
III. SUMMARY AND CONCLUSIONS
In summary, we have analyzed the conditions on the phase differences for the onset of complete synchronization at the critical coupling strength in a Kuramoto-like model with nearest neighbor coupling. Such a condition, which is 兩sin l*兩 = 1 共or cos *l = 0兲, allows us to solve the equations of the phase differences 关 Eq. 共2兲兴 analytically. We also found that full synchronization occurs always when the quantity A = 2 at Kc. Due to the diffusive nature of the LCKM, complete synchronization of all oscillators to a common value can be interpreted and understood once we have analytic ˙ l. However, it is still difficult to determine forms for l and analytically the number of oscillators in each cluster which merge into one at Kc. Therefore, one cannot allocate straightforwardly the two nearest neighbor oscillators which would have 兩sin l*兩 = 1. On the other hand, a detailed numerical study on the temporal evolution of phases, phase differences, and frequencies of oscillators at the borders of the clusters that merge into larger one at onset of complete synchronization helps us to determine the neighboring oscillators which have sin l* = ⫾ 1. Such analysis can also be used to understand the partial synchronization that leads to the formation of small clusters for coupling strengths below the critical coupling strength Kc. Of course, analysis of the simplest case of locally coupled phase oscillators can help to understand models with local interactions where amplitudes and phases are included.15,18–20 In such cases, a detailed study of the time evolution of amplitudes and phases can reveal a better understanding of the mechanism of synchronization. The present analysis can also be applicable to models in higher dimensions such as that for dislocations in solids which includes local nearest neighbor interactions.31 Furthermore, the present approach can be extended to understand the underlying mechanism in the case of locally coupled Kuramoto models with time delay6 共or phase delay兲 introduced between the coupled oscillators. In addition, the mechanism of synchronization in LCKM for open and fixed boundaries can be studied in a similar manner to the present work as well as for the case of unidirectional LCKM. We also want to mention that the scaling law given by Eq. 共5兲 has been found experimentally in a transition to phase synchronization in CO2 lasers32 and in electronic circuits.33,34 On the other hand, one cannot make a direct comparison between the mechanism of synchronization discussed here in LCKM and the scaling law that has been found in experiments since the physical systems are not necessarily the same. ACKNOWLEDGMENTS
H.F.E. thanks both of the School of Physics, Bharathidasan University, Tiruchirappalli, India and Abus Salam ICTP, Trieste, Italy, for hospitality during a part of this work. The work of P.M. is supported in part by Department of
Science and Technology, Government of India 共Ref. No. SR/ FTP/PS-79/2005兲, Conselho Nacional de Desenvolvimento Científico e Tecnológico 共CNPq兲, Brazil, and the Third World Academy of Sciences 共TWAS兲, Italy. F.F.F. acknowledges CNPq for financial support. A. T. Winfree, Geometry of Biological Time 共Springer, New York, 1990兲. C. W. Wu, Synchronization in Coupled Chaotic Circuits and Systems 共World Scientific, Singapore, 2002兲. 3 S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order 共Hyperion, New York, 2003兲. 4 C. M. Gray, P. Koenig, A. K. Engel, and W. Singer, Nature 共London兲 338, 334 共1989兲. 5 K. Otsuka, Nonlinear Dynamics in Optical Complex Systems 共Kluwer, Dordrecht, 2000兲. 6 H. Haken, Brain Dynamics: Synchronization and Activity Patterns in Pulse-Coupled Neural Nets with Delays and Noise 共Springer, Berlin, 2007兲. 7 Bifurcation, Patterns, and Symmetry, Special Issue of Physica D, edited by M. Golubitsky and E. Knobloch, Vol. 143 共2000兲. 8 Y. Kuramoto, Chemical Oscillations, Waves and Turbulences 共Springer, Berlin, 1984兲. 9 G. Hu, Y. Zhang, H. A. Cerdeira, and S. Chen, Phys. Rev. Lett. 85, 3377 共2000兲. 10 Y. Zhang, G. Hu, H. A. Cerdeira, S. Chen, T. Braun, and Y. Yao, Phys. Rev. E 63, 026211 共2001兲. 11 Y. Zhang, G. Hu, and H. A. Cerdeira, Phys. Rev. E 64, 037203 共2001兲. 12 I. A. Heisler, T. Braun, Y. Zhang, G. Hu, and H. A. Cerdeira, Chaos 13, 185 共2003兲. 13 P. A. Tass, Phase Resetting in Medicine and Biology 共Springer, Berlin, 1999兲. 14 J. A. Acebron, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, Rev. Mod. Phys. 77, 137 共2005兲. 15 Y. Ma and K. Yoshikawa, arXiv:0809.1697v2, 2008. 16 Y. Braiman, T. A. Kennedy, K. Wiesenfeld, and A. Khinik, Phys. Rev. A 52, 1500 共1995兲. 17 B. C. Daniels, S. T. M. Dissanayake, and B. R. Trees, Phys. Rev. E 67, 026216 共2003兲. 18 Z. Liu, Y.-C. Lai, and F. C. Hoppensteadt, Phys. Rev. E 63, 055201共R兲 共2001兲. 19 A. Khinik, Y. Braiman, V. Protopopescu, T. A. Kennedy, and K. Wiesenfeld, Phys. Rev. A 62, 063815 共2000兲. 20 D. Tsygankov and K. Wiesenfeld, Phys. Rev. E 73, 026222 共2006兲. 21 S. Manrubbia, A. Mikhailov, and D. Zanette, Emergence of dynamical Order: Synchronization Phenomena in Complex Systems 共World Scientific, Singapore, 2004兲. 22 J. Rogge and D. Aeyels, J. Phys. A 37, 11135 共2004兲. 23 S. H. Strogatz and R. E. Mirollo, Physica D 31, 143 共1988兲. 24 Z. Zheng, G. Hu, and B. Hu, Phys. Rev. Lett. 81, 81 共1998兲. 25 Z. Zheng, B. Hu, and G. Hu, Phys. Rev. E 62, 402 共2000兲. 26 H. F. El-Nashar, A. S. Elgazzar, and H. A. Cerdeira, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 2945 共2002兲. 27 H. F. El-Nashar, Y. Zhang, H. A. Cerdeira, and F. Ibyinka A., Chaos 13, 1216 共2003兲. 28 H. F. El-Nashar, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 3473 共2003兲. 29 P. Muruganandam, F. F. Ferreira, H. F. El-Nashar, and H. A. Cerdeira, Pramana, J. Phys. 70, 1143 共2008兲. 30 S. H. Strogatz, Nonlinear Dynamics and Chaos 共Perseus, New York, 2000兲. 31 A. Carpio and L. L. Bonilla, Phys. Rev. Lett. 90, 135502 共2003兲. 32 S. Bocaletti, E. Allaria, R. Meucci, and F. Arecchi, Phys. Rev. Lett. 89, 194101 共2002兲. 33 L. Zhu, A. Raghu, and Y.-C. Lai, Phys. Rev. Lett. 86, 4017 共2001兲. 34 G.-M. Kim, G.-S. Yim, J.-W. Ryu, Y.-J. Park, and D.-U. Hwang, Europhys. Lett. 71, 723 共2005兲. 1 2
CHAOS 19, 013104 共2009兲
Anticipating synchronization of chaotic systems with time delay and parameter mismatch Qi Han, Chuandong Li, and Junjian Huang College of Computer Science, Chongqing University, Chongqing 400030, People’s Republic of China
共Received 21 April 2008; accepted 15 October 2008; published online 14 January 2009兲 This paper studies the effect of parameter mismatch on anticipating synchronization of chaotic systems with time delay in the framework of the master-slave configuration. The convergence criteria for the error dynamical system under study are established by means of model transformation incorporated with Lyapunov functional and linear matrix inequality. The error bound of anticipating synchronization is estimated by rigorous theoretical analysis. Its accuracy is confirmed by numerical simulation results. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3013600兴 In the last decade, synchronization of chaos has triggered considerable interest mainly because of a wide range of applications in electronic circuits, physical, chemical, and biological systems. Recently, much effort has been devoted to the theoretical analysis of anticipating synchronization after the observation of this kind of synchronization by Voss. In general, parameter mismatch is inevitable in the synchronization of practical chaotic systems due to noise or other artificial factors. It is a relevant issue to investigate the effects of parameter mismatch on anticipating synchronization although there are only a few reports on this so far. In this paper, several criteria for the anticipating synchronization of coupled chaotic systems with time delay and parameter mismatch are established. Numerical simulations are presented to support the effectiveness of the theoretical analysis. I. INTRODUCTION
Since the seminal works of Pecora and Carroll,1,2 chaos synchronization has received a great deal of interest from various fields. Synchronization phenomena3 in coupled chaotic systems have been extensively studied in electronic circuits, laser systems, pairs of neurons, and biological systems. In general, chaos synchronization describes the gradual consistency of the trajectories of two chaotic systems starting from different initial conditions. Typical forms of synchronization include complete synchronization,1 generalized synchronization,4 lag synchronization,5 phase synchronization,6 projective synchronization,7 anticipating synchronization,8 etc. It was observed by Voss8 that dissipative chaotic systems with a time-delayed feedback can drive nearly identical systems in such a way that the state of one of the systems synchronizes with the future state of the other, i.e., y共t兲 → x共t + 兲 , t → ⬁, where x and y denote the states of the master and the slave systems, respectively, and the constant is the anticipation time. As usual, the anticipating synchronization error is denoted by e共t兲 = y共t − 兲 − x共t兲. A practical approach for analyzing chaos synchronization is to determine the stability of the studied error systems. This is because the asymptotical stability of the origin of these error systems 关i.e., e共t兲 → 0 as t → ⬁兴 implies the existence of chaos synchronization. For the case of anticipating synchronization, e共t兲 → 0 means y共t兲 → x共t + 兲. There are many re1054-1500/2009/19共1兲/013104/10/$25.00
ports on anticipating synchronization.9–16 For example, the anticipating synchronization of chaotic Lur’e time-delayed systems in a master-slave setting was studied in Ref. 9 by introducing three scenarios for anticipating synchronization. The sufficient conditions for the existence of anticipating synchronizing slave systems in terms of LMI were presented. In Ref. 10, the model of coupled multiple-delay feedback systems was investigated with the schemes of anticipating and projective-anticipating synchronizations. A new synchronization scheme was presented for the anticipating synchronization of discrete-time chaotic systems in Ref. 11. Research effort has also been devoted to the application of anticipating synchronization in both electrical and optical fields.12–14 The anticipating synchronization of chaotic semiconductor lasers with optical feedback was analyzed numerically in Ref. 12. In Ref. 13, the authors investigated the anticipating synchronization between two chaotic laser diodes subjected to incoherent optical feedback and incoherent optical injection, respectively, and demonstrated that anticipating synchronization could be applied for cryptographic purposes. Reference 14 reported the experimental results of anticipating synchronization using two diode lasers as transmitter and receiver, respectively. Clearly, the aforementioned contributions, among many others,15,16 are of theoretical and experimental importance. However, they did not address the case with parameter mismatch. Complete chaos synchronization is achieved for chaotic systems with well-matched parameters. However, parameter mismatch is inevitable in practical implementations of chaos synchronization because of noise or other artificial factors. The parameter mismatch may have the detrimental effect on the quality of synchronization. If the mismatch is small, the synchronization error may only have small fluctuations around zero. However, loss of synchronization may exist if the mismatch becomes large. Therefore it is important to study the effect of parameter mismatch on the synchronization of chaotic systems. More recently, there are some reports on chaos synchronization in the presence of parameter mismatch. In Refs. 17–19 the authors investigated the robustness of the synchronization with respect to parameter mismatches or noise. Shahverdiev et al. demonstrated in Ref. 20 that parameter mismatches might explain the fact that the lag time is equal
19, 013104-1
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to the coupling delay. The synchronization of a class of structurally nonequivalent chaotic systems with time delays was investigated in Ref. 21. In Ref. 22, the authors studied the synchronization between two nonidentical unidirectionally linearly coupled chaotic systems with time delay and showed that parameter mismatch is of crucial importance in achieving synchronization. Synchronization between two nonidentical unidirectionally coupled chaotic Ikeda systems was investigated in Ref. 23. The effect of parameter mismatch on the synchronization of a class of coupled chaotic systems with time delay was studied in Refs. 24 and 25. However, to the best of our knowledge, only a few studies have addressed the effects of parameter mismatches on anticipating synchronization theoretically.26 The objective of this paper is to analyze theoretically how the anticipating synchronization of chaotic systems with parameter mismatches can be achieved approximately for a general class of chaotic systems coupled by delayed feedback. For this purpose, we shall derive several criteria that ensure the convergence of anticipating-synchronization error to a small region by using the Lyapunov functional and LMI technique. Then the relationship between coupling strength, anticipating time, and parameter mismatch is investigated. Finally, we estimate the bound of anticipatingsynchronization error, which depends on coupling strength, anticipating time, and parameter mismatch. The rest of this paper is organized as follows: In the next section, we formulate the problem of anticipating synchronization of chaotic systems with parameter mismatch. In Sec. III, a general convergence criterion for the error dynamical systems is established, followed by some simplified versions. In Sec. IV, the relationship between the coupling strength k,
anticipating time and parameter mismatch is characterized while the error bound is estimated by the simplified criterion. Numerical simulation results are presented to show the validity of our analysis. Finally, conclusions are drawn in Sec. V. Notation: Throughout this paper, we denote by x˙共t兲 the time derivative of x共t兲; PT the transpose of matrix P; m共P兲 and M 共P兲 the minimal and maximal eigenvalues of a real symmetric matrix P, respectively; P ⬎ 0共⬍0兲 the symmetric and positive 共negative兲 definite matrix P and 储 · 储 the Euclidean norm of a vector or a square matrix. Moreover, ⍀ = 兵x 苸 Rn 兩 储x储 艋 其 represents a vector set including the strange attractor of the chaotic systems considered in this paper. II. PROBLEM FORMULATION AND PRELIMINARIES
Consider a class of delay systems defined by the following delay differential equations: x˙共t兲 = A1x共t兲 + B1 f关x共t兲兴 + C1g关x共t − 兲兴, t ⬎ 0, − 艋 t 艋 0,
x共t兲 = 共t兲,
where x共t兲 苸 Rn denotes the state vector, A1, B1, C1 苸Rn⫻n are constant matrices, is non-negative which represents the state delay of system 共1兲, f , g : Rn → Rn are nonlinear functions satisfying the following Lipschitz condition: There exist positive constants L f , Lg such that for all x , y 苸 Rn, 储 f共x兲
− f共y兲 储 艋 L f 储 x − y 储,
储 g共x兲
− 艋 t 艋 0,
where A2, B2, C2 苸Rn⫻n are constant matrices, k ⬎ 0 is the coupling strength to be determined, and denotes the anticipating time. In this paper, we focus on the case of A1 ⫽ A2 or B1 ⫽ B2 or C1 ⫽ C2, i.e., there exists parameter mismatch in the coupled systems. For the simplicity of analysis, we let ⌬A = A2 − A1, ⌬B = B2 − B1, ⌬C = C2 − C1 denote the parameter errors. It is easy to observe from coupled systems 共1兲 and 共2兲 that the anticipating-synchronization error e共t兲 is governed by
− g共y兲 储 艋 Lg储 x − y 储 .
To realize anticipating synchronization via feedback control in master-slave configuration with system 共1兲 as the master system, the corresponding slave system with the same model equation but different parameters is designed as
y˙ 共t兲 = A2y共t兲 + B2 f关y共t兲兴 + C2g关y共t − 兲兴 − k关y共t − 兲 − x共t兲兴, t ⬎ 0, y共t兲 = 共t兲,
共1兲
共2兲
The anticipating synchronization with anticipation time is achieved if the synchronization error e共t兲 stabilizes asymptotically at the origin as time t approaches infinity. But this is not the case in the presence of parameter mismatch because the origin e = 0 is not an equilibrium point of the error system 共3兲. However, as shown in the sequel, it is possible to anticipating-synchronize the master and slave systems up to a relatively small error bound which depends on the parameter errors between the two systems 共1兲 and 共2兲. To this end, our objective shall be to find such a bound that the difference between y共t − 兲 and x共t兲, i.e., e共t兲, converges asymptotically to it. Making use of the transformation
e˙共t兲 = A2e共t兲 + B2兵f关y共t − 兲兴 − f关x共t兲兴其 + C2兵g关y共t − − 兲兴 − g关x共t − 兲兴其 + ⌬Ax共t兲 + ⌬Bf关x共t兲兴 + ⌬Cg关x共t − 兲兴 − ke共t − 兲.
共3兲
d dt
冕
t
t−
e共s兲ds = e共t兲 − e共t − 兲,
共4兲
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Chaos 19, 013104 共2009兲
Anticipating synchronization
system 共3兲 is rewritten as
冋
d e共t兲 − k dt
冕
册
t
t−
e共s兲ds = 共A2 − kI兲e共t兲 + B2兵f关y共t − 兲兴 − f关x共t兲兴其 + C2兵g关y共t − − 兲兴
bound of the anticipating-synchronization error. Our method is different from the typical Lyapunov function/functional method whose main idea is to transform the original stability issue into the evolution of a Lyapunov function/functional with negative time-derivative in the Euclidian/function space. On the contrary, our method does not require the negative derivative of the Lyapunov-type functional.
− g关x共t − 兲兴其 + ⌬Ax共t兲 + ⌬Bf关x共t兲兴 + ⌬Cg关x共t − 兲兴. 共5兲 Obviously, system 共5兲 is equivalent to system 共3兲. In the sequel, we shall first construct a suitable Lyapunov-type functional, and then estimate the time derivative along the solution of error system 共5兲, which leads to a differential inequality. Based on this inequality, we shall estimate the bound of the Lyapunov-type functional and finally obtain the
III. MAIN RESULTS
In this section, we shall estimate the anticipatingsynchronization error bound by analyzing the exponential convergence of the error dynamical system. Theorem 1: Suppose that ⍀ = 兵x 苸 Rn 兩 储x储 艋 1其 and the parameter errors satisfy 储⌬A储 + l1储⌬B储 + l2储⌬C储 艋 2. Also, suppose that there exists a symmetric and positive definite matrix P ⬎ 0 and positive scalars ␣1, ␣2, 1, 2, 3, 4, 5, and 6 such that the following conditions hold:
2 T T T 2 2 2 2 2 2 PA2 + AT2 P + 1 PB2BT2 P + −1 1 l1I + 2 PC2C2 P + k P + 4k PB2B2 P + 5k PC2C2 P + 6k P + 3 P 2 + 共kI − A2兲T P共kI − A2兲 + −1 4 l1I − ␣1 P 艋 0,
共6兲
2 −1 2 −1 2 l2I + 5 l2I − ␣2 P 艋 0,
2k − ␣1 − ␣2 ⬎ 0.
Then the anticipating-synchronized error system 共3兲 converges exponentially to a small region D containing the origin, D = 兵e 苸 Rn 兩 储e储 艋 冑共3 + 6兲 / m共P兲r36 exp兵k其其, where = 12, and r is the unique positive solution of r = 共2k − ␣1兲 − ␣2er .
共7兲
Furthermore, for any arbitrary small positive number , there is a positive T such that, for any t 艌 T, 储e共t兲储 艋 +
冑
3 + 6 exp兵k其. m共P兲r36
共8兲
The explicit proof of this theorem in presented in the Appendix. Remark 1: From 储e共t兲储 艋 + 冑共3 + 6兲/m共P兲r36 exp兵k其, we know that the error bound depends on the parameter mismatches, anticipating time and the exponential convergence degree of the anticipating-synchronization error system.
Remark 2: When 2 = 0, namely, 储⌬A储 = 0 , 储⌬B储 = 0 , 储⌬C储 = 0, it is clear that parameter mismatches disappear and complete synchronization occurs. Theorem 1 indicates that the anticipating synchronization error with anticipation time 6 will converge to the region D if there exists the parameters P, ␣1, ␣2, 1, 2, 3, 4, 5, and 6 such that Eq. 共6兲 holds. Therefore the existence of these parameters should be proven before Theorem 1 can be applied to the systems under study. The size of region D can be estimated once the values of these parameters are known. However, complicated computation is required to obtain the coupling strength and the corresponding error bound. To simplify the computation, three corollaries are derived for the case of P = I 共identity matrix兲, 1 = l1, 2 = l2, 3 = 1, 4 = l1 / k, 5 = l2 / k, and 6 = 1. Although these corollaries are more conservative than Theorem 1, they are more convenient to use in estimating the bound and the coupling strength. Corollary 1: Suppose that ⍀ = 兵y 苸 Rn 兩 储y储 艋 1其 and the parameter errors satisfy 储⌬A储 + L f 储⌬B储 + Lg储⌬C储 艋 2. Let = 12. If there exist positive scalars ␣1, ␣2 such that the following conditions hold:
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max共A2 + AT2 + l1B2BT2 + l2C2CT2 + kB2BT2 + kC2CT2 − kA2 − kAT2 + A2AT2 兲 + l1 + 1 + 3k2 + kl1 − ␣1 艋 0, l2 + l2k − ␣2 艋 0,
共9兲
2k − ␣1 − ␣2 ⬎ 0,
then systems 共1兲 and 共2兲 are anticipating-synchronized with anticipation time and error bound 储e共t兲储 艋 + 冑共 + 1兲 / r exp兵k其, where r is defined in Theorem 1. Note that there are only two parameters ␣1 and ␣2 in Eq. 共9兲. If we further take
␣ 2 = l 2 + l 2 k
共10兲
and
␣1 = max共A2 + AT2 + l1B2BT2 + l2C2CT2 + kB2BT2 + kC2CT2
Remark 4: This corollary shows the mathematical relationship between the anticipating time and the coupling strength k. For a given example, one can explicitly compute the critical curve of the feasible region of 共 , k兲, as shown in the next section. Remark 5: Given the anticipating time , we can also characterize the relationship between the coupling strength k and the exponential convergence degree r. More specifically, from r = 共2k − ␣1兲 − ␣2er, Eqs. 共10兲 and 共11兲, one observes that
− kA2 − kAT2 + A2AT2 兲 + l1 + 1 + 3k2 + kl1 , 共11兲 then one observes that the condition 2k − ␣1 − ␣2 ⬎ 0 is equivalent to ␥1 ⬍ ␥2, where
␥1 = 3k + 2
共max共B2BT2
+
C2CT2
− A2 −
AT2 兲
+ l1 + l2兲k
+ max共A2AT2 兲
mk2 + nk + l = 0, where m = 3, n = max共B2BT2 + C2CT2 − A2 − AT2 兲 + l1 r + l2e − 2 and l = max共A2 + AT2 + l1B2BT2 + l2C2CT2 + A2AT2 兲 + l1 + 1 + l2er − r.
and
␥2 = 2k − max共A2 + AT2 + l1B2BT2 + l2C2CT2 兲 − l1 − 1 − l2 . From Corollary 1, the following result is immediate. Corollary 2: The master system 共1兲 and the slave system 共2兲 are anticipating-synchronized if the anticipating time satisfies one of the following conditions: 共i兲 共ii兲
␥
␥1 ⬎ 0 , ␥2 ⬎ 0, and 0 ⬍ ⬍ ␥21 ; ␥ ␥1 ⬍ 0 , ␥2 ⬍ 0, and ⬎ ␥21 .
Remark 3: This corollary indicates the mathematical relationship between the coupling constant k and the anticipating time . For a given example, one can explicitly compute the critical curve of the feasible region of 共k , 兲, as shown in the next section. Furthermore, let a = 3, b = max共B2BT2 + C2CT2 − A2 − AT2 兲 + l1 + l2 − 2, and c = max共A2 + AT2 + l1B2BT2 + l2C2CT2 + A2AT2 兲 + l1 + 1 + l2. One then observes that the condition 2k − ␣1 − ␣2 ⬎ 0 is also equivalent to ak2 + bk + c ⬍ 0.
共12兲
This leads to the following corollary. Corollary 3: The master system 共1兲 and the slave system 共2兲 are anticipating-synchronized within the error region D = 兵e 苸 R 兩 兩e兩 艋 冑共 + 兲1 / r exp兵k其其 if the coupling strength k satisfies one of the following conditions: 共i兲 共ii兲
⌬ = b2 − 4ac ⬎ 0, −b / 2a 艌 0, and max兵0 , 共−b − 冑b2 − 4ac兲 / 2a其 ⬍ k ⬍ 共−b + 冑b2 − 4ac兲 / 2a; ⌬ = b2 − 4ac ⬎ 0, −b / 2a ⬍ 0, 共−b + 冑b2 − 4ac兲 / 2a ⬎ 0, and 0 ⬍ k ⬍ 共−b + 冑b2 − 4ac兲 / 2a.
IV. NUMERICAL EXAMPLES
In this section, two examples are given to show the validity of our analysis made in the previous section. Example 1: We take the Ikeda oscillator28 as an example to justify the theoretical analysis. The dimensionless form of the Ikeda oscillator is given by
x˙共t兲 = − a1x共t兲 + b1 sin关x共t − 兲兴,
共13兲
where a1 = 1, b1 = 4, = 2. The chaotic attractor of Eq. 共13兲 is shown in Fig. 1. For numerical simulations, we assume that the slave system associated with Eq. 共13兲 is of the form
y˙ 共t兲 = − a2y共t兲 + b2 sin关y共t − 兲兴 − k关y共t − 兲 − x共t兲兴, where a2 = 1.001, b2 = 4.001. Note that l2 = 4, A2 = −1, C2 = 1, ␥1 ⬎ 0, and ␥2 ⬎ 0. Therefore, condition 共i兲 in both Corollary 2 and Corollary 3 is feasible. The critical curve of the feasible region of 共k , 兲 is plotted in Fig. 2, where the feasible region is the region below the curve. From Fig. 2, we select a point in the feasible region of 共k , 兲, such as, 共k , 兲 = 共8.19, 0.03兲, which implies that r = 0.15. Note also that 1 = 4 and 2 = 0.005. One can then estimate the error bound D = 兵e 苸 R 兩 兩e兩 艋 0.067其. To show the accuracy of the estimated error-bound, the error curve and the estimated error-bound are plotted in Fig. 3 for comparison.
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Chaos 19, 013104 共2009兲
FIG. 1. 共Color online兲 The chaotic attractor of the Ikeda system.
FIG. 2. 共Color online兲 The critical relationship curve between coupling constant k and anticipating time . The region below the curve is the feasible region of 共k , 兲.
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FIG. 3. 共Color online兲 Anticipating-synchronization error curve of the Ikeda system with the coupling strength k = 8.16 and anticipating-synchronization time = 0.0361. The initial values for drive and response systems are, respectively, x共兲 = 3, y共兲 = 2, 苸 关−2 , 0兴.
Example 2: Consider the Lu oscillator, which is given by29 x˙共t兲 = − E1x共t兲 + F1 tanh关x共t兲兴 + U1 tanh关x共t − 兲兴, where E1 =
U1 =
冉 冊 冉 冉 冊 1 0 0 1
,
F1 =
− 2.5
0.2
0.1
− 1.5
3.0 5.0 0.1 2.0
,
冊
,
and = 1.
Figure 4 shows the chaotic attractor of the Lu oscillator. We assume that the slave system is of the form
pling strength k and the anticipating time in Fig. 5. If = 0.02 is selected as a special case, we obtain 8.2716⬍ k ⬍ 21.7647 from condition 共i兲 of Corollary 3. The estimated and simulated error bounds are compared numerically for all feasible coupling strengths, as shown in Fig. 6. In particular, each point of simulated error bounds is obtained by averaging the absolute values of the anticipating-synchronization errors at 5000 points over the time interval 关0, 50兴. Those points are obtained by the corresponding value of k. Simulated error bounds is used to describe when the coupling strength satisfies 8.2716⬍ k 艋 14.5911, one observes that the estimated error bound 冑共 + 1兲 / r exp兵k其 decreases from Fig. 6. However, the bound increases when the coupling strength is 14.6582艋 k ⬍ 21.7647.
y˙ 共t兲 = − E2y共t兲 + F2 tanh关y共t兲兴 + U2 tanh关y共t − 兲兴 V. CONCLUSIONS
− k关y共t − 兲 − x共t兲兴, where E2 =
U2 =
冉 冉
1.001
0
0
1.001
冊
,
− 2.501
0.201
0.101
− 1.501
F2 =
冊
冉
3.001 5.001 0.101 2.001
冊
,
.
Note that l1 = 兩max共F2兲兩 = 3.37, l2 = 兩max共U2兲兩 = 2.5209, w2 = 储⌬E储 + l1储⌬F储 + l2储⌬U储 = 0.0128, and w1 = 5. Based on Corollary 3 共i兲, we plot the relationship curve between the cou-
The effect of parameter mismatch on anticipating synchronization of coupled chaotic systems has been studied by characterizing the complex relationship among coupling strength, anticipating time, and parameter errors. Our results show that a suitable choice of coupling strength and anticipating time leads to a small anticipating-synchronization error for the case of the same parameter errors. For given systems, it is easy to determine the coupling strength and anticipating time. Then the error bound of anticipating synchronization is estimated using the last corollary. However, many issues along the line of our work are still open. For
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Anticipating synchronization
Chaos 19, 013104 共2009兲
FIG. 4. 共Color online兲 The chaotic attractor of the Lu oscillator described by Eq. 共12兲 with initial value x1共兲 = 0.1, x2共兲 = −0.5, for 苸 关−1 , 0兴.
FIG. 5. 共Color online兲 The relationship curve between coupling constant k and anticipating time .
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Chaos 19, 013104 共2009兲
Han, Li, and Huang
FIG. 6. 共Color online兲 Theoretically estimated error bounds and numerically simulated error bounds. The initial values for drive and response systems are, respectively, x共兲 = 共0.1, −0.5兲T, y共兲 = 共−0.1, 0.4兲T, 苸 关−1 , 0兴.
example, a sufficiently small anticipating-synchronization error is often required in applications, such as, secure communication. The approach in finding such coupling strength and anticipating time leads to an optimization problem that is worth studying.
where ␣, k1, k2 are positive constants, and k1 ⬎ k2. Then y共t兲 艋 储y共0兲储e−rt +
␣ , r
t 艌 0,
共A2兲
where 储y共0兲储 = max−艋s艋0兩y共s兲兩 and r is the unique positive solution of r = k 1 − k 2e r .
ACKNOWLEDGMENTS
The authors are grateful to the associate editor and the reviewers for their constructive comments, based on which the presentation of the paper has been greatly improved. The work described in this paper was partially supported by the National Natural Science Foundation of China 共Grant No. 60574024兲 and Program for New Century Excellent Talents at the University of China.
共A3兲
Lemma 2: 共Ref. 27兲 Given any real matrices ⌺1, ⌺2, ⌺3 with appropriate dimensions and a scalar ⬎ 0 such that 0 ⬍ ⌺3 = ⌺T3 , the following inequality holds: ⌺T1 ⌺2 + ⌺T2 ⌺1 艋 ⌺T1 ⌺3⌺1 + −1⌺T2 ⌺−1 3 ⌺2 . Proof of Theorem 1: Consider the following Lyapunovtype functional: V关e共t兲兴 = V1关e共t兲兴 + V2关e共t兲兴,
APPENDIX: PROOF OF THEOREM 1
The proof of Theorem 1 makes use of the following results: Lemma 1: 共Ref. 24兲 Suppose that function y共t兲 is nonnegative when t 苸 共− , ⬁兲 and satisfies the following inequality:
冋
V1关e共t兲兴 = e共t兲 − k
t 艌 0,
共A1兲
t
t−
册冋 T
e共s兲ds
P e共t兲 − k
冕
t
册
e共s兲ds ,
t−
共A4兲 V2关e共t兲兴 =
dy共t兲 艋 − k1y共t兲 + k2y共t − 兲 + ␣, dt
冕
冕冕 t
t
t−
s
eT共u兲P1e共u兲duds,
where P1 = k2 P + 4k2 PB2BT2 P + 5k2 PC2CT2 P + 6k2 P2.
013104-9
Chaos 19, 013104 共2009兲
Anticipating synchronization
Differentiating V1 and V2 with respect to t along the trajectory of error system 共5兲 and using Lemma 1 and Lemma 2 yields
冋
V˙1关e共t兲兴 = 2 e共t兲 − k
冕
t
e共s兲ds
t−
册
T
P共共A2 − kI兲e共t兲 + B2兵f关y共t − 兲兴 − f关x共t兲兴其 + C2兵g关y共t − − 兲兴 − g关x共t − 兲兴其 + ⌬Ax共t兲
+ ⌬Bf关x共t兲兴 + ⌬Cg关x共t − 兲兴兲 2 T T T 艋 eT共t兲关P共A2 − kI兲 + 共A2 − kI兲T P兴e共t兲 + 1eT共t兲PB2BT2 Pe共t兲 + −1 1 l1e 共t兲e共t兲 + 2e 共t兲PC2C2 Pe共t兲 2 T −1 2 T 2 2 + −1 2 l2e 共t − 兲e共t − 兲 + 3e 共t兲P e共t兲 + 3 + k
+ 4k 2
冕
冕
t−
t
t−
2 T 2 eT共s兲PB2BT2 Pe共s兲ds + −1 4 l1e 共t兲e共t兲 + 5k
2 T 2 + −1 5 l2e 共t − 兲e共t − 兲 + 6k
V˙2关e共t兲兴 = eT共t兲P1e共t兲 −
冕
冕
t
冕
eT共s兲Pe共s兲ds + eT共t兲共kI − A2兲T P共kI − A2兲e共t兲 t
t−
eT共s兲PC2C22 Pe共s兲ds
t
t−
2 eT共s兲P2e共s兲ds + −1 6 ,
t
t−
eT共s兲P1e共s兲ds.
Therefore,
2 T T T 2 2 2 V˙关e共t兲兴 = V˙1关e共t兲兴 + V˙2关e共t兲兴 艋 eT共t兲关PA2 + AT2 P + 1 PB2BT2 P + −1 1 l1I + 2 PC2C2 P + k P + 4k PB2B2 P + 5k PC2C2 P 2 T + 6k2 P2 + 3 P2 + 共kI − A2兲T P共kI − A2兲 + −1 4 l1I − ␣1 P兴e共t兲 + 共␣1 − 2k兲e 共t兲Pe共t兲 2 −1 2 −1 −1 T 2 + eT共t − 兲关−1 2 l2I + 5 l2I − ␣2 P兴e共t − 兲 + ␣2e 共t − 兲Pe共t − 兲 + 共3 + 6 兲 −1 2 艋 − 共2k − ␣1兲V关e共t兲兴 + ␣2V关e共t − 兲兴 + 共−1 3 + 6 兲 .
Then, we obtain a differential inequality
−1 2 V˙关e共t兲兴 艋 − 共2k − ␣1兲V关e共t兲兴 + ␣2V关e共t − 兲兴 + 共−1 3 + 6 兲 .
Obviously, the time derivative of V is not always negative in its evolution, as stated in Sec. II. Our aim is to use the differential inequality derived above to estimate the bound of V, and to further estimate the bound of error e共t兲. Since 2k − ␣1 − ␣2 ⬎ 0, by Lemma 1, one observes
V关e共t兲兴 艋 储V关e共0兲兴储e−rt +
−1 −1 3 + 6 2 , r
共A5兲
where 储V关e共0兲兴储 = max−艋s艋0兩关共s兲 − 共s兲兴T P关共s兲 − 共s兲兴兩. Note that
冋
V关e共0兲兴 = e共0兲 − k
冐
冕
0
−
艋 M 共P兲 e共0兲 − k
冋
册冋 T
e共s兲ds
冕
P e共0兲 − k
0
−
e共s兲ds
冐
冕
册
0
−
2
冕冕 0
+ M 共P1兲
册
−
冕冕 0
e共s兲ds +
−
0
eT共u兲P1e共u兲duds
s
0
储e共u兲储2duds
s
1 艋 M 共P兲共1 + k兲2 + M 共P1兲2 兩兩2 2 ⬅ M 2兩 兩 2 ⬍ ⬁,
共A6兲
013104-10
Chaos 19, 013104 共2009兲
Han, Li, and Huang
where 共兲 = e共兲 and 兩兩 = sup关兩共兲兩兴, 苸 关−¯ , 0兴. Note also that the Lyapunov-type functional 共A4兲 satisfies
冐
m共P兲 e共t兲 − k
冕
t
e共s兲ds
t−
冐
2
艋 V关e共t兲兴.
共A7兲
冐
冕
t
e共s兲ds
t−
冐
艋 V关e共t兲兴 艋 M 2兩兩2 e−rt +
−1 −1 3 + 6 2 , r
which implies
冕 冑 t
储e共t兲储 艋 k
+
t−
储e共s兲储ds +
M兩兩
冑m共P兲 e
3 + 6 , m共P兲r36
−rt/2
t ⬎ 0.
From the Gronwall inequality, one derives
储e共t兲储 艋
冋
M兩兩
冑m共P兲 e
−rt/2
+
冑
册
3 + 6 exp兵k其. m共P兲r36 共A8兲
This implies that the synchronization error system 共3兲 converges exponentially to the small region D containing the origin. For any arbitrary small positive number , from inequality 共A8兲, there is a positive T such that, for any t 艌 T,
3 + 6 exp兵k其. m共P兲r36
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2
冑
Therefore, the anticipating-synchronization with error bound + 冑共3 + 6兲 / m共P兲r36 exp兵k其 for any arbitrary small positive number is achieved. This completes the proof. 1
From Eqs. 共A5兲–共A7兲, one obtains
m共P兲 e共t兲 − k
储e共t兲储 艋 +
CHAOS 19, 013105 共2009兲
Enhanced synchronizability in scale-free networks Maoyin Chen,1 Yun Shang,2 Changsong Zhou,3 Ye Wu,4 and Jürgen Kurths4 1
Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China and Department of Automation, Tsinghua University, Beijing 100084, China 2 Institute of Mathematics, AMSS, Academia Sinica, Beijing 100080, China 3 Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong 4 Institut für Physik, Potsdam Universität, Am Neuen Palais 10, D-14469, Germany
共Received 11 August 2008; accepted 8 December 2008; published online 16 January 2009兲 We introduce a modified dynamical optimization coupling scheme to enhance the synchronizability in the scale-free networks as well as to keep uniform and converging intensities during the transition to synchronization. Further, the size of networks that can be synchronizable exceeds by several orders of magnitude the size of unweighted networks. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3062864兴 Works on synchronizability in networks with a given topology can be divided into two classes according to the coupling matrix. One class is the static mechanism, where the coupling matrix remains fixed during the transition to synchronization. This mechanism includes the degree and load based weighted networks. The other class is the dynamical mechanism, where the coupling matrix evolves in time by introducing adaptive strengths between connected oscillators. The adaptation process can enhance synchronization by modifying the coupling matrix, but the resulting networks have nonuniform intensities even for networks with homogeneous degrees. In this paper, we introduce a modified dynamical optimization mechanism to enhance the synchronizability in the scale-free networks as well as to keep uniform and converging intensities during the transition to synchronization. Further, the size of networks that can be synchronizable exceeds by several orders of magnitude the size of unweighted networks. I. INTRODUCTION
In the past few years, the dynamics of complex networks has been extensively investigated.1–4 As a typical dynamical process on networks, synchronization, especially the ability of networks to obtain synchronization 共synchronizability兲, has attracted a lot of interest.5–22 Recent studies have revealed that unweighted small-world and scale-free networks are more synchronizable than unweighted regular networks.5,6 But the assumption that local units are symmetrically coupled with undirected couplings does not match the properties of real networks 共such as unequal connection weights and asymmetry of the couplings兲.7,8 Recent efforts have been focused on achieving efficient synchronization by introducing connection weights and directionality into networks.9–15,18–21 From Ref. 20, works on the synchronizability in networks with a given topology can be divided into two classes according to the coupling matrix. One class is the static mechanisms, where the coupling matrix is invariant.6,9–17 For randomly enough unweighted and weighted networks, the synchronizability is controlled by Smax / Smin, where Smax and 1054-1500/2009/19共1兲/013105/5/$25.00
Smin are, respectively, the maximum and minimum of intensity Si, which is defined by the sum of the coupling strengths of oscillator i.14 For unweighted Barabási–Albert 共BA兲 networks,14 Smax / Smin = kmax / kmin ⬃ N1/2, where kmax and kmin are the maximal and minimal degrees, respectively. Hence, the synchronizability can be enhanced if intensities become more homogeneous. From the degree based weighted networks,11,13 one necessary condition for the optimal synchronizability Ropt is that the intensities become uniform. The other class is the dynamical mechanisms, where the coupling matrix is variant by introducing adaptive strengths into networks of identical oscillators18 and nonidentical oscillators.19 The adaptation process can enhance the synchronization by modifying the coupling matrix, but the resulting networks have heterogeneous intensities due to heterogeneous degrees. For BA networks, after the adaptation, the synchronizability is characterized by Smax / Smin ⬃ N/2 with  = 1 − and ⬃ 0.5.18 Inspired by the static mechanisms,11,13 one necessary condition for the optimal synchronizability is that intensities become uniform. However, even for networks with homogeneous degrees, the mechanisms18,19 cannot ensure uniform intensities due to different initial conditions of oscillators.20 Therefore, a problem naturally arises: By the dynamical mechanism, how can we realize the synchronization in networks as well as ensure uniform intensities during the transition to synchronization and enhance the synchronizability, regardless of heterogeneous degrees and initial conditions of oscillators? II. THE MODIFIED DYNAMICAL OPTIMIZATION MECHANISM
Recently, we have already obtained some results on the above problem. For different variants of the Kuramoto model, we have proposed a dynamical gradient network 共DGN兲 approach to realize phase synchronization.21 It is shown that all the oscillators have uniform intensities during the transition to synchronization. However, the DGN approach is very special in two aspects. One is that it should assign a scale potential to each oscillator within any time interval, which depends on the extent of the local synchronization among itself and its neighbor oscillators. The other is
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that the adjustment of the respective link by the DGN approach is often mostly ineffective. Inspired by the DGN approach,21 we have further introduced the original dynamical optimization 共DO兲 mechanism for small-world networks 共SWNs兲.20 The main idea in the original DO mechanism is to increase the coupling strength of only one incoming link of oscillator i by a small value in different intervals with a fixed length. It reflects the “winner-take-all” strategy, where the incoming link to be adjusted is always chosen as a pair of oscillators with the weakest synchronization. This means that the original DO mechanism is more effective than the DGN approach. We previously showed that the original DO mechanism has much better synchronizability in SWNs.20 Unfortunately, there exists one main shortcoming in the original DO mechanism.20 That is, the coupling strength between two connected oscillators is an increasing function of time as well as the intensities are diverging to infinity. Basically, this means that full synchronization is trivially obtained for some kinds of networks, such as any variant of the Kuramoto model21 and networks of Rössler oscillators coupled through full states. The above networks always converge to a fully synchronized regime if the couplings 共or intensities兲 are sufficiently large. However, for some kinds of networks such as networks of Rössler oscillators coupled through partial states,22 the synchronization cannot be realized if the couplings 共or intensities兲 are largely enough. In our recent work,20 we have to end the original DO mechanism provided that the synchronization error is small enough. If not, the couplings 共or intensities兲 are so large that the synchronization can be destroyed and the synchronization error becomes large again. Obviously, it is reasonable to introduce one dynamical mechanism with limited couplings 共or intensities兲 even if time increases to infinity. Here we modify the original DO mechanism such that the intensities are converging and the ultimate intensity can be adjusted. We consider networks consisting of N coupled oscillators N
x˙ i = F共xi兲 +
兺
Gij共H共x j兲 − H共xi兲兲,
1 艋 i 艋 N,
共1兲
j⫽i,j=1
where xi is the state, F is the dynamics of individual oscillator, H is the output function, and G = 共Gij兲 is the coupling matrix. Gij = AijWij, where A = 共Aij兲 is the binary adjacency matrix, Wij is the coupling strength of the incoming link 共i , j兲 pointing from oscillator j to oscillator i if they are connected, Gii = −兺 j苸KiAijWij, and Ki is the neighbor set of oscillator i. In unweighted networks, Wij = 1 is uniform for all the incoming links. In the original DO mechanism,20 we increase the coupling strength of only one incoming link of oscillator i by a small value, at the time step tn = t0 + n, where n 艌 1 is the positive integer, t0 is the transient time, and ⬎ 0 is the duration time. This adaptation results from the competition between neighbor oscillators within the interval 关tn−1 , tn兲. For oscillator i and one neighbor j 苸 Ki, a total synchronization difference, i.e., En共i , j兲 = 兰ttn 共xi , x j兲dt, within the interval n−1 关tn−1 , tn兲 is evaluated, where is a non-negative error function, and satisfies 共xi , x j兲 = 0 if oscillators i , j are synchro-
nized. For oscillator i, the incoming link with the weakest synchronization, i.e., 共i , jnmax兲, is the winner within the interval 关tn−1 , tn兲, where the index j nmax is decided by the optimization problem jnmax = arg max En共i, j兲. j苸Ki
共2兲
If several neighbors have the same synchronization difference, we choose only one randomly. In the original DO mechanism, the coupling strength is adjusted dynamically by20 n+1
n
Wijn = Wijn + , max
max
n n Wn+1 ij = Wij, j ⫽ j max ,
共3兲
where the incremental coupling ⬎ 0 is a small value, and Wnij is the coupling strength in the interval 关tn−1 , tn兲. Obviously, the intensities are diverging as time tends to infinity. In order to ensure that the intensities converge to a limited value as time tends to infinity, we adjust the coupling strength by n+1
n
Wijn = Wijn + n , max
max
n n Wn+1 ij = Wij, j ⫽ j max ,
共4兲
where n ⬎ 0 is the incremental coupling. Here we give some basic rules for choosing the incremental coupling n, which make the ultimate intensities be uniform and convergent. 共i兲 The incremental couplings n for all oscillators are identical at the time step tn, which make the intensities Si be uniform during the transition to synchronization. 共ii兲 The incremental coupling n is limited by the fixed constant , which implies that at the time step tn the incremental coupling n should not be large. 共iii兲 The incremental coupling n is a nonincreasing function on the time step n, and the ultimate intensity ¯S = 兺⬁i=1i exists. This requirement means that after the time step tn, the total intensity 兺ni=1i is convergent and n approaches zero as the time step n tends to infinity. 共iv兲 The ultimate intensity ¯S can be adjusted. This is consistent with realistic cases where the intensities 共or couplings兲 for synchronization are in a certain range 共such as networks of Rössler oscillators coupled through partial states兲. We can further discuss the relationship between network synchronization and network topology by adjusting the ultimate intensity ¯S. Summing up the above analysis, the term e−n/n0 is one suitable choice of the incremental coupling n. Hence, we choose n = e−n/n0. We then adjust the coupling strength by n+1
n
Wijn = Wijn + e−n/n0 , max
max
n n Wn+1 ij = Wij, j ⫽ j max ,
共5兲
where n0 is a suitable positive integer. Here we call this mechanism 关namely, Eqs. 共2兲 and 共5兲兴 the modified DO mechanism. In this paper, the initial coupling strengths in networks are assumed to be zero.23 Hence, the intensities are uniform
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at the time step tn, since the intensity of each oscillator increases by the same amount e−n/n0 at the time step tn. Further, the intensity Si for oscillator i is bounded by the limit ¯S = lim n→⬁Si, where 共6兲
We can adjust the ultimate intensity by the suitable parameter n0. For a fixed , when n0 is larger 共smaller兲, the intensity ¯S is larger 共smaller兲. III. ENHANCED SYNCHRONIZABILITY IN SCALE-FREE NETWORKS
We briefly review the stability of networks
E
¯S = e−1/n0/共1 − e−1/n0兲.
10 5 0 −2 0
1000
t
2000
3000
FIG. 1. The average error E as a function of time t. The parameters are N = 1000, M = 5, = 1, = 0.001, and n0 = 1200.
N
G0ij共H共x j兲 − H共xi兲兲, 兺 j⫽i,j=1
1 艋 i 艋 N, 共7兲
where is the overall strength. For a generally asymmetric matrix G0 = 共G0ij兲, the variational equation on the synchronous state 兵xi = s , ∀ i其 is u˙ i = 关DF共s兲 − lDH共s兲兴ui, where D is the Jacobian operator, and l is the complex eigenvalue of the Laplacian matrix L 共=−G0兲, satisfying Re共1兲 艋 Re共2兲 艋 ¯ 艋 Re共N兲. The largest Lyapunov exponent 共LLE兲, i.e., ⌳共⑀ , 兲, of the master stability equation v˙ = 关DF共s兲 − 共⑀ + i兲DH共s兲兴v is a function of ⑀ and , which is the master stability function 共MSF兲.22 Let R be the region in the complex plane where the MSF provides a negative LLE. The synchronization condition is that the set 兵l : l ⫽ 0其 is entirely contained in R.22 Here we only consider the case where the region R is bounded, which is shown by the dashed line in Figs. 4共a兲 and 4共c兲. A better synchronizability is achieved if simultaneously the ratio Re共N兲 / Re共2兲 and max兩Im共l兲兩 are minimized.10,12 In this paper, we have two aims based on networks 共1兲 and 共7兲. One is to realize the synchronization in network 共1兲, in which all the oscillators have uniform intensities during the transition to synchronization. The other is to examine the synchronizability in network 共7兲 when the coupling matrix G0 is assigned by the coupling matrix from the synchronization in network 共1兲, during or after the adaptation. Our analysis and simulation are based on BA networks.4 Initially, M oscillators with labels i = 1 , . . . , M are fully connected. At every time step a new oscillator is introduced to be connected to M existing oscillators. The probability that the new oscillator is connected to oscillator i depends on degree ki, i.e., ⌸i = ki / 兺 jk j. Here we choose Rössler networks to illustrate the effectiveness of our mechanism: xi = 共xi , y i , zi兲, H共xi兲 F共xi兲 = 共−0.97y i − zi , 0.97xi + 0.15y i , zi共xi − 8.5兲 + 0.4兲, = 共xi , 0 , 0兲, and 共xi , x j兲 = 兩xi − x j兩 + 兩y i − y j兩 + 兩zi − z j兩. In order to measure the synchronization, we define the average error as E = 共1 / N兲兺Ni=1兩xi − ¯x兩, where ¯x = 共1 / N兲兺Ni=1xi is the global mean field. In our simulations the initial conditions for oscillators are randomly chosen from Rössler attractor 共here, t0 = 0兲. The parameter n0 in Eq. 共5兲 is n0 = 1200. Hence, the limit ¯S is about 1.2 if the value = 0.001. From Fig. 1, the synchronization in network 共1兲 is realized effectively. From Eqs. 共2兲 and 共5兲, all the oscillators have uniform intensities during the
transition to synchronization, regardless of heterogeneous degrees and initial conditions. It is consistent with the necessary condition for the optimal synchronizability in the static mechanisms.11,13 But this is totally different from the dynamical mechanisms.18,19 The average intensity S共k兲 over oscillators with degree k increases as S共k兲 ⬃ k with  ⬃ 0.5.18 During the transition to synchronization, the ratio Re共N兲 / Re共2兲 in network 共7兲 with G0 = G decreases towards the optimal synchronizability Ropt ⬇ 3.8 共Fig. 2兲. The value Ropt is decided by the eigenratio of the Laplacian matrix where G⬘共␣兲 = 共G⬘ij共␣兲兲 with G⬘ij共␣兲 of G⬘共0兲, = 共kik j兲␣ / 兺 j苸Ki共kik j兲␣ and G⬘ii共␣兲 = −1.11 From Eqs. 共2兲 and 共5兲, the incoming link to be adjusted for each oscillator is always chosen to be the pair of oscillators with the maximal synchronization difference in the previous time interval, which greatly decrease the ratio Re共N兲 / Re共2兲. However, there still exists the discrepancy between the ultimate value of Re共N兲 / Re共2兲 and Ropt. Now we explain the reason for the discrepancy. Due to the “winner-take-all” strategy inherent in the DO mechanism, the coupling strengths Wij for oscillator i are almost uniform statistically as the time step n approaches infinity; namely, Wij ⬃ k−1 i . Unfortunately, the exact uniform coupling strength Wij = k−1 i cannot be realized by dynamical mechanisms. In order to show it, we define the average standard deviation Esd共k兲 = 共1 / lk兲兺E0l between G0 3
10 Re(λN)/Re(λ2)
x˙ i = F共xi兲 +
2
10
1
10
0
10
0
1000
n
2000
3000
FIG. 2. The ratio Re共N兲 / Re共2兲 as a function of the adjustment step n. Solid line: the ratio by the modified DO mechanism; dashed line: Ropt. The parameters are the same as those in Fig. 1.
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−1
10 Re(λN)/Re(λ2)
10
−2
10
2
Esd
10
−3
10
1
10
−4
10
0
1
10
10
2
3
10
k
0
10
FIG. 3. Standard deviation Esd共k兲 as a function of degree k. The parameters are the same as those in Fig. 1.
given by the following Eq. 共8兲 and G⬘共0兲, where lk is the number of oscillators with degree k and E0l = 共1 / k兲冑兺 j⫽i共G0ij − 1 / k兲2 共Fig. 3兲. From this figure, the exact uniform coupling strength Wij = k−1 i cannot be realized by dynamical mechanisms. This may be the reason for the discrepancy between the ultimate value of Re共N兲 / Re共2兲 and Ropt. We assign the coupling matrix G0 in network 共7兲 by ¯, G0 = Gnorm = Gend/S
共8兲
where Gend is the coupling matrix of network 共1兲 after the adaptation. Since all the oscillators have uniform intensities, the Laplacian matrices of Gnorm and Gend have equal ratios Re共N兲 / Re共2兲. When = 1.5, all the nonzero eigenvalues of the Laplacian matrix of Gnorm are located in a very narrow region around the real axes in the region R, and the absolute values of imaginary parts are sufficiently small 关Figs. 4共b兲 and 4共c兲兴. The ratio Re共N兲 / Re共2兲 in network 共7兲 with G0 = Gnorm increases slightly with increasing the network size N, and this can be well fitted by a power-law dependence, which means the synchronizability decreases slightly 共Fig. 5兲. From the fitting and the value R, we find that the network 共7兲 is still synchronizable until N ⬇ 10.11 The size of the network 共7兲 that can be synchronizable exceeds by several orders of magnitude the size of unweighted networks 共⬇103兲 and networks with adaptive coupling 共⬇8 ⫻ 105兲.18 Obviously, this is a great enhancement of the synchronizability in
6
10
8
N
10
10
10
12
10
networks, compared with unweighted networks and networks with adaptive coupling.18 It should be pointed out that for different size of networks, max兩Im共l兲兩 is sufficiently small 共the maximal value is less than 0.1兲. For the coupling matrix G0 = Gnorm, all the eigenvalues are fully contained within the unit circle centered at 1.24 Thus, 0 艋 Re共l兲 艋 2, 兩Im共l兲兩 艋 1, and the largest Re共N兲 never diverges, independently of the network size N.10 During the transition to synchronization in network 共1兲, Smax / Smin is always equals to 1. But in Refs. 14 and 18, the synchronizability decreases with the increasing of Smax / Smin, and Smax / Smin increases with the increasing of the size N. Hence, the synchronizability here is better than Ref. 18, whose main aim is to reduce the heterogeneity of the intensities adaptively. For the fixed n0 and N, we discuss the effect of parameters and on the synchronizability in network 共7兲 with G0 = Gnorm 关Figs. 6共a兲 and 6共b兲兴. The value can be chosen in a wide range, and the length can be arbitrary large. In our simulations, the value belongs to 关0.0001, 0.005兴 From Figs. 6共a兲 and 6共b兲, the ratio Re共N兲 / Re共2兲 is almost independent of the values of and .
(b)
2
0.05
(c)
l
l
Im(λ )
2
0.1
(a)
4
10
FIG. 5. The ratio Re共N兲 / Re共2兲 for different size of network 共7兲. Diamond: unweighted networks; square: networks with adaptive coupling 共see Ref. 18兲; circle: the ratio by the modified DO mechanism; solid line: fitting; dashed line: R = ␦2 / ␦1 ⬇ 40. The parameters M , , , n0 are the same as those in Fig. 1. All the estimates are averaged over 20 realizations of networks.
Im(λ )
3
10 2 10
0
0 −0.05
−2 −3
0 2 4 6 Re(λ ) l
0 −2
−0.1 0.1 1 2 3 Re(λ ) l
δ
−0.50
δ2
1
1
2 3 4 Re(λl)
5
6
FIG. 4. 共a兲 The stability region R bounded by the dashed line. 共b兲 Distribution of nonzero eigenvalues l of the Laplacian matrix of Gnorm. 共c兲 The location of nonzero eigenvalues l in the region R. Circles: nonzero eigenvalues by the modified DO mechanism; ␦1 ⬇ 0.144 and ␦2 ⬇ 5.76 are the minimum and maximum of real parts in the region R, respectively. The parameters N , M , , are the same as those in Fig. 1, and = 1.5.
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6 Re(λN)/Re(λ2)
able exceeds by several orders of magnitude the size of unweighted networks.
5
ACKNOWLEDGMENTS
4 3
(a)
2 0
5 τ
10
Re(λN)/Re(λ2)
2
5 4 3
(b)
S. H. Strogatz, Nature 共London兲 410, 268 共2001兲. R. Albert and A. L. Barabási, Rev. Mod. Phys. 74, 47 共2002兲. 3 D. J. Watts and S. H. Strogatz, Nature 共London兲 393, 440 共1998兲. 4 A. L. Barabási and R. Albert, Science 286, 509 共1999兲. 5 M. Barahona and L. M. Pecora, Phys. Rev. Lett. 89, 054101 共2002兲. 6 T. Nishikawa, A. E. Motter, Y. C. Lai, and F. C. Hoppenstead, Phys. Rev. Lett. 91, 014101 共2003兲. 7 G. A. Polis, Nature 共London兲 395, 744 共1998兲. 8 V. Latora and M. Marchioric, Phys. Rev. Lett. 87, 198701 共2001兲. 9 M. Chavez, D. Huang, A. Amann, H. G. E. Hentschel, and S. Boccaletti, Phys. Rev. Lett. 94, 218701 共2005兲. 10 M. Chavez, D. Huang, A. Amann, and S. Boccaletti, Chaos 16, 015106 共2006兲. 11 A. E. Motter, C. Zhou, and J. Kurths, Phys. Rev. E 71, 016116 共2005兲. 12 D. Huang, M. Chavez, A. Amann, and S. Boccaletti, Phys. Rev. Lett. 94, 138701 共2005兲. 13 X. Wang, Y. Lai, and C. Lai, Phys. Rev. E 75, 056205 共2007兲. 14 C. Zhou, A. E. Motter, and J. Kurths, Phys. Rev. Lett. 96, 034101 共2006兲. 15 T. Nishikawa and A. E. Motter, Phys. Rev. E 73, 065106 共2006兲. 16 L. Donetti, P. I. Hurtado, and M. A. Munoz, Phys. Rev. Lett. 95, 188701 共2005兲. 17 B. Wang, H. Tang, T. Zhou, and Z. Xiu, “Optimizing synchronizability of networks,” arXiv:cond-mat/0512079, v2, 23 March 2007. 18 C. Zhou and J. Kurths, Phys. Rev. Lett. 96, 164102 共2006兲. 19 Q. Ren and J. Zhao, Phys. Rev. E 76, 016207 共2007兲. 20 Y. Wu, Y. Shang, M. Chen, C. Zhou, and J. Kurths, Chaos 18, 037111 共2008兲. 21 M. Chen, Y. Shang, Y. Zou, and J. Kurths, Phys. Rev. E 77, 027101 共2008兲. 22 L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 80, 2109 共1998兲. 23 The adjustment time n is counted when the number of zero eigenvalues of matrix G is 1. 24 R. S. Varga, Gersgorin and His Circles 共Springer, Heidelberg, 2004兲. 1
6
2 0
M.C. was supported by NSFC project 共No. 60804046兲, Special Doctoral Fund in the University by Ministry of Education 共No. 20070003129兲, and the Alexander von Humboldt Foundation. Y.S. thanks the partial support by NSFC Project Nos. 60736011 and 60603002, and 863 Project No. 2007AA01Z325. J.K. was supported by SFB 555 共DFG兲 and BRACCIA 共EU兲.
1
2
ε
3
4 5 −3 x 10
FIG. 6. The ratio Re共N兲 / Re共2兲 for different 共a兲 and 共b兲. Solid line: the ratio by the modified DO mechanism. Dashed line: Ropt. The parameters N , M , n0 are the same as those in Fig. 1. All the estimates are averaged over 20 realizations of networks.
IV. CONCLUSION
In this paper, we introduce a modified dynamical optimization coupling scheme to enhance the synchronizability in the scale-free networks as well as to keep uniform and converging intensities during the transition to synchronization. Moreover, the size of networks that can be synchroniz-
CHAOS 19, 013106 共2009兲
Outer synchronization of coupled discrete-time networks Changpin Li,1 Congxiang Xu,1 Weigang Sun,2 Jian Xu,3 and Jürgen Kurths4 1
Department of Mathematics, Shanghai University, Shanghai, China The School of Science, Hangzhou Dianzi University, Hangzhou, China 3 Department of Engineering Mechanics and Technology, Tongji University, Shanghai, China 4 Institute of Physics, Humboldt University, Berlin, Germany and Potsdam Institute for Climate Impact Research, Potsdam, Germany 2
共Received 12 September 2008; accepted 17 December 2008; published online 29 January 2009兲 In this paper, synchronization between two discrete-time networks, called “outer synchronization” for brevity, is theoretically and numerically studied. First, a sufficient criterion for this outer synchronization between two coupled discrete-time networks which have the same connection topologies is derived analytically. Numerical examples are also given and they are in line with the theoretical analysis. Additionally, numerical investigations of two coupled networks which have different connection topologies are analyzed as well. The involved numerical results show that these coupled networks with different connection matrices can reach synchronization. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3068357兴 Synchronization inside a coupled network with/without time delays, or called “inner synchronization” for convenience, has been recently intensively and extensively studied.1 Roughly speaking, inner synchronization inside a network, constituted by numbers of identical nodes, indicates that each node of the network approaches an asymptotical steady state. Obviously, this kind of synchronization reflects one aspect of the real world. In effect, there exist other kinds of network synchronization in the world, for example, “outer synchronization,” i.e., synchronization between two coupled networks. An outstanding example is the Acquired Immune Deficiency Syndrome, AIDS for brief,2 which was originally infected among gorillas, afterwards was contagious to human beings unexpectedly. This fatal infectious disease is spread between two different mammalian communities: gorillas and human beings. People can still recall mad cow disease.3 Cows and human beings can also be regarded as two different networks in terms of network language. There are two more recent examples of coupled species: avian influenza (bird flu),4 and severe acute respiratory syndrome, or SARS for brevity.5 These cited examples show the strong importance and challenge to study the dynamics between different coupled networks.
example, the infectious diseases, such as AIDS, mad cow disease, bird flu, SARS, were originally spread between two communities 共or networks兲. This means that to study the dynamics between two coupled networks is necessary and important. Li et al. have studied the synchronization between two coupled continuous-time networks, where a synchronization criterion and numerical simulations were presented.7 Shortly after, Li et al. and Tang et al., further studied outer synchronization.8 All these studies are for continuous-time networks. In this article, we focus on studying the discretetime case. The outline of the rest of the paper will be organized in the following. Theoretical analysis is derived in Sec. II. The illustrated examples are given in Sec. III. And the last section includes comments and conclusions. II. THEORETICAL ANALYSIS
Throughout this paper, the following notations always work. 共1兲 R denotes the real numbers. For u 苸 Rn, uT denotes its transpose. M 苸 Rn⫻n denotes a matrix of order n; 共2兲 I denotes an identity matrix; 共3兲 M ⬎ 0 means that M is positive definite. The coupled equations of two discrete-time networks can be expressed as follows:
I. INTRODUCTION
N
Synchronization inside a coupled network with/without time delays, or called “inner synchronization” for convenience, which is a hot topic, has been recently intensively and extensively studied.1 This kind of synchronization reveals one aspect of the real world.6 In reality, synchronization between two coupled networks, or “outer synchronization”7 共by the way, the first paper of this reference is an early work in this respect, where two identical oscillators can be regarded as two networks; whilst the second paper is the early paper where “outer synchronization” was clearly proposed兲, always does exist in our lives. For 1054-1500/2009/19共1兲/013106/7/$25.00
xi共t + 1兲 = f关xi共t兲兴 + c 兺 aij⌫x j共t兲 + CYX共Y,X兲,
共1兲
j=1 N
y i共t + 1兲 = g关y i共t兲兴 + c 兺 bij⌫y j共t兲 + CXY 共X,Y兲,
共2兲
j=1
where c ⬎ 0 is the coupling strength of the network, xi , y i 苸 Rn, i = 1 , 2 , . . . , N , f , g : Rn → Rn are continuously differentiable functions which determine the dynamical behavior of the nodes in networks X and Y, respectively. ⌫ 苸 Rn⫻n is a constant 0-1 matrix linking coupled variables. For simplicity,
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known that the closed-loop 共or “feedback”兲 and open-loop control methods each have their advantages and disadvantages. To overcome their shortcomings, Jackson and Grosu first combined these two methods to derive a new control method 共OPCL method兲 and applied to complex dynamical systems; for details, see the first paper of Ref. 9. In the present paper, we use the OPCL method to construct the following scheme for a drive-response network system: N
xi共t + 1兲 = f共xi共t兲兲 + c 兺 aij⌫x j共t兲,
共3兲
j=1 N
y i共t + 1兲 = g共y i共t兲兲 + c 兺 bij⌫y j共t兲 j=1
冉
+ H−
冊
f共xi兲 共y i共t兲 − xi共t兲兲. xi
共4兲
Network 共3兲 is regarded as a drive one and network 共4兲 a response one. Definition: Networks 共3兲 and 共4兲 are said to attain 共complete兲 synchronization if lim 储y i共t兲 − xi共t兲储 = 0,
共5兲
i = 1,2, . . . ,N.
t→+⬁
In the following, we start with the simple case with the synchronization between systems 共3兲 and 共4兲, assuming that they both have the same topological structures, i.e., A = 共aij兲N⫻N = B = 共bij兲N⫻N 共they do not need to be symmetric兲 and same dynamics 共f = g兲. Letting ei = y i − xi yields N
ei共t + 1兲 = Hei共t兲 + c 兺 aij⌫e j共t兲,
i = 1,2, . . . ,N,
共6兲
j=1
by linearizing the error system around xi. Henceforth, Eq. 共6兲 can be rewritten as FIG. 1. Examples of actions 共denoted by arrows兲 between two networks. 共a兲 Unidirectional action; 共b兲 bidirectional action; 共c兲 special node interaction.
one often assumes that ⌫ = diag共r1 , r2 , . . . , rn兲 艌 0 is a diagonal matrix. A = 共aij兲N⫻N and B = 共bij兲N⫻N represent the coupling configurations of both networks, whose entries aij and bij are defined as follows: if there is a connection between node i and node j共j ⫽ i兲, then set aij ⬎ 0 and bij ⬎ 0, otherwise aij = 0, bij = 0共j ⫽ i兲; the matrices A and B can be symmetric or asymmetric, as usual, each line sum of A and B is assumed to be zero. CXY 共X , Y兲共CYX共Y , X兲兲 represents the interaction from network X共Y兲 to network Y共X兲. There are lots of active forms between networks, for instance, Fig. 1 gives the actions between 3-nodes networks. In this paper, we only study unidirectional coupling between networks 共1兲 and 共2兲. Here we choose CXY 共X , Y兲 = 关H − f共xi兲 / xi兴关y i共t兲 − xi共t兲兴 and CYX共X , Y兲 = 0, where H 苸 Rn⫻n is a constant matrix to be set. This chosen interaction is based on the open-plus-closed-loop 共OPCL兲 method.9 It is
共7兲
e共t + 1兲 = He共t兲 + c⌫e共t兲AT ,
where e = 关e1 , e2 , . . . , eN兴 苸 Rn⫻N and AT denote the transpose of A. As is well known, the coupling matrix can be decomposed into AT = SJS−1, where J is a Jordan form with complex eigenvalues 苸 C and S contains the corresponding eigenvectors s. Denoting = eS gives
共t + 1兲 = H共t兲 + c⌫共t兲J,
共8兲
by multiplying Eq. 共7兲 from the right-hand side with S, where J = diag共J1 , J2 , . . . , Jl兲 is a block diagonal matrix, and Jk 苸 RNk⫻Nk is the block corresponding to the Nk multiple eigenvalue k of A, that is,
J=
冤
J1 Jl
k = 1, . . . ,l.
冥
,
Jk =
冤
¯
0 0
]
k 1 ¯ ]
]
0
0
¯ k
1
0
0
¯
k
k 1 0
0
0
冥
,
共9兲
013106-3
Chaos 19, 013106 共2009兲
Coupled discrete-time networks
Let = 关1 , 2 , . . . , l兴 and k = 关k,1 , k,2 , . . . , k,Nk兴. Since the sum of each line of the matrix A is assumed to be zero, we can assume 1 = 0 and J1 = 0. It immediately follows that 1共t + 1兲 = H1共t兲. Its zero solution is asymptotically stable if HTH − I is negative definite where I is the identity matrix. Now we rewrite Eq. 共8兲 in component form,
k,1共t + 1兲 = 共H + ck⌫兲k,1共t兲, k,p+1共t + 1兲 = 共H + ck⌫兲k,p+1共t兲 + c⌫k,p共t兲,
共10a兲
networks 共3兲 and 共4兲 is equivalent to that in Synchronization Criterion II for networks 共11兲 and 共12兲 provided that all f i and Hi are the same is a puzzle and remains unsolved. And we cannot make a decision which Synchronization Criterion is more convenient in analyzing outer synchronization of coupled networks. On the other hand, Synchronization Criteria I and II also hold for the following bidirectional coupled networks:
冉
xi共t + 1兲 = f共xi共t兲兲 + ␣ H −
1 艋 p 艋 Nk − 1,
冊
f共xi兲 共xi共t兲 − y i共t兲兲 xi
N
where k = 2 , 3 , . . . , l. For the first system of Eq. 共10a兲, its zero solution is asymptotically stable if there exists a matrix H 苸 Rn⫻n such that PTk Pk − I are negative definite, where Pk =
冋
H + c ␣ k⌫
− c  k⌫
c  k⌫
H + c ␣ k⌫
册
,
k = 2, . . . ,l.
共10b兲
For the second system of Eq. 共10a兲, its zero solution is asymptotically stable if QTk Qk − I are negative definite, in which Qk = diag共c⌫,c⌫, Pk兲,
共10c兲
k = 2, . . . ,l.
The proofs are given in the Appendix. So we can assert the Synchronization Criterion I if 共a兲 A = B, and each line sum of them is zero; 共b兲 f = g; and 共c兲 HTH − I, PTk Pk − I, QTk Qk − I, k = 2 , . . . , l, are negative definite, in which Pk and Qk are given in Eqs. 共10b兲 and 共10c兲, respectively, then networks 共3兲 and 共4兲 can be synchronized. If we set LA共X兲 = c共A 丢 ⌫兲X, LB共Y兲 = c共B 丢 ⌫兲Y, where 丢 denotes the Kronecker product, f共xi兲 → f i共xi兲 in Eqs. 共3兲 and 共4兲, g共y i兲 → f i共y i兲, H → Hi in Eq. 共4兲, then Eqs. 共3兲 and 共4兲 can be rewritten in a compact form X共t + 1兲 = F共X共t兲兲 + LA共X共t兲兲,
共11兲
冉
Y共t + 1兲 = F共Y共t兲兲 + LB共Y共t兲兲 + ⌳ −
冊
F共X兲 共Y共t兲 − X共t兲兲, X 共12兲
where X = 共xT1 , . . . , xTN兲T, Y = 共y T1 , . . . , y TN兲T 苸 RnN, F共X兲 = 共f 1共x1兲T , . . . , f N共xN兲T兲T, F共Y兲 = 共f 1共y 1兲T , . . . , f N共y N兲T兲T 苸 RnN, ⌳ = diag共H1 , H2 , . . . , HN兲. If A = B, then the error system between Eqs. 共11兲 and 共12兲 reads e共t + 1兲 = 共⌳ + cA 丢 ⌫兲e共t兲,
共13兲
in which e = 共eT1 , eT2 , . . . , eTN兲T 苸 RnN. Compared to the error system 共13兲 with system 共7兲, the former is a system of equations in RnN, and the latter is a matrix equation in Rn⫻N. From error system 共13兲, it is immediately seen that Synchronization Criterion II if 共a兲 A = B and 共b兲 all the eigenvalues of ⌳ + cA 丢 ⌫ lie inside the unit circle, then networks 共11兲 and 共12兲 can be synchronized. Since the error systems 共7兲 and 共13兲 共here provided that all Hi are identical兲 have different expressions, the derived synchronization conditions are different. Whether or not the synchronization condition in Synchronization Criterion I for
+ c 兺 aij⌫x j共t兲,
i = 1,2, . . . ,N,
共14兲
j=1
冉
y i共t + 1兲 = f共y i共t兲兲 + 共1 − ␣兲 H −
冊
f共xi兲 共y i共t兲 − xi共t兲兲 xi
N
+ c 兺 bij⌫y j共t兲,
i = 1,2, . . . ,N,
共15兲
j=1
where ␣ is a parameter. Here we do not further consider synchronization between two bidirectional networks. Here and throughout, we only use Synchronization Criterion I. If networks 共3兲 and 共4兲 关or Eqs. 共11兲 and 共12兲兴 have different connection topologies, i.e., A ⫽ B, then generalized synchronization10 between them often appears, but no any theoretical analysis regarding synchronization is available until now due to technical difficulties 共see the reviews in Refs. 7 and 10兲. We hope that this puzzle can be solved in the near future. Here we will also present numerical studies for the case with A ⫽ B in the following. Next we do some numerical experiments. In the driveresponse networks 共3兲 and 共4兲, the node systems are chosen as the one-dimensional Logistic map and the twodimensional Hénon map. III. ILLUSTRATIVE EXAMPLES
The logistic map x共t + 1兲 = x共t兲共1 − x共t兲兲,
共16兲
has as the adjustable parameter. The Hénon map is as follows: x共t + 1兲 = 1 + y共t兲 − ax2共t兲,
y共t + 1兲 = bx共t兲.
共17兲
If a = 1.4 and b = 0.3, this two-dimensional map has a strange attractor. Here, we always let ⌫ be the unit matrix for convenience. We proceed to our numerical studies for two situations: A = B, and A ⫽ B. A. Equivalent topological structure, i.e., A = B
First, we study the situation A = B. Through the LMI toolbox in MATLAB, we choose 共H , c兲, 共i兲 generate A which has the small-world network property11 共this case is that A is symmetric兲, then let B = A; 共ii兲 generate A at random which makes the network 共3兲 directed 共this case is that A is nonsymmetric兲, then set B = A. These H , c , A make the condition
013106-4
Chaos 19, 013106 共2009兲
Li et al. 1.4
0.12
1.2
−− H=1.1 desynchronization − H=0.9 synchronization
0.1
1 0.08
e
e(t)
0.8 0.06
0.6 0.04
0.4
0.02
0.2
0 0
(a)
20
40
60
80
100
t
(b)
0 1
1.5
2
2.5 H
3
3.5
4
FIG. 2. Synchronization-desynchronization diagram between Eqs. 共3兲 and 共4兲, c = 0.001, = 3.9, ⌫ = 1, A with order N = 50 共p = 0.1, m = 3兲 is generated in case 共i兲, and B = A. 共a兲 Synchronization for H = 0.9, desynchronization for H = 1.1. 共b兲 Bifurcation diagram of e vs H.
of Synchronization Criterion I for networks 共3兲 and 共4兲 be satisfied. In the following, we study cases 共i兲 and 共ii兲. 共i兲 A共=B兲 is symmetric. For the logistic map with = 3.9, choose 共H , c兲 and A such that the condition of the synchronization criterion holds. Once A with order N = 50 共the connection probability and average degree for matrix A are chosen as p = 0.1, m = 3兲 is taken, using the LMI toolbox in MATLAB, we find 共H , c兲 such that the condition of the synchronization criterion is satisfied, e.g., if H = 0, then c 苸 共0 , 0.007兲; if c = 0.001, then −1 ⬍ H ⬍ 1, for this value c, Eqs. 共3兲 and 共4兲 cannot be synchronized if 兩H兩 ⬎ 1. In Fig. 2共a兲, when H = 0.9, networks 共3兲 and 共4兲 can be synchronized but they cannot be synchronized when H = 1.1, where c = 0.001, = 3.9, and 储e共t兲储 = max1艋i艋50兩y i共t兲 − xi共t兲兩. Although desynchronization happens for this case with c = 0.001 and H ⬎ 1 共the case H ⬍ −1 is omitted here兲, the error of each
node is dramatically stabilized to nonzero constant共s兲, denoted as e, when t approaches +⬁, and interesting bifurcations appear. In Fig. 2共b兲, the bifurcation diagram for e in dependence on H is displayed. Furthermore, we investigate the relationships among the network size N, the average degree m of the network are the connection probability p which are used in generating the network, and the maximum coupling strength c available for synchronization. We fix the network size N = 1000 and = 3.9, ⌫ = 1. We find that the maximum coupling strength c decreases with the increase of the average degree m for fixed p, see Fig. 3共a兲. Next, we fix = 3.9, ⌫ = 1, p = 0.1. We find that the maximum coupling strength c decreases with the increase of network size N for the fixed m共艋20兲, see Fig. 3共b兲.
−3
6
−3
x 10
7
5
x 10
N=100
6
p=0.1
N=1000
p=0.9
5
4
c
c
4 3
3 2
2
1
0 0
(a)
1
5
10
15
20
m
25
30
35
40
0 0
(b)
5
10
15
20
m
25
30
35
40
FIG. 3. Relation diagram among the network size N, the average degree m, the connection probability p, and the maximum coupling strength c, where = 3.9, ⌫ = 1, A is generated in case 共i兲, and B = A. 共a兲 The maximum coupling strength c available in dependence on the average degree m of matrix A with N = 1000 for different connection probability p = 0.1 and p = 0.9. 共b兲 The maximum coupling strength c available in dependence on the average degree m of matrix A with order N = 100 and N = 1000 for fixed p = 0.1.
013106-5
Chaos 19, 013106 共2009兲
Coupled discrete-time networks
0.1
0.12
0.09 0.08
0.1
0.08
0.06
e(t)
e(t)
0.07
0.05
0.06
0.04 0.04
0.03 0.02
0.02
0.01 0
(a)
0
10
20
t
30
40
50
0
(b)
0
10
20
30
t
40
50
60
FIG. 4. Synchronization diagrams for case 共iii兲, where A is chosen as that in case 共i兲 and B is chosen as that in case 共ii兲, c = 0.000009. 共a兲 The node system is the logistic map. 共b兲 The node system is the Hénon map.
For the Hénon map 共17兲 with a = 1.4 and b = 0.3, we take the same A used in Fig. 1, B = A, ⌫ = diag共1 , 1兲. By almost the same procedure, we can get 共H , c兲 such that the condition of the synchronization criterion holds. In our numerical studies, if we choose H = diag共0.5, 0.5兲 for brevity, we find when c 苸 共0 , 0.001兲 such that networks 共3兲 and 共4兲 can achieve synchronization. The corresponding figures are omitted here. Next, we study case 共ii兲. 共ii兲 A共=B兲 is asymmetric. We generate A with order N = 50 randomly which makes the network 共3兲 directed 共this case is that A is nonsymmetric兲, and set B = A. In networks 共3兲 and 共4兲, we let ⌫ be the identity matrix and H be a zero matrix. If we choose Eq. 共16兲 with = 3.9 and Eq. 共17兲 with a = 1.4, b = 0.3 as node systems in networks 共3兲 and 共4兲, we find that Eqs. 共3兲 and 共4兲 can be synchronized when the coupling strengths c lie in 共0,0.002兲 and 共0 , 10−4兲, respectively. By comparing cases 共i兲 and 共ii兲, we find that the coupling strength c relates to the connection matrix A for the situation A = B under suitable conditions, i.e., the maximum coupling strength value c concerning synchronization between Eqs. 共3兲 and 共4兲 for symmetric matrix A is larger than that for the nonsymmetric matrix A. The associate figures are also left out here.
B. Different topological structure, i.e., A Å B
Although the synchronization criterion is not derived for the case A ⫽ B, i.e., the drive network 共3兲 and the response one 共4兲 have different connection topologies. However, we can still numerically study the coupled networks 共3兲 and 共4兲. The case A ⫽ B can be divided into four subcases: 共iii兲 A is symmetric but B is not; 共iv兲 B is symmetric but A is not; 共v兲 A , B are both symmetric but they are not equal; 共vi兲 A , B are both nonsymmetric and they are not equal. By numerical simulations, we indeed observe synchronization between Eqs. 共3兲 and 共4兲. If A is chosen as that in case 共i兲 and B is chosen as that in case 共ii兲; B is chosen as that in case 共i兲 and
A is chosen as that in case 共ii兲; A , B are chosen as those in case 共i兲 but they are not equal; A , B are chosen as those in case 共ii兲 but they are not equal. Set still H be a zero matrix for convenience. The networks 共3兲 and 共4兲 with node systems 共16兲 with = 3.9 and Eq. 共17兲 with a = 1.4, b = 0.3 can reach synchronization. For the logistic map with = 3.9, the coupling strength c lies in 共0 , 10−5兲 in 共iii兲, 共0 , 10−5兲 in 共iv兲, 共0 , 10−4兲 in 共v兲 and 共0 , 10−4兲 in 共vi兲. For the Hénon map, the coupling strength c lies in 共0 , 10−5兲 in cases 共iii兲–共vi兲. Here we only present the synchronization diagrams for case 共iii兲, see Fig. 4. From numerical simulations, one can see that the synchronization interval regarding c for networks 共3兲 and 共4兲 in situation A ⫽ B is narrower than that in A = B.
IV. COMMENTS AND CONCLUSIONS
In this paper, outer synchronization between two coupled discrete-time networks is theoretically and numerically studied. If networks 共3兲 and 共4兲, or the more general case 共11兲 and 共12兲, have the same topological connections, i.e., A = B, then a theoretical analysis on synchronization between them is derived. Numerical examples are also given, which fit the analytical results. From our numerical simulations, we can especially see that the maximum coupling strength c has relations to the symmetric properties of matrices A and B. In detail, the maximum coupling strength value c of the case with symmetric matrices A = B is larger than that for the case with nonsymmetric ones A = B. If A ⫽ B, such a value c is the smallest, nearly equal to zero. Besides, the scale-free network12 generally has a smallworld property,11 so we investigate here only two coupled small-world networks by numerical calculations. From our numerical computations available, we can see that the maximum coupling strength value c for synchronization relates also to the network size N, linking the probability p and the average degree m. Through the numerical experiments, we
013106-6
Chaos 19, 013106 共2009兲
Li et al.
find that 共1兲 the maximum coupling strength value c decreases with increasing average degree m for the fixed linking probability p and the networks size N. 共2兲 Such a value c decreases with the increase of the average degree p for the given average degree m and the network size N. 共3兲 This maximum value c decreases with the increase of network size N for the known average degree m and linking probability p. Although outer synchronization between coupled networks, which is different from the intranetwork synchronization, exists in our lives, and such a 共de兲synchronization has potential applications in the real world, and studies on such synchronization between coupled networks are very limited. We hope that the investigation and studies in this regard will appear elsewhere.
vk,1共t + 1兲 = 共H + c␣k⌫兲vk,1共t兲 + ck⌫uk,1共t兲.
Here, we construct the Lyapunov function as the following form: V共t兲 = uTk,1共t兲uk,1共t兲 + vTk,1共t兲vk,1共t兲. Therefore we get ⌬V共t兲 = V共t + 1兲 − V共t兲 =
where Pk =
−
vk,p共t兲 uk,p+1共t兲 vk,p+1共t兲
= 关uk,p
vk,p
H + c ␣ k⌫
− c  k⌫
c  k⌫
H + c ␣ k⌫
v共t兲兴关PTk Pk
u共t兲 u共t兲 − v共t兲 v共t兲
− I兴关u共t兲
H + c ␣ k⌫
− c  k⌫
c  k⌫
H + c ␣ k⌫
册
,
T
u共t兲 v共t兲
v共t兲兴T ,
k = 2, . . . ,l.
共A1兲
1 艋 p 艋 Nk − 1,
vk,p+1共t + 1兲 = 共H + c␣k⌫兲vk,p+1共t兲 + ck⌫uk,p+1共t兲
+ c⌫vk,p共t兲,
1 艋 p 艋 Nk − 1.
Choose the Lyapunov function as ¯V共t兲 = uT 共t兲u 共t兲 + vT 共t兲v 共t兲 + uT 共t兲u k,p k,p k,p+1共t兲 k,p k,p k,p+1 T 共t兲vk,p+1共t兲, + vk,p+1
therefore,
¯ 共t兲 = V共t + 1,u共t + 1兲兲 − V共t,u共t兲兲 ⌬V
uk,p共t兲
H + c ␣ k⌫
+ c⌫uk,p共t兲,
uk,1共t + 1兲 = 共H + c␣k⌫兲uk,1共t兲 − ck⌫vk,1共t兲,
c⌫
0
0
0
0
c⌫
0
0
0
0
H + c ␣ k⌫
− c  k⌫
0
0
c  k⌫
H + c ␣ k⌫
T
c  k⌫
T
uk,p+1共t + 1兲 = 共H + c␣k⌫兲uk,p+1共t兲 − ck⌫vk,p+1共t兲
Now we study the stability of the zero solution, i.e., the synchronized regime, of system 共10a兲. Let k = ␣k + ık, k,1 = uk,1 + ıvk,1, where ı is the imaginary unit, ␣k , k 苸 R , uk,1 , vk,1 苸 Rn. Then the first equation of 共10a兲 can be decomposed into
vk,p+1共t兲
− c  k⌫
If there exists a matrix H 苸 Rn⫻n such that PTk Pk − I are negative definite, then the zero solution of the first system of Eq. 共10a兲 is asymptotically stable. Next, define k,p+1 = uk,p+1 + ıvk,p+1, the second system of Eq. 共10a兲 reads as
APPENDIX: THE PROOF OF STABILITY OF ZERO SOLUTION TO SYSTEM „10a…
冤 冥冤 冤 冥冤 冥
冋
册 册冋 册 冋 册 冋 册
H + c ␣ k⌫
v共t兲
= 关u共t兲
The authors wish to thank two anonymous reviewers for their careful reading and providing beneficial correction suggestions. This work was partially supported by the National Natural Science Foundation under Grant No. 10872119, Shanghai Leading Academic Discipline Project under Grant No. J50101, Key Disciplines of Shanghai Municipality under Grant No. S30104, Systems Biology Research Foundation of Shanghai University, SFB 555 共DFG兲 and EC project BRACCIA Contract No. 517133 NEST.
T
T
u共t兲
⫻
ACKNOWLEDGMENTS
uk,p共t兲 vk,p共t兲 = uk,p+1共t兲
冋 册冋 冋
冥冤 T
c⌫
0
0
0
0
c⌫
0
0
0
0
H + c␣ k⌫
− c  k⌫
0
0
c  k⌫
H + c ␣ k⌫
冥冤 冥 uk,p共t兲 vk,p共t兲 · uk,p+1共t兲
vk,p+1共t兲
uk,p共t兲
vk,p共t兲 uk,p+1共t兲 vk,p+1共t兲
uk,p+1
vk,p+1兴关QTk Qk − I兴关uk,p
vk,p
uk,p+1
vk,p+1兴T ,
in which Qk = diag共c⌫,c⌫, Pk兲,
共A2兲
k = 2, . . . ,l.
The zero solution to the second system of Eq. 共10a兲 is asymptotically stable if
QTk Qk − I
are negative definite.
013106-7
Coupled discrete-time networks
So if PTk Pk − I and QTk Qk − I, k = 2 , . . . , l, are negative definite, then the zero solution to system 共10a兲 is asymptotically stable. The proof is thus complete. 1
S. Boccaletti, V. Latora, Y. Morento, M. Chavez, and D.-U. Hwang, Phys. Rep. 424, 175 共2006兲; Q. Y. Wang, Q. S. Lu, and G. Chen, Europhys. Lett. 77, 10004 共2007兲; J. Zhou, L. Xiang, and Z. Liu, Physica A 385, 729 共2007兲; L. Huang, Y.-C. Lai, and R. A. Gatenby, Phys. Rev. E 77, 016103 共2008兲. 2 V. Berridge, AIDS and Contemporary History 共Cambridge University Press, Cambridge, 2002兲. 3 For mad cow disease, visit the webpage http://www.newstarget.com/ mad_cow_disease.html. 4 M. Small, D. M. Walker, and C. K. Tse, Phys. Rev. Lett. 99, 188702 共2007兲. 5 For SARS in details, explore http://english.peopledaily.com.cn/zhuanti/ Zhuanti_335.shtml. 6 R. Steuer, T. Gross, J. Selbig, and B. Blasius, Proc. Natl. Acad. Sci. U.S.A.
Chaos 19, 013106 共2009兲 103, 11868 共2006兲; W. H. Deng, Y. J. Wu, and C. P. Li, Comput. Math. Appl. 54, 671 共2007兲; W. G. Sun, C. X. Xu, C. P. Li, and J. Q. Fang, Commun. Theor. Phys. 47, 1073 共2007兲; W. G. Sun, Y. Chen, C. P. Li, and J. Q. Fang, ibid. 48, 871 共2007兲; W. G. Sun, Y. Y. Yang, C. P. Li, and J. Q. Fang, “Synchronization in delayed map lattices with scale-free interactions,” Int. J. Non-Linear Mech. 共in press兲. 7 E. Montbrió, J. Kurths, and B. Blasius, Phys. Rev. E 70, 056125 共2004兲; C. P. Li, W. G. Sun, and J. Kurths, ibid. 76, 046204 共2007兲. 8 Y. Li, Z. R. Liu, and J. B. Zhang, Chin. Phys. Lett. 25, 874 共2008兲; H. Tang, L. Chen, J. Lu, and C. K. Tse, Physica A 387, 5623 共2008兲. 9 E. A. Jackson and I. Grosu, Physica D 85, 1 共1995兲; A. I. Lerescu, N. Constandache, S. Oancea, and I. Grosu, Chaos, Solitons Fractals 22, 599 共2004兲; I. Grosu, E. Padmanaban, P. K. Roy, and S. K. Dana, Phys. Rev. Lett. 100, 234102 共2008兲. 10 S. Boccaletti, J. Kurths, G. Osipov, D. L. Vauadares, and C. S. Zhou, Phys. Rep. 366, 1 共2002兲. 11 D. J. Watts and S. H. Strogatz, Nature 共London兲 393, 440 共1998兲. 12 A. L. Barabási and R. Albert, Science 286, 509 共1999兲.
CHAOS 19, 013107 共2009兲
Yet another 3D quadratic autonomous system generating three-wing and four-wing chaotic attractors L. Wanga兲 Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
共Received 15 October 2008; accepted 19 December 2008; published online 5 February 2009兲 This paper introduces a new three-dimensional quadratic autonomous system, which can generate a pair of double-wing chaotic attractors. More importantly, this new system can generate three-wing and four-wing chaotic attractors with very complicated topological structures over a large range of parameters. Several issues, such as some basic dynamical behaviors, bifurcations, and the dynamical structure of the new chaotic system, are investigated either analytically or numerically. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3070648兴 For nearly 40 years, many simple chaotic flows have been found and further studied within the framework of threedimensional (3D) quadratic autonomous systems. There are one-, two-, three-, and four-wing attractors that have been found in the 3D smooth quadratic autonomous chaotic systems. However, how to establish an effective theory frame to generate three-wing chaotic attractors in 3D smooth autonomous quadratic system is still a difficult task. A new 3D smooth autonomous quadratic system is designed in this paper. It is found that this new system can generate three- and four-wing attractors. It is expected that our results will stimulate a great deal of interest in the studies of multiwing chaotic attractors from a unified point of view. It is believed that there are still many chaotic prototype flows to be explored. Therefore, the topic on generating three-wing chaotic attractors from the 3D smooth autonomous quadratic system deserves further detailed investigation. I. INTRODUCTION
It came as a big surprise to most scientists when Loren discovered chaos in a simple system of smooth threedimensional quadratic autonomous ordinary differential equations in 1963.1 The celebrated Lorenz system is a thirdorder autonomous system with only two quadratic nonlinearities. However, the Lorenz system can display very complex dynamical behaviors,2 especially the well-known twoscroll butterfly chaotic attractor. In 1999,3,4 another similar but topologically nonequivalent chaotic system was found by Chen. The Chen system can be considered to be dual to the Lorenz system.5 In 2002, Lü and Chen6 found an intermediate chaotic system, called the Lü system, and the system represents the transition between the Lorenz system and the Chen system. Soon thereafter, a very large and rich family that so-called Lorenz system family7 was constructed, showing the convex combination of the Lorenz, Chen, and Lü systems. Recently, Telephone: ⫹86 27 87543837. Fax: ⫹86 27 87543238. Electronic mail: wanglinfl
[email protected].
a兲
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Li and Yin5 found another new system, which connects the Lorenz and Chen systems via nonlinear control. It is not surprising, therefore, the papers on the Lorenz system family is extensive and constantly expanding. Various Lorenz-type systems 共e.g., Refs. 8–12兲 were constructed and analyzed, showing some interesting dynamical behaviors. According to the simple classification of 3D autonomous chaotic systems with smooth quadratic terms presented by Lü et al.,12 that is, from the viewpoint of the chaotic attractor solutions to the dynamical systems, one can clearly see the fact: in the past decades, among the multiscroll chaotic attractors generated in 3D autonomous systems by 共non兲parametric polynomial transformation,13,14 there are one-wing attractors, two-wing attractors, and four-wing attractors12 that have been found in the 3D smooth quadratic autonomous chaotic systems. Very recently, a 3D smooth autonomous quadratic chaotic system with three equilibria was designed by Li,15 which can produce a three-wing 共scroll兲 attractor. More recently, Wang16 designed a new chaotic system with five equilibria. This new chaotic system, again, can display a three-wing attractor. Although, few 3D smooth autonomous quadratic chaotic systems have been found to display three-wing attractors 共see, e.g., Refs. 15 and 16兲, how to establish an effective theory frame to generate three-wing 共or three-scroll兲 chaotic attractors in 3D smooth autonomous quadratic system is still a difficult task. Therefore, the topic on generating three-wing chaotic attractors from the 3D smooth autonomous quadratic system deserves further detailed investigation. This paper introduces another new 3D smooth autonomous system with five equilibria, in which each equation contains a single quadratic term. The system can generate a pair of double-wing chaotic attractors. It is amazing to find that four-wing and three-wing chaotic attractors can be generated in some way. Basic properties of the new system are analyzed, including, such as, stability of equilibria and bifurcations. Finally, the dynamical structure of this new system is discussed.
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FIG. 1. 共Color online兲 Phase portraits of the upper-attractor and the lower-attractor.
II. THE NEW 3D CHAOTIC SYSTEM
First consider the following 3D quadratic autonomous system, designed by Lü et al.,12 which can display chaotic attractors, x˙ = −
ab x − yz, a+b
y˙ = ay + xz,
共1兲
z˙ = bz + xy, where a and b are real constants. As discussed in Ref. 12, if ab ⬎ 0 then system 共1兲 has five equilibria. It has demonstrated that this system can display a pair of double-wing chaotic attractors. For more details, the reader is referred to Ref. 12. However, this system cannot generate four-wing chaotic attractors as well as threewing chaotic attractors. In this paper, by adding a simple linear state feedback term to system 共1兲, we obtain another new 3D smooth quadratic autonomous system described by
A. Some basic properties
System 共2兲 does not share the main properties with some known 3D smooth quadratic autonomous systems, such as the Lorenz system. These are listed in the following subsections. 1. Symmetry and invariance
Note the invariance of the system under the transform 共x , y , z兲 → 共−x , y , −z兲. That is, system 共2兲 is symmetrical about the coordinate axes y and is not symmetrical about the coordinate axes x or z. 2. Dissipativity and existence of attractor
For system 共2兲, it is easily found that ⵜV =
共a + 21 b兲2 + 43 b2 ab x˙ y˙ z˙ + + =− +a+b= . x y z a+b a+b 共3兲
Thus, when a + b ⬍ 0, system 共2兲 is dissipative, with an exponential contraction rate, dV a2 + ab + b2 = V. dt a+b
ab x − yz, x˙ = − a+b y˙ = ay + xz,
III. DYNAMICAL BEHAVIORS OF THE NEW CHAOTIC SYSTEM
共2兲
z˙ = bz + cx + xy in which, similarly, a, b, and c are real constants. Solutions of Eqs. 共2兲 were obtained by using a fourthorder Runge–Kutta integration algorithm, with a step size of 0.001; unless otherwise stated, the initial conditions employed were 共2.2, 2.8, 8兲. This new system is found to be chaotic in a wide parameter range and has many interesting complex dynamical behaviors. For example, it is chaotic for the parameters a = −7.87, b = −4, c = 0; in this case, system 共2兲 displays two double-wing chaotic attractors as shown in Fig. 1. It should be noted that the initial conditions employed were 共2.2, 2.8, −8兲 for the lower attractor.
共4兲
That is, a volume element V0 is contracted by the flow into a 2 2 volume element V0e共a +ab+b 兲t/共a+b兲 in time t. This also implies that each volume containing the system trajectory shrinks to zero as t → ⬁ at an exponential rate, a2 + ab + b2 / 共a + b兲. Therefore, all system orbits are ultimately confined to a specific subset of zero volume, and the asymptotic motion settles onto an attractor. B. Equilibria and bifurcations
It is noticed that if ab ⬎ 0, then system 共2兲 has five equilibria, S1共0,0,0兲, S2
冉
冊
b共cA1/2 + B1/2兲 共cA1/2 + B1/2兲 ab ,− , 1/2 , 2 2A1/2 A
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冊
b共cA1/2 − B1/2兲 共cA1/2 − B1/2兲 ab ,− , 1/2 , 2 2A1/2 A
S4 −
冉
b共cA1/2 − B1/2兲 共cA1/2 − B1/2兲 ab ,− ,− 1/2 , 1/2 2 2A A
冉
b共cA1/2 + B1/2兲 共cA1/2 + B1/2兲 ab ,− ,− 1/2 , 2 2A1/2 A
S5 −
冊 冊
where A = 共a + b兲b and B = abc2 + c2b2 + 4ab3. However, if ab 艋 0, then system 共2兲 has a unique equilibrium, S1共0 , 0 , 0兲. Moreover, S3 and S4, S2 and S5 are symmetric with respect to the y-axis. Obviously, the origin is always a saddle point in the 3D space for any nonzero real numbers a, b, and c. About the null solution S1, linearization of system 共2兲 gives three eigenvalues: 1 = −ab / 共a + b兲, 2 = a, and 3 = b. About the nonzero equilibria S2 , S3 , S4 , S5 linearization of system 共2兲 yields the following same characteristic equation: f共兲 = 3 −
a2 + ab + b2 2 a2b2 − = 0. a+b a+b
共5兲
For the above characteristic equation, it has been verified that Hopf bifurcation will not occur at the five equilibria. The interested reader is referred to Ref. 12 for more details on the proof of this conclusion. C. Dynamical structure of the new chaotic attractors
As mentioned in the above, if c = 0, then system 共2兲 is reduced to system 共1兲, designed by Lü et al.12 In this case, it is also noted that system 共2兲 can display a pair of doublewing chaotic attractors, as shown in Fig. 1. In what follows, it is of interest to investigate the dynamical structure of the new system 共2兲 for various values of c. For that purpose, extensive calculations have produced the bifurcation diagrams of Fig. 2, for signals of x, y, and z, with initial conditions 共2.2, 2.8, 8兲 and a = −7.87, b = −4. In Fig. 2, it can be seen that the variable parameter is c, varied from −4 to 4. Generally, there exist two divisions in the parameter region of c, that is, periodic and chaotic regions. It can be observed that the system is periodic in the regions of 3.17 ⬍ 兩c兩 ⬍ 4 and 0.44⬍ 兩c兩 ⬍ 0.63. Therefore, in the regions of 3.17艌 兩c兩 艌 0.63 and 0.44艌 兩c兩 艌 0, the system undergoes a chaotic regime. However, there are relatively small subregions of periodic solutions embedded within the chaotic region, e.g., for 1.34⬍ 兩c兩 ⬍ 1.36 there is what appears to be a periodic subregion. For large values of 兩c兩, numerical calculations have shown that the trajectory will gravitate towards to a stable fixed point. It is instructive to look at phase portraits associated with various values of c, corresponding to different dynamical behavior as discussed in the foregoing. Sampling results are shown in Figs. 3 and 4. In Fig. 3, it is seen that the trajectories are periodic. Furthermore, in the region 0.44⬍ 兩c兩 ⬍ 0.63, it can be observed that this is a region with perioddoubling bifurcations. In the region −0.63⬍ c ⬍ −0.44 the
FIG. 2. Bifurcation diagrams of system 共2兲 for 共a兲 signals x, 共b兲 signals y, and 共c兲 signals z, for various c.
period-doubling bifurcations for the lower-wing will appear, as displayed in Figs. 4共a兲 and 4共b兲; in the region 0.44⬍ c ⬍ 0.63, however, the period-doubling bifurcations for the upper-wing will appear, as shown in Figs. 4共c兲 and 4共d兲. It ought to be stressed that the effect of initial conditions on the period-doubling bifurcations is not visible in the calculations. If the initial conditions were chosen to be 共2.2, 2.8, −8兲, that is, z0 ⬍ 0, again, the same period-doubling bifurca-
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FIG. 3. 共Color online兲 Phase portraits of periodic orbits of system 共2兲 with c = −3.7.
FIG. 4. 共Color online兲 Phase portraits of periodic orbits of system 共2兲 for 共a兲 c = −0.45, 共b兲 c = −0.448, 共c兲 c = 0.45, and 共d兲 c = 0.448.
FIG. 5. The four-wing chaotic attractor of system 共2兲 for c = −0.25.
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FIG. 6. The four-wing chaotic attractor of system 共2兲 for c = −2.8, a = −6, and b = −4.
tions will occur. This means that in the region 0.44⬍ c ⬍ 0.63 the period-doubling bifurcations for the lower-wing will appear and in the region −0.63⬍ c ⬍ −0.44 the perioddoubling bifurcations for the upper-wing will appear. As mentioned in Ref. 12, when c = 0 system 共2兲 has five equilibria: S1 , S2 , S3 , S4 , S5, in which S2 , S3 are above the plane z = 0 while S4 , S5 are below this plane. Furthermore, there is a close correlation between the equilibria S1 , S2 , S3 and the upper attractor. Also, there is a close correlation between the equilibria S1 , S4 , S5 and the lower attractor. It is therefore interesting to ask if the linear feedback term 关the added term, cx, in the third equation of system 共2兲兴 can connect the upper-attractor and lower-attractor when c ⫽ 0. More extensive calculations give a positive answer to this question.
In fact, a linear feedback term works well and can connect the upper-attractor and lower-attractor to form a four-wing chaotic attractor. Figure 5 displays a four-wing chaotic attractor, for c = −0.25. It can be clearly seen that the upper attractor and the lower attractor are connected. It is recalled that in the bifurcation diagram of Fig. 2共c兲, the signals of z共t兲 are both positive and negative in a large region of c. In this parameter region, the trajectory may run around the upper equilibria 共S2 and S3兲 as well as the lower equilibria 共S4 and S5兲, thus yielding the four-wing chaotic attractors. Hence, system 共2兲 has a different dynamical structure if compared with system 共1兲.
FIG. 7. The three-wing chaotic attractor of system 共2兲 for c = −2.8. 共a兲 3D phase portrait, 共b兲 x-y plane projection, 共c兲 x-z phase projection, and 共d兲 y-z plane projection.
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(a)
(b) FIG. 8. Poincaré maps on the Poincaré section ⌺ with two different values of c.
In fact, if a or b is the only varied parameter and the values of two other parameters are fixed, system 共2兲 also can display four-wing attractors. For example, if we choose a = −6, b = −4, and c = −2.8, a four-wing chaotic attractor can be detected, as shown in Fig. 6. Thus, there are two-wing attractors and four-wing attractors that have been found in system 共2兲. Now, a natural question is whether system 共2兲 can display a three-wing chaotic attractor? This study will give a positive answer to this interesting question. From Fig. 7, one can clearly see that the 3D quadratic autonomous chaotic system 共2兲 can exhibit an attractor with three wings for a = −7.87, b = −4, and c = −2.8. To distinguish a chaotic response from a regular one, the Poincaré mapping technique, as is well known, is proved to be very informative. A Poincaré section is often used to reduce a higher continuous system to a discrete map of lower dimension. The strength behind this tool is that these sections have the same topological properties as their continuous counterparts. For drawing the Poincaré maps we use an appropriate Poincaré section 兺 defined by
兺 = 兵共y,z兲 苸 R2,x = 0其. Figures 8共a兲 and 8共b兲 show the Poincaré mapping on the 兺 section with two values of c in the chaotic bands. It is observed that the Poincaré maps can be separated into several sheets in Figs. 8共a兲 and 8共b兲. In Fig. 8共a兲, the Poincaré map represents a three-wing attractor defined in Fig. 7. It is obviously seen that the Poincaré map consists of an infinite number of points along three dashed lines 共or three sheets兲. In Fig. 8共b兲, the Poincaré map represents a four-wing attractor defined in Fig. 5. In this case, however, four sheets 共dashed lines兲 are clearly visible. The Poincaré map consists of infinite number of points along these four dashed lines. Thus, for a three-wing attractor, three sheets appear in the Poincaré map; for a four-wing attractor, four sheets appear in the Poincaré map. According to the results obtained above, the new chaotic attractors have two wings, three wings or four wings. Notably, system 共2兲 has a more complex topological structure than the three-scroll attractor generated from the Li system,15 since system 共2兲 has five equilibria while the Li system has only three equilibria.
IV. CONCLUSIONS
This paper has presented and studied a new chaotic system of 3D quadratic smooth autonomous equations, which can generate two double-wing chaotic attractors simultaneously with five equilibria. Dynamical behaviors of this new chaotic system, including some basic dynamical properties, bifurcations, periodic region, and dynamical structure of the new attractors have been investigated both theoretically and numerically. Of particular interest is the fact that this chaotic system can generate a complex four-wing chaotic attractor as well as a three-wing chaotic attractor. The obtained results clearly show this is a new chaotic system and deserves further detailed investigation on its topological structure in the near future. ACKNOWLEDGMENTS
This work is supported by the Scientific Research Foundation of Huazhong University of Science and Technology through Grant No. 2006Q003B. The author thanks the anonymous reviewers for their helpful suggestions. E. N. Lorenz, J. Atmos. Sci. 20, 130 共1963兲. C. Sparrow, The Lorenz Equations: Bifurcations; Chaos; and Strange Attractors 共Springer-Verlag, New York, 1982兲. 3 G. Chen and T. Ueta, Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, 1465 共1999兲. 4 T. Ueta and G. Chen, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 1917 共2000兲. 5 D. Li and Z. Yin, Commun. Nonlinear Sci. Numer. Simul. 14, 655 共2009兲. 6 J. Lü and G. Chen, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 659 共2002兲. 7 J. Lü, G. Chen, D. Cheng, and S. Čelikovský, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 2917 共2002兲. 8 C. Liu, T. Liu, L. Liu, and K. Liu, Chaos, Solitons Fractals 22, 1031 共2004兲. 9 J. Lü, G. Chen, and S. Zhang, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 1001 共2002兲. 10 L. Wang, Q. Ni, P. Liu, and Y. Huang, J. Dyn. Control Syst. 3, 1 共2005兲 共in Chinese兲. 11 J. Lü, G. Chen, and S. Zhang, Chaos, Solitons Fractals 14, 669 共2002兲. 12 J. Lü, G. Chen, and D. Cheng, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1507 共2004兲. 13 R. Miranda and E. Stone, Phys. Lett. A 178, 105 共1993兲. 14 S. Yu, J. Lü, K. Wallace, S. Tangb, and G. Chen, Chaos 16, 033126 共2006兲. 15 D. Li, Phys. Lett. A 372, 387 共2008兲. 16 L. Wang, “3-scroll and 4-scroll chaotic attractors generated from a new 3D quadratic autonomous system,” Nonlinear Dyn. 共in press兲. 1 2
CHAOS 19, 013108 共2009兲
State and parameter estimation of spatiotemporally chaotic systems illustrated by an application to Rayleigh–Bénard convection Matthew Cornick,1 Brian Hunt,1 Edward Ott,1 Huseyin Kurtuldu,2 and Michael F. Schatz2 1
University of Maryland, College Park, Maryland 20742, USA Center for Nonlinear Science and School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
2
共Received 2 November 2008; accepted 29 December 2008; published online 5 February 2009兲 Data assimilation refers to the process of estimating a system’s state from a time series of measurements 共which may be noisy or incomplete兲 in conjunction with a model for the system’s time evolution. Here we demonstrate the applicability of a recently developed data assimilation method, the local ensemble transform Kalman filter, to nonlinear, high-dimensional, spatiotemporally chaotic flows in Rayleigh–Bénard convection experiments. Using this technique we are able to extract the full temperature and velocity fields from a time series of shadowgraph measurements. In addition, we describe extensions of the algorithm for estimating model parameters. Our results suggest the potential usefulness of our data assimilation technique to a broad class of experimental situations exhibiting spatiotemporal chaos. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3072780兴 It is often desirable to predict the future state of a chaotic system, i.e., to forecast the system. Before one can hope to estimate the future state, one must know the current state. For many systems this is not always possible, either because measurements are too noisy, or because not all system variables can be measured. This is especially true in spatiotemporally chaotic systems (chaotic systems which are spatially extended) in which it may not be possible to achieve a measurement density high enough to reconstruct the state with sufficient accuracy. For example, in weather forecasting, measurements occur at weather stations which, in remote locations, are sparsely distributed. Traditional algorithms such as the extended Kalman filter “assimilate” current and previous measurements, using a model for the system dynamics, to estimate the current system state. However, these traditional algorithms do not scale well to high-dimensional systems with many degrees of freedom, a hallmark of spatiotemporal chaos. Recent developments in the field of numerical weather prediction have demonstrated algorithms capable of handling high-dimensional systems. Although originally developed for weather prediction, these algorithms can be applied to any spatiotemporally chaotic system. Here we present the first successful application of a recently developed data assimilation algorithm to a spatiotemporally chaotic laboratory experiment. We have chosen a commonly studied Rayleigh–Bénard convection laboratory experiment exhibiting a form of spatiotemporal chaos known as spiral defect chaos. I. INTRODUCTION
Numerous systems exhibit spatiotemporal chaos. Examples of this complex behavior with many dynamical degrees of freedom occur in optics,1 chemical and biological media,2 and hydrodynamics,3,4 including geophysical flows in the ocean and atmosphere. Estimation of the state of an 1054-1500/2009/19共1兲/013108/10/$25.00
evolving dynamical system from measurements is often a prerequisite for prediction and control. However, obtaining the system state is a common experimental difficulty for many systems exhibiting spatiotemporal chaos, where available measurements may be incomplete and noisy. When an approximate model for the system is available, it can be used in conjunction with incoming measurements to estimate the evolving system state, a process referred to as “data assimilation.” The Kalman filter5,6 optimally solves the data assimilation problem for systems with linear dynamics 共and Gaussian measurement noise兲. Several methods extending the Kalman filter methodology to nonlinear systems have been proposed, including the extended Kalman filter 共EKF兲,7 and the class of ensemble Kalman filters 共EnKF兲.8 Straightforward application of these methods to large spatiotemporally chaotic systems is often completely infeasible. In particular, the EKF requires inversion of N ⫻ N matrices, where N is the number of model variables.9 For spatiotemporally chaotic systems, N can be very large 共e.g., in the millions兲 making such matrix inversions impossible in practice. Despite these difficulties, recent developments10–12 from the field of numerical weather prediction13–18 suggest the possibility of achieving good accuracy 共as in a Kalman filter兲, but in a way that is computationally feasible for large spatiotemporally chaotic systems. In this paper we test the efficacy of a new method, the local ensemble transform Kalman filter 共LETKF兲.12 Although originally motivated by application to weather prediction, the LETKF is potentially broadly applicable to any large spatiotemporally chaotic system. It is motivated by two observations: 共i兲 When the state is examined in a local region that is small compared to the system size, it has been shown that it can be accurately described by a relatively few degrees of freedom;19 and 共ii兲 many spatiotemporally chaotic systems exhibit finite space/time correlation scales. Thus, by 共ii兲 we
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expect that the system state at the space/time coordinate 共x , t兲 is significantly correlated only in locations x⬘ at a previous time t − ⌬t that lie within some region, say 兩x − x⬘兩 ⱗ l⌬t 共where l⌬t might be expected to increase with ⌬t兲. With these points in mind, the LETKF uses a process that we refer to as localization. By this we mean that we employ many independent data assimilations in a set of overlapping local regions. We choose the size of these regions empirically, increasing them until our results are no longer further improved. 共This procedure may be thought of as an operational means of estimating the effective average value of l⌬t.兲 Because these regions are relatively small, individual computations associated with them are not prohibitive. Furthermore, by use of a simple example10,11 it was indicated that, by exploiting localization in this way, state estimates with accuracies virtually the same as those for a classical Kalman filter technique 共thus presumably of near optimal accuracy兲 can be achieved 共here we use the specific implementation described in Ref. 12兲. The purpose of this paper is to test the localization methodology on a system which represents a realizable laboratory experiment. Assimilation schemes have been used on laboratory experiments before,20 but never using a localization technique. We have chosen to investigate a common laboratory experiment that exhibits an especially high dimensionality, Rayleigh–Bénard convection. Flows such as spiral defect chaos3,21 in the Rayleigh–Bénard problem are, perhaps, the best studied experimental examples of spatiotemporal chaos; nevertheless, many general aspects of spatiotemporal chaos remain poorly understood. What follows is an introduction to Rayleigh–Bénard convection 共Sec. II兲 and data assimilation 共Sec. III兲, followed by tests of the accuracy of the LETKF 共Secs. IV and V兲. We also investigate performance with extremely sparse/noisy measurements and test extensions of the LETKF for estimating model parameters. Details of the LETKF algorithm are described in the Appendix. II. RAYLEIGH–BÉNARD CONVECTION
In Rayleigh–Bénard convection, a horizontal fluid layer of thickness d is confined between a heated lower plate and a cooled upper plate. For a temperature difference ⌬T between the plates that is sufficiently small, the fluid is at rest and heat is transported by conduction 共resulting in a temperature T which varies linearly with vertical distance兲. As ⌬T is raised above a critical value ⌬Tc there is an onset of fluid motion when buoyancy overcomes viscous dissipation and thermal diffusion. Rayleigh–Bénard convection is typically modeled using the Boussinesq equations,22 which are commonly nondimensionalized with temperature scaled by ⌬T, length scaled by d, and time scaled by the vertical diffusion time tv = d2 / , where is the thermal diffusivity. This system of units is used throughout the paper. The temperature deviation from the conducting static solution is denoted as . We solve the Boussinesq equations in the disk shaped region x2 + y 2 艋 ⌫2, 兩z 兩 艋 21 , with Dirichlet boundary conditions u = 0, = 0 on all walls. ⌫ is called the aspect ratio and denotes the radius of
the disk in units of d. In terms the fluid’s velocity u, temperature deviation , and pressure p, the Boussinesq equations take the form
冉
+ u · ⵜ u = − ⵜp + Pr ⵜ2u + Pr Rzˆ , t
冊
冉
+ u · ⵜ = ⵜ2 + u · zˆ , t
冊
共1兲
ⵜ · u = 0. These equations have two dimensionless parameters, the Rayleigh number R and the Prandtl number Pr, R=
g␣d3⌬T ,
Pr =
.
共2兲
Here ␣ is the thermal expansion coefficient, is the kinematic viscosity, and g is gravitational acceleration. The critical Rayleigh number for convective onset is Rc ⬇ 1707. The reduced Rayleigh number
⑀=
R − Rc ⌬T − ⌬Tc = Rc ⌬Tc
共3兲
measures the amount above onset. Fluid convection arises when ⑀ ⬎ 0. It is important to note that the Boussinesq equations are an approximation to the full Navier–Stokes equations. This approximation assumes small deviations of the density from its average value and neglects any temperature dependence of the transport coefficients. While the Boussinesq approximation is fairly good for the situation we will apply it to, it can be expected that it does lead to some model error. We have investigated the parameter region near ⑀ = 1, Pr= 1. At these values of ⑀ and Pr, the spatiotemporally chaotic state known as spiral defect chaos can arise;3,21 however, in our studies using ⌫ ⬇ 20, the region is too small to support the large spirals typically seen in spiral defect chaos. Nevertheless, the convective flows in our studies exhibit complex behavior in both space and time 共see Fig. 6 below, for an example of the spatial structure of the evolving state兲. In experiments, Rayleigh–Bénard flows are visualized using the shadowgraph method,23 an indirect measurement of the fluid’s spatial temperature variation. Time series of twodimensional shadowgraph images are typically collected with sampling periods ⬍1tv and with high spatial resolution 共⬃105 – 106 pixels per image兲. Due to its difficulty, measurement of the fluid velocity is not performed in typical experiments. We note, however, that the so-called mean flow has been shown, through the use of simulations, to play a significant role in the dynamics.24 Here we define the mean flow as ¯u共x , y兲 ⬅ 兰u⬜共x , y , z兲dz, where u = u⬜ + uzzˆ . Because of its physical importance, it would be desirable to be able to estimate the mean flow field ¯u. We connect the shadowgraph light intensity I共x , y兲 to the temperature field using the following relation:
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I共x,y兲 =
Iⴰ共x,y兲 . 1 − aⵜ2 ¯共x,y兲
共4兲
⬜
Equation 共4兲 is derived from geometric optics23,25 under the approximation that 兩aⵜ2⬜¯兩 Ⰶ 1. In Eq. 共4兲, ⵜ2⬜ ⬅ 2 / x2 + 2 / y 2 is the horizontal Laplacian, and the temperature field is vertically averaged: ¯共x , y兲 ⬅ 兰共x , y , z兲dz. Iⴰ共x , y兲 is the incident light intensity and a = 2z1兩dn / dT兩, where n is the index of refraction of the fluid, z1 is the optical path length from the midplane of the fluid layer to the image plane 共in units of d兲, and the temperature coefficient of the index of refraction 兩dn / dT兩 is evaluated at the average temperature of the fluid layer. We have checked the validity of the geometric optics approximation by computing I共x , y兲 for the simple state of straight convection rolls using both geometric optics and physical optics26 for our setup under the conditions of our experiment. Under these conditions, we find the geometric optics approximation yields results that are in good agreement with the more exact results from physical optics. III. DATA ASSIMILATION A. Outline of method
Our goal is to determine the full fluid state, given by the temperature and velocity fields 关共x , y , z兲 and u共x , y , z兲兴 from a time series of shadowgraph measurements, and we view this as a test case investigation of the general usefulness of the LETKF technique for laboratory experiments on spatiotemporal chaos. Moreover, we place particular emphasis on the ability of the state estimate to produce accurate forecasts. We begin by considering a system state vector with N components, for which we have a dynamical model, j+1 = G共 j兲. Here, G is an integration of the Boussinesq equations 共1兲 from a time t j to t j+1 = t j + ⌬t, where the t j are the times at which we wish to construct an estimate of the system state 共also the times at which measurements are assumed to be made兲. Our Boussinesq integration is performed using the pseudospectral method described in Ref. 27 and the state consists of the variables and u defined on the grid points 共rm , n , zl兲 of a cylindrical mesh;28 symbolically,
=
冋册
. u
Most data assimilation algorithms are iterative, cycling between a predict and update step once every time interval ⌬t. In the update step, current measurements are used to update 共or correct兲 the prediction. The prediction step then propagates the updated state, via the model, to the next measurement time 共i.e., it is a short term forecast兲. The aim of this process is to synchronize the experiment and the model by coupling them via the measurements. The LETKF assimilation method is based on the ensemble Kalman filter 共EnKF兲, in which the update and predict steps take place for an ensemble of k system states.8 This ensemble gives a finite sampling approximate representation of the probability distribution function 共PDF兲 of the system
state. The updated ensemble 兵u,1 . . . u,k其 results from an update of the predicted ensemble 兵 p,1 . . . p,k其, update step: p,k u,1 u,k 兵 p,1 j . . . j 其 + 兵measurements其 → 兵 j . . . j 其
u,i predict step: p,i j+1 = G共 j 兲
i = 1 . . . k.
共5兲 共6兲
The details of the update step are specific to the type of EnKF used, but in all cases it is based on the original Kalman filter equations. This iterative procedure begins with an initial predicted ensemble 兵0p,1 . . . 0p,k其 consisting of states randomly sampled from the system attractor. The maximum likelihood estimate of the system’s state after an update step is the center of the updated ensemble, ¯u = 1 / k兺ii,u. When no localization is used, as the system size grows and the dynamical degrees of freedom increase, the necessary number of ensemble members k must increase so as to span the space of possible system states. This is a major drawback of ensemble methods, preventing their use for spatiotemporal chaos in large domains which would require an infeasibly large k. For example, in our numerical experiments we found for the Rayleigh–Bénard problem 共with ⌫ = 20兲 that, using the EnKF, it was not computationally feasible to use large enough ensembles to obtain results of any use.29 The LETKF method, which localizes the update step, is advantageous since the number of ensemble members required is independent of the system size, making the method applicable to large domains.10,11 An explanation of the LETKF’s update step 共5兲, including the method of localization, is given in the Appendix.
B. Measurements
At times t j共j = 1 , 2 . . . 兲 we assume that several scalar measurements are taken, so that at each time we can represent the set of measurements by an s-component vector y. In the context of Rayleigh–Bénard convection, the elements of the vector y are the intensities of shadowgraph pixels, y = 关I共x1 , y 1兲I共x2 , y 2兲 . . . I共xs , y s兲兴T, where I共xl , y l兲 is the light intensity at the location 共xl , y l兲 of pixel l. Note that the location of these intensity measurements need not occur on a uniform mesh; we assume that their location is fixed but arbitrary. Measurement noise is assumed to be normally distributed with mean zero and 共s ⫻ s兲 covariance matrix R. We assume for simplicity that R = 2I, i.e., a multiple of the identity matrix, so that measurement noise is homogeneous and uncorrelated with a standard deviation of . In general y is ideally 共i.e., without noise兲 a function of the system state, y = H共兲; H is known as the observation operator. H共兲 outputs the vector of pixel intensities y using a finite resolution approximation to Eq. 共4兲, where for ⵜ2⬜ we use a finite difference on the cylindrical mesh. Note that, since we require 兩aⵜ2⬜¯兩 Ⰶ 1 for Eq. 共4兲 to be a good approximation, H is only weakly nonlinear. Both R and H are critical components for the update step 共5兲 of the ensemble-based methods.
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C. Parameter estimation
There is a straightforward extension of the ensemble methods for cases in which some model parameters are unknown. Consider the model
j+1 = G共 j,p兲,
共7兲
where p is a vector of model parameters. We now formally extend the system state to include the model parameters, ␥ = 关 p 兴, where the extended state evolves as
冋 册冋
册
G共 j,p j兲 j+1 ˆ 共␥ 兲. ␥ j+1 = = =G j p j+1 pj
IV. RESULTS: PERFECT MODEL
共8兲
Estimates of ␥ 共and therefore of the parameters p兲 result from an implementation in the same way as for , but in the space of ␥ vectors. In general, observation operator parameters may also be estimated in exactly the same way as ˆ 共␥兲. model parameters, by replacing H共兲 by H共 , p兲 ⬅ H Here p is a concatenation of model and observation operator parameters. D. Direct insertion
In order to assess how well the LETKF method is performing, we will compare it to a more naive approach that we call direct insertion 共DI兲. With shadowgraph measurements, no state variables are measured directly; however, there is a one to one correspondence between a shadowgraph and the vertically averaged field ¯共x , y兲. With this in mind, the DI update step adjusts the 共t = t j兲 predicted vertically averaged temperature field ¯ pj 共x , y兲 to reflect the current measurement exactly. At the time t j of the shadowgraph measurement I j共x , y兲, the DI method updates the predicted temperature field pj 共x , y , z兲 by adding a correction ␦ j共x , y , z兲 which is the unique field that is quadratic in z, matches the boundary conditions at 兩z兩 = 21 , and for which the updated field uj 共x , y , z兲 = pj 共x , y , z兲 + ␦ j共x , y , z兲 satisfies I j共x,y兲 =
The z-dependence of the predicted temperature field is only slightly affected by the update since, if measurements are sufficiently frequent, the correction ␦ j共x , y , z兲 is small. This method is the most successful data assimilation technique we have tested that does not use an update based on the Kalman filter. It is meant to represent what one might try when measurements are sufficiently dense and frequent, in which case DI is a reasonable alternative to more sophisticated data assimilation techniques.
Iⴰ共x,y兲 . 1 − aⵜ2 ¯u共x,y兲 ⬜ j
A. Setup of the numerical experiments
In this section we describe so-called perfect model tests in which a time series of states, generated from a Boussinesq simulation 共⌫ = 20, ⑀ = 1, Pr= 1兲 of one particular initial condition, serves as the “true” system. Simulated shadowgraph measurements of this time series are generated every ⌬t = 1 / 4 by using Eq. 共4兲 with the parameters a = 0.08, Iⴰ共x , y兲 = 0.5. By this technique we generate a situation in which the “true state” to be estimated and the model used to estimate it both evolve under exactly the same dynamical rules. In Sec. V we use real 共not simulated兲 observations of a physical system for which the model dynamics is surely not an exact description. To reproduce the effects of measurement noise we add to each pixel a small random error that is an uncorrelated normally distributed number with mean zero and standard deviation . Measurements are made sparse by removing shadowgraph pixels, leaving only those which lie on observation locations. We introduce the measurement density ⬅ s / 共⌫2兲 as a measure of sparseness, where s is the number of observation locations. For 艌 4 we randomly and uniformly distribute observation locations over the disk, for ⬍ 4 the observation locations are placed on a Cartesian grid covering the disk 共giving more repeatable results when using sparse measurements兲. We apply the LETKF and DI methods to our simulated shadowgraphs to approximately reconstruct the original time series of true states. Here we document their performance as a function of measurement noise and measurement density . Performance is quantified via the temperature and mean flow RMS relative error,
This gives the update E共t兲 =
␦ j共x,y,z兲 = 共¯uj 共x,y兲 − ¯ pj 共x,y兲兲共 23 − 6z2兲 , where ¯uj 共x , y兲 is found by solving a Poisson equation, ⵜ2¯uj 共x,y兲 =
冋
册
1 Iⴰ共xc,y c兲 1− , a I j共xc,y c兲
E¯u共t兲 = 共9兲
and 共xc , y c兲 is the location of the closest pixel to 共x , y兲 that is observed. Note that with DI the velocity is not updated, uuj 共x , y , z兲 = u pj 共x , y , z兲, rather it develops through coupling with the temperature during the simulation step,
冋
pj+1共x,y,z兲 u pj+1共x,y,z兲
册 冉冋 =G
uj 共x,y,z兲 uuj 共x,y,z兲
册冊
.
冑
具兩共x,y,z,t兲 − t共x,y,z,t兲兩2典 , 具兩t共x,y,z,t兲兩2典
冑
¯ 共x,y,t兲 − ¯ut共x,y,t兲兩2典 具兩u , ¯ t共x,y,t兲兩2典 具兩u
where t共x , y , z , t兲 and ¯ut共x , y , t兲 are the temperature and mean flow from the “true” time series of states, and 具·典 indicates a spatial average. Simulated shadowgraphs are assimilated at times t j, j = 1 . . . J. During this process we converge on an estimate of the system state 共J chosen large enough to ensure convergence兲. At time tJ assimilation is turned off and the final updated state estimate is used as an initial condition for a long term forecast. Three measures of the quality of a state
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FIG. 1. Typical temperature error E共t兲 of forecasts with = 0.01 and = 127. The inset shows E共t兲 as each method converges on a state. Assimilation is turned off at time tJ = 3.25 in the small graph, corresponding to time t = 0 in the large graph. The dashed line is our chosen threshold, E共t兲 艋 0.15, below which we consider the forecasts “good.” The fluid parameters are ⌫ = 20, ⑀ = 1, Pr= 1.
estimate are used: the predictability time , defined as the time when E共t兲 first crosses the 共somewhat arbitrary兲 value of 0.15, and the minimum values attained by E共t兲 and E¯u共t兲 and E¯umin. The latter two during a forecast, denoted as Emin measures are used because the initial state estimate does not attain the minimum error, instead it occurs about 1 tv into the forecast. This is a result of the simulation rapidly balancing the fields by strongly suppressing field errors outside the Busse balloon. This effect is very slight in the LETKF forecasts, but can be quite strong in DI forecasts. B. Performance with noise/sparseness
We define a “standard” ideal scenario as measuring a shadowgraph every tv / 4 with = 127 共corresponding to a 451⫻ 451 shadowgraph image兲 and = 0.01 共this situation can be achieved in an experiment兲. Under these conditions the DI and LETKF 共with k = 18 ensemble members兲 typically converge on a state estimate within ⬃tv and ⬃4tv, respectively 共observing ⬃4 and ⬃16 shadowgraphs, respectively兲. Under these conditions, measurements are sufficiently dense and frequent for DI to perform well; hence both DI and the LETKF are effective for estimation of the 共unobserved兲 mean flow ¯u共x , y兲. However, the LETKF typically achieves a minimum error E¯umin that is less than half that of DI. The forecast errors E共t兲 and E¯u共t兲 versus the forecast lead time t for a typical state estimate are shown in Figs. 1 and 2. The general character of the forecasts is an initial shadowing of the true state, followed by rapid divergence. When divergence begins, the spatial structure of the error is concentrated near defects. This behavior is expected, as the magnitude of the Lyapunov vector associated with the largest Lyapunov exponent has maximum magnitude at the location of defects.30 Under nonideal conditions the LETKF proves much more robust than DI. Results for sparse measurements, shown in Fig. 3, demonstrate the large range of for which the LETKF converges. One can observe the existence of a
Chaos 19, 013108 共2009兲
FIG. 2. Typical mean flow error E¯u共t兲 of forecasts with = 0.01 and = 127. The inset shows E¯u共t兲 as each method converges on a state. Assimilation is turned off at time tJ = 3.25 in the small graph, corresponding to time t = 0 in the large graph. The fluid parameters are ⌫ = 20, ⑀ = 1, Pr= 1.
critical density of observations above which the LETKF does not substantially improve and below which it fails to converge. By adjusting the parameters of the LETKF’s update step 共as described in the Appendix兲 we have been able to push the critical density as low as = 1.3 without a significant loss of quality in the state estimate. DI on the other hand min exhibits a steady increase in Emin and E¯u as is decreased, as well as a rapidly deteriorating forecast when even a few observation locations are removed. Just as there is a critical measurement density, we have also found evidence of a critical measurement frequency. This frequency lies somewhere between 1 and 2 shadowgraph images per vertical diffusion time for repeatable convergence of the LETKF under ideal conditions. This corresponds to about 1 Hz in a typical experiment.
min FIG. 3. Emin , E¯u , and the predictability time as the density of observations is reduced with = 0.01.
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min FIG. 4. Emin , E¯u , and the predictability time as measurement noise is increased with = 127.
The magnitude of measurement noise is characterized by normalizing it to the RMS intensity variation of a typical shadowgraph, denoted sg. In other words, the meaningful signal to noise ratio is sg / . The variance 2sg is obtained by averaging 具兩I j共x , y兲 − 具I j共x , y兲典兩2典 over many shadowgraph images 关sg ⬇ 0.12 when a = 0.08 and Iⴰ共x , y兲 = 0.5兴. Results, shown in Fig. 4, indicate that DI forecasts are useful for low to moderate noise levels; whereas the LETKF operates up to and exceeding = sg. We note that all results are from one particular realization of the possible “true” time series, generated from one particular initial condition. These results are typical of what one can expect; however, variability can be expected 共particularly in 兲 for different data sets. C. Parameter estimation
In our numerical parameter estimation experiments reported below we take ⌫ = 20, Pr= 1 and we estimated p = 关R a兴T,31 where a is the observation operator parameter in Eq. 共4兲. The initial ensemble 兵␥0p,1 . . . ␥0p,k其 is constructed from states sampled from the attractor in the component, while the p component is sampled from a normal distribution 共with ¯ ¯a兴T and a diagonal covariance matrix with elemean ¯p = 关R 2 ments R and 2a兲. A typical convergence process is demonstrated in Fig. 5. In this example the ensemble converges in 8tv on p = 关R a兴T = 关3414.26 0.07979兴T ⫾ 关1.61 0.000072兴T, compared to the true value p = 关3414.0 0.08兴T 共the error estimates for R and a are the standard deviations of the ensemble after the last update at t = tJ兲. Remarkably, even when measurements are sparse 共 = 3.6, near the critical measurement density兲 the parameter estimates are very good, p = 关3416.71 0.07976兴T ⫾ 关9.9 0.00044兴T. When estimating the state and parameters simultaneously, the eventual values of Emin and E¯umin are similar to those shown in Figs. 3 and 4. That is, the ability to estimate the system state is not adversely affected when parameters
FIG. 5. Simultaneously estimating the parameters a 共with true value 0.08兲 and R 共with true value 3414兲. The error bars give a visual representation of the ensemble spread, extending one standard deviation up and down. The thick bars represent the case = 127 and = 0.01, while the thin lines represent the sparse measurement case, = 3.6 and = 0.01. The initial distri¯ = 3073, ¯a = 0.07兲 and standard deviation 共 bution was given mean 共R R = 683, a = 0.02兲.
are simultaneously estimated. It is important to note that estimating parameters 共in ␥ space兲 requires more ensemble members than when parameters are known; thus parameter estimation tests were performed with k = 20. V. RESULTS: EXPERIMENT
The experiment differs from a perfect model scenario in that G and H are now approximations, requiring robustness to model error as well as observation operator error. In particular, the Boussinesq model is an approximation to the more exact Navier–Stokes equations and our geometric optics treatment is an approximation to a more involved physical optics treatment. For example, the Boussinesq equations do not treat the temperature dependence of the fluid viscosity, thermal expansion coefficient, or thermal conductivity; each of which varies by 5%–10% over the temperature range ⌬T of the experiment. The geometry, parameter values, and boundary conditions are closely matched between experiments and simulations. For our experiments, the fluid is a thin 共d = 0.602⫾ 0.002 mm兲 layer of carbon dioxide gas compressed at a gauge pressure 31.58⫾ 0.06 bar. The layer is surrounded
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by a circular boundary of radius 12.50⫾ 0.02 mm. In the experiment, the top, bottom, and lateral boundaries are composed of sapphire, aluminum, and polyethersulfone, respectively; the thermal conductivities of the boundaries exceed that of the gas by at least an order of magnitude. For this fluid, the critical temperature difference for convection onset is ⌬Tc = 6.02 ° C and the vertical diffusion time is tv = 1.66⫾ 0.01 s. A fixed temperature difference ⌬T = 10.23⫾ 0.09 ° C is imposed across the layer at a fixed mean temperature of 22.6⫾ 0.1 ° C. These conditions correspond to R = 2902⫾ 26 共⑀ = 0.7兲, Pr= 0.97, and ⌫ = 20.76⫾ 0.08. The temperature difference and pressure were stable to within the indicated uncertainties. DI and the LETKF were used to assimilate shadowgraph images from the experiment. Images were taken every ⌬t = tv / 5 共3.0 Hz兲 as 395⫻ 395 bit maps 共 = 90兲 having sg = 0.059. The measurement noise distribution was characterized by taking the difference between two images below onset. The distribution of pixel noise was normally distributed with a standard deviation of 0.0032, a signal to noise ratio of approximately 18.4 共 / sg ⬇ 0.054兲.32 In experiments, the true fluid state is not available for directly ascertaining the accuracy of state estimates. Instead, we generate a forecast of the state estimate and compare the predicted shadowgraph sequence to subsequent measurements. We measure the forecast error by a technique which emphasizes the location of rolls and defects. Shadowgraphs are first filtered by removing high frequency components 共wavelengths less than d / 2兲. We then threshold the image such that half the pixels are set to 1 共the remaining half are 0兲. This filtering/threshold procedure is applied to both the predicted and measured shadowgraph time series. The natural error measure is then the fraction of pixels incorrectly predicted, denoted EI. The LETKF was given as 4tv to converge on state and parameter estimates; this is sufficient for both ideal 共 = 90兲 and sparse observation 共 = 4兲 cases. Figure 6 shows a typical state estimate from the LETKF. In particular, Fig. 6共d兲 shows the vorticity potential from the extracted mean flow, a quantity not directly observed. Typical examples of the forecast error are shown in Fig. 7 for both methods. To the eye, DI state estimates look nearly identical to the LETKF estimates. However, DI forecasts are significantly worse than the LETKF forecasts, which use their respective R estimates. Forecasts demonstrate an approximately linear forecast error growth up to the saturation point near EI = 0.5. To our knowledge, this is the first direct comparison of the Boussinesq equations 共using accurate boundary conditions兲 with an experiment on a one-to-one forecast basis. The Rayleigh number can be accurately measured directly in experiments; thus parameter estimation is unnecessary for the purpose of determining R. However, we place an emphasis on the ability of state and parameter estimates to generate good forecasts. Thus we allowed the LETKF to estimate R, as the model error can typically be compensated for, to some extent, by adjustment of model parameters off their measured values. In the dense measurement case 共 = 90兲, the LETKF converges on the parameter estimate R = 2625 共the experimentally measured value is R = 2902⫾ 26兲.
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FIG. 6. An estimate of the fluid state after assimilating for 4tv 共J = 20 frames兲. 共a兲 The t = tJ shadowgraph measurement indicating columns of warm rising fluid 共dark兲 and cold descending fluid 共light兲. 共b兲 Temperature profile ¯共x , y兲 from the state estimate. 共c兲 The modeled shadowgraph H共¯共x , y兲兲 of the state estimate for comparison to 共a兲. 共d兲 The inferred vorticity potential 共x , y兲 which solves ⵜ2共x , y兲 = −zˆ · 共ⵜ ⫻ ¯u兲 and indicates regions of clockwise rotating 共dark兲 and counterclockwise rotating 共light兲 mean flow.
When = 4 the LETKF converges on the estimate R = 2491. These estimates are obtained consistently 共with slight variation兲 throughout the experimental data set. In fact, forcing the LETKF to use the measured R value harms the forecast, bringing it up to the level of the DI forecast 共which uses the true value R = 2902兲. This indicates that the advantage of the LETKF in this case lies in its ability to estimate parameters which are optimal 共in the sense of producing the best forecasts when the forecast model is not exact兲. Figure 8 shows how the forecasts of Fig. 7 compare with typical perfect model forecasts using the same parameters as the experiment 共R = 2902, ⌫ = 20.8, Pr= 0.97兲 as well as the same measurement frequency 共⌬t = tv / 5兲, density 共 = 90兲, and approximately the same noise level 共 / sg = 0.083兲.
FIG. 7. Forecast error EI for DI and LETKF methods are shown for high and low measurement densities. The LETKF forecast uses its parameter estimate 共R = 2625 for = 90, R = 2491 for = 4兲 while the DI forecast uses the measured value R = 2902.
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FIG. 9. Two local regions are shown on a reduced resolution mesh. Every grid point 共m , n兲 is the center of a local region. Associated with each local region 共m , n兲 is the local state vector mn consisting of state variables on the indicated horizontal grid points and all vertical grid points associated with them. FIG. 8. Forecast error EI for DI and LETKF methods in perfect model 共PM兲 tests and when using experimental data 共E兲. The LETKF forecasts use the estimated value of R, while DI forecasts use the true value. The parameters in all cases are R = 2902, ⌫ = 20.8, and Pr= 0.97. Noise levels are low 共 / sg = 0.083 关PM兴 and / sg = 0.054 关E兴兲, and the density of measurements is high 共 = 90兲.
Forecasts in the experimental situation are seen to be less accurate than those in our perfect model tests. We note that, among the uncertainties in the experiment, it is the uncertainty in the aspect ratio which has the largest potential to produce forecast error. Sensitivity tests with the model showed that the measured uncertainties in all quantities, including the aspect ratio, were too small to account for a significant portion of the forecast error. Hence, we attribute the discrepancy between the perfect model and experimental results to non-Boussinesq effects.33
APPENDIX: THE LETKF ALGORITHM
We now describe the LETKF’s update step 共5兲. This appendix is an adaptation to our Rayleigh–Bénard problem of the technique developed in Ref. 12. Because of the measurement noise we cannot know the system state exactly. Thus, we seek the PDF for . In order to apply the Kalman filter methodology, we assume that this PDF is Gaussian, i.e., it is proportional to exp兵−共 − ¯兲TP−1共 − ¯兲 / 2其. The center of this distribution ¯ is the most likely state, while the error covariance matrix P characterizes the uncertainty of that estimate. For a given ensemble i the mean and covariance are, respectively, estimated by ¯ ⬅ 1 兺 i , k i
VI. CONCLUSIONS
We have investigated two methods for estimating the fluid state in Rayleigh–Bénard convection experiments, DI and the LETKF. Both methods are effective for this purpose, with the LETKF outperforming DI both when using experimental data and in perfect model tests, especially when data are sparse/noisy. One purpose of this paper is the introduction of data assimilation methodology to a community for which DI-type techniques are the only techniques known, and to demonstrate that more involved techniques can be worth the effort. We have demonstrated that techniques developed for weather forecasting 共such as the LETKF兲 can be successfully applied to a real laboratory system. We believe this is important as a means of introducing consideration of data assimilation techniques to the large community of researchers investigating spatiotemporal chaos in laboratory experiments. ACKNOWLEDGMENTS
We are grateful to Laurette Tuckerman for the use of her code for simulating the Boussinesq equations in a cylindrical geometry. Support for this work by National Science Foundation Grant Nos. 04-34225 and 04-34193 and the Office of Naval Research 共Physics兲 is gratefully acknowledged. FORTRAN
P⬅
1 Z共Z兲T , k−1
共A1兲
共A2兲
where the columns of Z are the ensemble perturbations, Z ⬅ 关␦1兩␦2兩 ¯ 兩␦k兴, with ␦i = i − ¯. The ensemble is to be constructed to represent the Gaussian PDF with mean 共A1兲 and covariance 共A2兲. Let mn be a vector whose components consist of the collection of all elements of that lie on grid points within a horizontal distance L of the point 共rm , n兲 of the mesh used by the model. We call mn a local state and L the local region radius. There are as many local regions as horizontal grid points 共rm , n兲, hence these regions are heavily overlapping 共see Fig. 9兲. When the center of the local region 共rm , n兲 is near the radial boundary, the local region is the intersection of a disk having radius L centered at 共rm , n兲 and the diskshaped domain with radius ⌫ centered on the origin. Note that, since the problem of interest is essentially two dimensional, local regions are indexed by two indices 共m , n兲. The three-dimensional nature of the system is reflected in the fact that, for each horizontal grid point, the vector mn contains the state at all z levels. Associated with the updated and predicted global ensemble members u,i and p,i are the local u,i p,i ensemble members mn and mn 共all local states, global states, and ensemble states have an implied time index j兲. The predicted observation ensemble of shadowgraphs
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State estimation of spatiotemporal chaos
兵y p,1 . . . y p,k其 is defined as y p,i = H共 p,i兲 共the projection of the p,i be predicted ensemble into the observation space兲. Let ymn p,i all elements of y within the local region 共m , n兲. If there are smn measurements made within the local region 共m , n兲, then p,i p has dimension smn. We form the matrix Ymn the vector ymn p,1 p,2 p,k p,i p,i p p ⬅ 关␦ymn 兩␦ymn 兩 ¯ 兩␦ymn兴 where ␦ymn = ymn −¯ymn and ¯ymn is defined as in Eq. 共A1兲. The local measurements ymn have an associated local smn ⫻ smn covariance matrix Rmn, which is equal to 2 multiplied by the smn ⫻ smn identity matrix. We modify this matrix by forming the tapered diagonal covari2 ii ⬅ 关e共r / r f 兲 兴2, ance matrix Qmn having i, ith element Qmn where r is the 共horizontal兲 distance from the grid point 共m , n兲 to the measurement location associated with the ith element of ymn, and r f is some falloff distance. This modification effectively weighs measurements further from the grid point 共m , n兲 less heavily when estimating the state at the point 共m , n兲. This type of distance-dependant modification to covariance matrices has been investigated previously.17 We also weigh current measurements more heavily than prior ones by the method of multiplicative variance inflation in which the predicted covariance matrix is inflated by a factor ⍀2 ⬎ 1, to lessen the influence of prior measurements on the current state, and to compensate in some rough way for model error and nonlinearities.12,16 Ordinarily ⍀ is chosen empirically. The perfect model tests reported used variance inflation ⍀ = 1.0– 1.1, whereas results from experimental data in Sec. V used an inflation factor of ⍀ ⬇ 1.4. We proceed to compute the updated ensemble. As derived and discussed in Refs. 11 and 12, the procedure listed below is followed. The inputs are the global predicted ensemble p,i and the measurement y. The output is the global updated ensemble u,i. Compute each y p,i = H共 p,i兲 and ¯y p. Form the matrix Y p with columns ␦y p,i. Compute ¯ p and form the matrix Z p with columns ␦ p,i. For each grid point 共m , n兲 perform steps 共1兲–共7兲: 共1兲 Form ymn from the elements of the current measurement y, along with the tapered covariance matrix Qmn. p p and Ymn from the relevant elements of ¯y p and 共2兲 Form ¯ymn p Y. 共3兲 Compute the updated k ⫻ k covariance matrix, ˜Pu = 关共k − 1兲⍀−2I + 共Y p 兲TQ−1 Y p 兴−1 . mn mn mn mn
共A3兲
共4兲 Next compute u p T −1 p 共Ymn 兲 Qmn共ymn − ¯ymn 兲. wmn = ˜Pmn
共A4兲
共5兲 Calculate the matrix ˜ u 兴1/2 + w , Wmn = 关共k − 1兲P mn mn
共A5兲
where, by adding a vector to a matrix we mean adding it to each column of the matrix. The 1 / 2 power here indicates taking the positive symmetric matrix square root. p,1 p,2 p,k p 共6兲 Form the matrix Zmn ⬅ 关␦mn 兩 ␦mn 兩 ¯ 兩␦mn 兴 from the p p ¯ relevant elements of Z . Also form mn from ¯ p. 共7兲 Finally, compute the local updated ensemble perturbations,
u p Zmn = Zmn Wmn .
共A6兲
u,1 u,2 u,k u ⬅ 关␦mn 兩 ␦mn 兩 ¯ 兩␦mn 兴, and the local As before, Zmn u,i ¯ p u,i . updated ensemble is given by mn = mn + ␦mn
To complete the update step, components of the global updated ensemble member u,i at each horizontal grid point u,i 共m , n兲 are taken to be equal to the elements of mn at the center of local region 共m , n兲. Note that each local region is assimilated independently, allowing for massive parallelization. To estimate parameters, simply replace with ␥ everywhere in the above steps. This formulation assumes state variables are spatially extended. Thus, when adding global parameters to the state space we must assume that they are spatially dependant. That is, when estimating both the Rayleigh number and a of Eq. 共4兲 共p = 关R a兴T兲, the state ␥ is and pˆ , where pˆ the concatenation of = 关R11 . . . Rmn . . . a11 . . . amn . . . 兴T. The LETKF is then augmented by averaging these parameter values over the grid, after the update step, to form global parameters. This average is performed for each global ensemble member ␥u,i by set¯ i ¯Ri . . . ¯ai ¯ai . . . 兴T, where ¯Ri and ting its pˆ component to pˆ u,i = 关R i ¯a are the spatial averages of R and a for the ith ensemble member. The model G and observation operator H then use ¯ i and ¯ai when applied to ensemble member the parameters R i. If the model allows for spatially dependent parameters, then this last averaging step is not necessary. The LETKF formulation is advantageous since the number of ensemble members required for convergence is independent of the system size,10 making the method applicable to large domains. The number of ensemble members will presumably scale with the number of dynamical degrees of freedom in a local region. We used a local region radius of L = 2.6d and a falloff distance of r f = 1.4d in the perfect model section. For the experimental data, we found that a local region radius of L = 2.6d and a falloff distance of r f = 1.0d worked well. In both cases, L is comparable to the correlation length of spiral defect chaos of 2.7d when ⑀ = 0.7 and 2.3d when ⑀ = 1.0.34 As the ensemble converges, it tends to confine itself to a space of dimension lower than k, indicating that one could optimize by “pruning” the ensemble size as it converges. All the results in this paper are for a constant k = 18 共or k = 20 when estimating parameters兲, but we have found that starting with k = 18 and reducing to k = 8 linearly within 10 measurement times gives similar results with a significant reduction in computation time. In addition, the strength of the model nonlinearities is largest when the ensemble spread is large 共during the first few assimilation steps兲, thus one can begin assimilation with a large ⍀ and reduce it linearly to speed convergence. This procedure was found to be successful, but was not performed in the results reported here. F. T. Arecchi, S. Boccaletti, and P. Ramazza, Phys. Rep. 318, 1 共1999兲. H. L. Swinney and V. I. Krinsky, Waves and Patterns in Chemical and Biological Media 共MIT Press, Cambridge, 1991兲. 3 S. W. Morris, E. Bodenschatz, D. S. Cannell, and G. Ahlers, Phys. Rev. Lett. 71, 2026 共1993兲. 1 2
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M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 共1993兲. The Kalman filter is a form of recursive Bayesian estimation. Essentially, it uses the model to find an optimal fitting of the current system state to current and past measurements. 6 R. Kalman, J. Basic Eng. 82, 35 共1960兲. 7 D. Simon, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches 共Wiley-Interscience, New York, 2006兲. 8 G. Evensen, Data Assimilation: The Ensemble Kalman Filter 共Springer, Berlin, 2006兲. 9 For example, for a discretized partial differential model evolving M scalar spatial fields in time, the number of model variables is M multiplied by the number of grid points. 10 E. Ott, B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M. Corazza, E. Kalnay, D. J. Patil, and J. A. Yorke, Phys. Lett. A 330, 365 共2004兲. 11 E. Ott, B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M. Corazza, E. Kalnay, D. J. Patil, and J. A. Yorke, Tellus, Ser. A 56, 415 共2004兲. 12 B. Hunt, E. Kostelich, and I. Szunyogh, Physica D 230, 112 共2007兲. 13 J. S. Whitaker and T. M. Hamill, Mon. Weather Rev. 130, 1913 共2002兲. 14 M. K. Tippett, J. L. Anderson, C. H. Bishop, T. M. Hamill, and J. S. Whitaker, Mon. Weather Rev. 131, 1485 共2003兲. 15 C. H. Bishop, B. J. Etherton, and S. J. Majumdar, Mon. Weather Rev. 129, 420 共2001兲. 16 J. Anderson, Mon. Weather Rev. 129, 2884 共2001兲. 17 J. W. T. M. Hamill and C. Snyder, Mon. Weather Rev. 129, 2776 共2001兲. 18 P. L. Houtekamer and H. L. Mitchell, Mon. Weather Rev. 129, 123 共2001兲. 19 D. J. Patil, B. R. Hunt, E. Kalnay, J. A. Yorke, and E. Ott, Phys. Rev. Lett. 86, 5878 共2001兲. 20 M. Galmiche, J. Sommeria, E. Thivolle-Cazat, and J. Verron, Meccanica 12, 331 共2003兲. 21 E. Bodenschatz, W. Pesch, and G. Ahlers, Annu. Rev. Fluid Mech. 32, 709 共2000兲. 22 F. H. Busse, Rep. Prog. Phys. 41, 1929 共1978兲. 23 W. Merzkirch, Flow Visualization 共Academic, New York, 1974兲, p. 258. 4 5
Chaos 19, 013108 共2009兲
Cornick et al. 24
K.-H. Chiam, M. R. Paul, M. C. Cross, and H. S. Greenside, Phys. Rev. E 67, 056206 共2003兲. 25 S. Rasenat, G. Hartung, B. L. Winkler, and I. Rehberg, Exp. Fluids 7, 412 共1989兲. 26 S. Trainoff and D. S. Cannell, Phys. Fluids 14, 1340 共2002兲. 27 L. S. Tuckerman, J. Comput. Phys. 80, 403 共1989兲. 28 The grid points are located at the Chebyshev points in r and z, rm = ⌫ cos共共m − 1兲 / 2共Nr − 1兲兲, n = 2n / N, zl = 1 / 2 cos共共l − 1兲 / Nz − 1兲. In the investigated parameter region we find Nr = 151, N = 451, Nz = 9 共for a total of roughly 6 ⫻ 105 grid points兲, and a simulation time step of tv / 100 sufficient. 29 We also investigated the smaller domain ⌫ = 6 with ⑀ = 2.0 and found that the use of the EnKF required k ⬇ 100 ensemble members. 30 D. A. Egolf, I. V. Melnikov, W. Pesch, and R. E. Ecke, Nature 共London兲 404, 733 共2000兲. 31 Due to limitations in our simulation, we were unable to estimate Pr in the full ⌫ = 20 system. Instead we estimated Pr, R, and a together in small aspect ratio tests 共⌫ = 4, R = 8540, Pr= 1兲. The results of these tests indicated that all three parameters can be simultaneously estimated with high accuracy. 32 We found empirically that using larger values for R 共 ⬇ 0.01– 0.02兲 in the LETKF’s update step 共rather than the measured noise level of = 0.0032兲 resulted in better forecasts. 33 We attempted to model some non-Boussinesq effects. However, there are several ways the fluid can be non-Boussinesq, many of which are not easy to model. For example, our numerical technique relies heavily on the assumption of incompressibility, and deviations from this behavior are beyond our modeling capabilities. Some non-Boussinesq effects were included 共temperature dependence of conductivity and specific heat兲, but did not significantly affect forecast error when compared with experimental data. 34 S. Morris, E. Bodenschatz, D. Cannell, and G. Ahlers, Physica D 97, 164 共1996兲.
CHAOS 19, 013109 共2009兲
Generalized outer synchronization between complex dynamical networks Xiaoqun Wu,1,2,a兲 Wei Xing Zheng,2,b兲 and Jin Zhou1,3 1
School of Mathematics and Statistics, Wuhan University, Hubei 430072, China School of Computing and Mathematics, University of Western Sydney, Penrith South DC, NSW 1797, Australia 3 Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong 2
共Received 12 October 2008; accepted 29 December 2008; published online 10 February 2009兲 In this paper, the problem of generalized outer synchronization between two completely different complex dynamical networks is investigated. With a nonlinear control scheme, a sufficient criterion for this generalized outer synchronization is derived based on Barbalat’s lemma. Two corollaries are also obtained, which contains the situations studied in two lately published papers as special cases. Numerical simulations further demonstrate the feasibility and effectiveness of the theoretical results. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3072787兴 Synchronization of complex networks have been extensively investigated in many research and application fields. Most of this research has been focused upon a coherent behavior within a network, where each node of the network arrives at the same steady state. This kind of synchronization, which was called “inner synchronization” in Ref. 20 has attracted broad attention. As a matter of fact, in real-world situations, there also exist other kinds of synchronization for complex networks, such as “outer synchronization” between two networks as considered in Refs. 20 and 23 below, where under the assumption that all individuals in two networks have completely identical behaviors a “complete outer synchronization” was studied. However, this kind of assumption may not seem practical. Take the predator-prey interactions in ecological communities as an example, where predators and preys influence one another’s evolution. Without preys there would not be predators, while too many predators would bring the preys into extinction. The communities of predators and preys will finally reach harmonious coexistence without man made sabotage. It is worth noting that inside the networks of predators or preys, one individual always behaves differently from another. Thus it is more practical to assume that each node has different dynamics. Furthermore, the interactions of predators themselves usually differ from that of preys, that is, the topological structure of the predators community is different from that of the preys community. Therefore, synchronization between two different complex networks, where the difference results from node diverseness as well as topological difference, is a more practical and significant problem worth investigating. I. INTRODUCTION
Complex networks have received rapidly increasing attention from different fields in recent years. From the internet to the world wide web, from communication networks to social organizations, from food webs to ecological commua兲
Electronic mail:
[email protected]. Electronic mail:
[email protected].
b兲
1054-1500/2009/19共1兲/013109/9/$25.00
nities, etc., complex networks widely exist in our life and are presently prominent candidates to describe sophisticated collaborative dynamics in many sciences.1–6 So far, the dynamics of complex networks has been extensively investigated, in which synchronization is a typical topic that has attracted lots of interests. Synchronization is a fundamental phenomenon that enables coherent behavior in networks as a result of interactions. Pecora et al. used the master stability function approach to determine the stability of the synchronous state in coupled systems.7,8 Chen et al. imposed constraints on the coupling strengths to ensure stability of the synchronized states in arbitrarily coupled dynamical systems based on the master stability function together with Gershörin disk theory.9 Wu et al. investigated synchronization in linearly coupled identical dynamical systems by the Lyapunov direct method and proved that strong enough mutual diffusive coupling will synchronize an array of identical cells.10 The Lyapunov function method was also employed in some of Wu’s later works on synchronization of coupled systems.11–14 Wang and Chen studied synchronization in two specific kinds of networks: Scale-free networks and small-world networks.15,16 Lü and Chen introduced a time-varying dynamical network and further investigated its synchronization criteria.17 Zhou et al. and Lu considered synchronization in networks by integrating network models and an adaptive technique.18,19 Researches on synchronization of networks mentioned above focused on the phenomenon that all nodes in a network achieve a coherent behavior, which was called inner synchronization,20 as it is a collective behavior within a network. In reality, there exist other kinds of network synchronization, for example, synchronization between two or more networks, which was termed outer synchronization in Ref. 20. A representative illustration is predator-prey interactions in ecological communities,21 where predators and preys can influence one another’s evolution. For example, plant-eating animals, such as mice, rats, and rabbits, would soon strip the land bare without the controlling effect of predators. Areas where large predators have been reduced through trapping, shooting, and other predator-control methods often develop
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large populations of mice, rats, and rabbits that can destroy the plants needed by other wildlife species for both food and shelter. If the predators had been allowed to remain, the prey species probably would have been kept under control. In a word, the relationship between the network of predators and that of preys is important in maintaining balance among different animal species. Mankind has been trying every means to maintain this balance. Another example is the balance of intestinal microflora for human beings.22 A vast amount of good 共or beneficial兲 bacteria living inside our digestive system which perform very important functions exist alongside with bad 共or pathogenic兲 bacteria which produce harmful substance and create serious problems. The bacteria are essentially competing with one other for space and nutrients. A good balance of the two communities of good and bad bacteria provides protection against a broad range of pathogens while discomforts and symptoms of disease can result when factors like antibiotics, poor diet, and stress cause this balance to be disrupted. There are a great many examples about relationships between different networks, which also indicates that it is necessary and significant to investigate the dynamics between different networks. Recently, Li et al.20 pioneered in studying outer synchronization between two unidirectionally coupled networks and derived a criterion for the synchronization between two networks with identical topological structures. Shortly after, Tang et al. analyzed outer synchronization between two complex networks with nonidentical topologies using adaptive controllers.23 In these two papers, it is assumed that each node in both networks has identical dynamics, and the corresponding nodes in two networks manifest completely the same dynamics, so strictly speaking, it is complete outer synchronization between two networks. However, nodes in different networks usually have different dynamics 共parameter mismatch or structural discrepancy兲, while the two networks may still behave in a synchronous way. This kind of synchronization is called generalized synchronization,24–27 which represents another degree of coherence. For instance, in the aforementioned predator and prey networks, predators and preys may finally reach a synchronous state even though they have entirely different behaviors 共even individuals inside a network may behave in quite diverse ways兲. Motivated by the above discussions, in this paper, we introduce the concept of generalized outer synchronization between two complex dynamical networks, where nodes in one network synchronize with their counterparts in the other network through some smooth functions. A criterion on generalized outer synchronization is derived based on Barbalat’s lemma, and then two corollaries are drawn. In our study, each network can be undirected or directed, connected or disconnected, and nodes in either network may have identical or different dynamics. As an extension of complete synchronization, the generalized outer synchronization studied here has a much wider application range than complete outer synchronization. The rest of this paper is organized as follows: Network models and some preliminaries are introduced in Sec. II. In Sec. III, using nonlinear control, we present a criterion for generalized outer synchronization between two networks
with arbitrary node dynamics and topological structures. Some numerical simulations are provided to illustrate the feasibility and effectiveness of the proposed approach in Sec. IV. Finally, concluding remarks are given in Sec. V. II. PROBLEM DESCRIPTION A. Network models
Some necessary notations that will be used throughout this paper are first introduced. denotes the transpose of a matrix or a vector. 储储 indicates the 2-norm of a vector , i.e., 储储 = 冑. Ii 苸 Ri⫻i represents the identity matrix with dimension i. 丢 denotes the Kronecker product of two matrices.28 m共A兲 represents the maximum eigenvalue of a square matrix A. Consider a weighted general complex dynamical network consisting of N dynamical nodes with linear couplings, which is characterized by N
x˙ i共t兲 = fi共xi共t兲兲 + 兺 cij Px j共t兲,
i = 1,2, . . . ,N.
共1兲
j=1
Here xi共t兲 = 共xi1共t兲 , xi2共t兲 , . . . , xin共t兲兲 苸 Rn is the state vector of the ith node, and fi : Rn → Rn is a smooth nonlinear vectorvalued function governing the evolution of xi共t兲 in the absence of interactions with other nodes. P 苸 Rn⫻n is an innercoupling matrix determining the interaction of variables. C = 共cij兲N⫻N 苸 RN⫻N is the coupling configuration matrix representing the coupling strength and the topological structure of the network, in which cij is defined as follows: if there is a connection from node i to node j共j ⫽ i兲, cij ⫽ 0; otherwise, cij = 0. The diagonal elements of matrix C are defined as N
cii = −
兺
cij,
i = 1,2, . . . ,N.
j=1,j⫽i
Consider another complex dynamical network containing N dynamical nodes as follows: N
y˙ i共t兲 = Gi共yi共t兲兲 + 兺 dijQy j共t兲,
i = 1,2, . . . ,N,
共2兲
j=1
where yi共t兲 = 共y i1共t兲 , y i2共t兲 , . . . , y im共t兲兲 苸 Rm is the state vector of the ith node, and Gi : Rm → Rm is a smooth nonlinear vector-valued function governing the evolution of the ith isolated node yi共t兲. Q 苸 Rm⫻m is the inner-coupling matrix, and D = 共dij兲N⫻N 苸 RN⫻N is the coupling configuration matrix, which has the same meaning as that of matrix C. In the following, we will take Eq. 共1兲 as the drive network and Eq. 共2兲 as the response network. It is well-known that many systems, such as the Lorenz system, Chen system, Lü system, Rössler system, Chua’s circuit, hyperchaotic Rössler system, hyperchaotic Chen and Lü system, can be written in the following form: y˙ = Ay + g共y兲, where A 苸 Rm⫻m is the Jacobian matrix of the system at the origin, and g共y兲 is the nonlinear part. Therefore, without loss of generality, we can describe the response network 共2兲 with control as
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Generalized outer synchronization N
y˙ i共t兲 = Aiyi共t兲 + gi共yi共t兲兲 + 兺 dijQy j共t兲 + ui,
i = 1,2, . . . ,N,
j=1
共3兲 where Aiyi and gi are, respectively, the linear and nonlinear part of the ith node, and ui is the ith controller to be designed according to the specific node dynamics and topological structures of the drive and response networks.
e˙ i = y˙ i − Di共xi兲 · x˙ i N
= Aiei − kei + gi共yi兲 − gi共i共xi兲兲 + 兺 dijQe j, j=1
共5兲
i = 1,2, . . . ,N.
mN Let e = 共e 1 共t兲 , e2 共t兲 , . . . , eN 共t兲兲 苸 R , and consider the following Lyapunov candidate function: N
1 1 V共t兲 = ee = 兺 e 共t兲ei共t兲. 2 2 i=1 i
B. Preliminaries
In this subsection, we will first give a definition of generalized outer synchronization between two networks, followed by an assumption and a lemma which will be needed in the subsequent study. Definition 1: Let i : Rn → Rm共i = 1 , 2 , . . . , N兲 be continuously differentiable vector maps. Network 共1兲 is said to achieve generalized outer synchronization with network 共3兲 if lim 兺 储yi共t兲 − i共xi共t兲兲储 = 0.
t→⬁ i=1
=兺
e i A ie i
N
− k兺
e i ei
i=1
i=1
N
N
i=1
j=1
+ 兺 e i 关gi共yi兲 − gi共i共xi兲兲兴 i=1
N
N
i=1
i=1
艋 兺 e i A ie i − k 兺 e i e i + 兺 h ie i e i i=1
N
N
i=1
j=1
+ 兺 e i 兺 dijQe j = e Ae − ke e + 兺 hie i ei + e Qe
III. GENERALIZED OUTER SYNCHRONIZATION CRITERIA
冉冉
艋 m
A+A 2
冊
i=1
ui = Di共xi兲 · fi共xi兲 − Aii共xi兲 − gi共i共xi兲兲 − kei i = 1,2, . . . ,N, 共4兲 where Di共xi兲 is the Jacobian matrix of the map i共xi兲, ei = yi − i共xi兲, and k is a sufficiently large positive constant. Proof: Since ei = yi − i共xi兲, from networks 共1兲 and 共3兲, together with the control scheme 共4兲, we obtain the error dynamical network described by
− k + max兵hi其 + m i
冉
Q + Q 2
冊冊
ee,
where A = diag共A1 , A2 , . . . , AN兲 苸 RmN⫻mN 共i.e., the ith diagonal square block of A is Ai兲, and Q = D 丢 Q. Taking k 艌 k* = maxi兵hi其 + m共A + A / 2兲 + m共Q + Q / 2兲 + 1, we get 兩V˙共t兲兩共5兲 艋 − eTe.
With the network models and the definition given previously, we arrive at the following main theorem. Theorem 1: Suppose that Assumption 1 holds. The drive network 共1兲 can achieve generalized outer synchronization with the response network 共3兲 under the following control law:
j=1
N
N
holds for any z1 and z2. Lemma 1: 共Barbalat’s lemma29兲 If f共t兲 is non-negative, integrable, and uniformly continuous on 关a , + ⬁兲, then f共t兲 → 0 as t → ⬁.
j=1
i=1
N
储gi共z1兲 − gi共z2兲储 艋 hi储z1 − z2储
− 兺 dijQ j共x j兲 + Di共xi兲 兺 cij Px j,
N
˙i 兩V˙共t兲兩共5兲 = 兺 e i e
N
Usually, function gi共·兲 is globally Lipschitz continuous, i.e., the following assumption is satisfied. Assumption 1: For function gi共z兲共i = 1 , 2 , . . . , N兲, there exists a positive constant hi共i = 1 , 2 , . . . , N兲 such that
N
Calculating its derivative along the trajectories of Eq. 共5兲, under Assumption 1, we obtain
+ 兺 e i 兺 dijQe j
N
N
共6兲
共7兲
Obviously, V˙共t兲 艋 0, so V共t兲 is uniformly continuous. Furthermore, we have V共t兲 艋 V共0兲exp共− 2t兲,
共8兲
thus limt→⬁ 兰t0V共兲d exists, namely, V共t兲 is integrable on 关0 , + ⬁兲. According to Barbalat’s lemma, we obtain limt→+⬁ V共t兲 = 0, which implies limt→+⬁ ei共t兲 = 0 for i = 1 , 2 , . . . , N. Therefore, networks 共1兲 and 共3兲 asymptotically achieve generalized outer synchronization. This completes the proof. 䊐 Remark 1: In the theorem, the configuration matrices C and D need not be symmetric or irreducible, which means that networks 共1兲 and 共3兲 can be undirected or directed networks, and they may also contain isolated nodes and clusters. In addition, there is not any constraint imposed on the innercoupling matrices P and Q. Moreover, each node may have different node dynamics. Therefore, our method is applicable to a large variety of complex dynamical networks.
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Wu, Zheng, and Zhou
Remark 2: The feedback gain k can be chosen properly to adjust the synchronization speed. Theoretically, a larger k may lead to faster synchronization. However, it is noted that k 艌 k* is only a sufficient condition but not a necessary one. Later simulations will show that a small value of k can also lead to generalized outer synchronization quickly. Based on Theorem 1, we can easily derive the following corollaries: Corollary 1: Suppose that Assumption 1 holds. If networks 共1兲 and 共3兲 have the same topological structures and uniform inner-coupling matrices, then the two networks can achieve complete outer synchronization under the following control scheme: ui = fi共xi兲 − Aixi − gi共xi兲 − kei,
共9兲
i = 1,2, . . . ,N,
where k 艌 k* = maxi兵hi其 + m共A + A / 2兲 + m共Q + Q / 2兲 + 1. Corollary 2: Suppose that Assumption 1 holds. If the ith nodes in networks 共1兲 and 共3兲 have identical dynamics, namely, fi = Gi共i = 1 , 2 , . . . , N兲, then the two networks can achieve complete outer synchronization under the following control scheme: N
ui = − kei + 兺 共cij P − dijQ兲x j,
i = 1,2, . . . ,N,
共10兲
j=1
where k 艌 k* = maxi兵hi其 + m共A + A / 2兲 + m共Q + Q / 2兲 + 1. Remark 3: This corollary presents a control scheme on complete outer synchronization for the case as studied in Ref. 23. In that paper, an adaptive technique was employed, and for two networks of size N, N2 + N additional adaptive controllers have to be utilized, which immensely increases the control cost. However, according to our Corollary 2, only N simple linear controllers are needed, which are much easier to implement. In particular, if the two networks 共1兲 and 共3兲 have identical topological structures and inner-coupling matrices, then the controllers 共10兲 are further simplified into ui = − kei
共i = 1,2, . . . ,N兲
for complete outer synchronization. This is an extension of the case as studied in Ref. 20. Note that in Ref. 20 under the additional assumption that fi = f = Gi for i = 1 , 2 , . . . , N, a criterion for local synchronization was derived. In contrast, the synchronization obtained herein is global synchronization. IV. NUMERICAL SIMULATIONS
In this section, illustrative examples will be provided to verify the effectiveness of the control scheme obtained in the preceding section. For this purpose, we consider several benchmark chaotic systems, such as Lorenz system, Chen system, and Lü systems. Lorenz system is known to be a simplified model of several physical systems. It was originally derived from a model of the Earth’s atmospheric convection flow heated from below and cooled from above.30 Furthermore, it has been reported that Lorenz equations may describe such different systems as laser devices, disk dynamos, and several
problems related to convection.31 Later on, the Lorenz attractor was mathematically confirmed to exist.32 The Lorenz system is represented by
冢
a共x2 − x1兲
冣
共11兲
x˙ = cx1 − 0x1x3 − x2 , x1x2 − bx3
which has a chaotic attractor when a = 10, b = 8 / 3, c = 28. Chen system is a typical chaos anticontrol model, which has a more complicated topological structure than the Lorenz attractor.33 It has been implemented by circuitry,34 and has wide application potential in secure communications. The nonlinear differential equations that describe the Chen system are
冢
冣
a共x2 − x1兲 x˙ = 共c − a兲x1 − x1x3 + cx2 , x1x2 − bx3
共12兲
which has a chaotic attractor when a = 35, b = 3, c = 28. Lü system is a typical transition system, which connects the Lorenz and Chen attractors and represents the transition from one to the other.35 Lü system is described by
冢
a共x2 − x1兲
冣
共13兲
x˙ = − x1x3 + cx2 , x1x2 − bx3
which has a chaotic attractor when a = 36, b = 3, c = 20. Later on Lü et al. proposed a unified chaotic system,36 which contains Lorenz system and Chen system as two extremes and Lü system as a special case. The unified chaotic system is described by
x˙ =
冢
共25 + 10兲共x2 − x1兲 共28 − 35兲x1 − x1x3 + 共29 − 1兲x2
+8 x3 x 1x 2 − 3
冣
,
共14兲
where 苸 关0 , 1兴. Obviously, system 共14兲 is the original Lorenz system for = 0 while it reduces to the original Chen system for = 1. When = 0.8, system 共14兲 is just the critical system, Lü system. In fact, system 共14兲 bridges the gap between Lorenz system and Chen system. Especially, system 共14兲 is always chaotic over the whole interval 苸 关0 , 1兴. As is known, there are many hyperchaotic systems discovered in the high-dimensional social and economical systems. Typical examples are the four-dimensional 共4D兲 hyperchaotic Rössler system,37 hyperchaotic Lorenz–Haken system,38 hyperchaotic Chua’s circuit,39 and hyperchaotic Chen system.40 Since hyperchaotic systems have the characteristics of high capacity, high security, and high efficiency, they have broad application potential in secure communications, nonlinear circuits, biological systems, neural networks, etc. In Ref. 41, Chen et al. presented the hyperchaotic Lü system,41 which is described by
013109-5
Chaos 19, 013109 共2009兲
Generalized outer synchronization
A. Complete synchronization between two networks
Consider the unified chaotic system 共14兲 as isolated node dynamics. Let for the ith node be 兩sin i兩 in the drive network and 兩sin i2兩 in the response network, namely, each node in the two networks are different but all chaotic. Thus we have
fi共xi兲 =
冢
Ai =
冢
共25兩sin i兩 + 10兲共xi2 − xi1兲 共28 − 35兩sin i兩兲xi1 − xi1xi3 + 共29兩sin i兩 − 1兲xi2 xi1xi2 −
兩sin i兩 + 8 xi3 3
− 共25兩sin i2兩 + 10兲 25兩sin i2兩 + 10 28 − 35兩sin i2兩
0
29兩sin i2兩 − 1
0
0
兩sin i2兩 + 8 3
0
冣
冣
,
共16兲
, 共17兲
and
冢 冣 0
gi共y兲 = g共y兲 = − y 1y 3 . y 1y 2
共18兲
For any vectors y and z of the unified chaotic system 共14兲, there exists a positive constant M = 57 such that 储y p储 艋 M, 储z p储 艋 M for 1 艋 p 艋 3 since the unified chaotic system is bounded in a certain region.42 Therefore, one has 储g共y兲 − g共z兲储 = 冑共y 1共y 3 − z3兲 + z3共y 1 − z1兲兲2 + 共y 1共y 2 − z2兲 + z2共y 1 − z1兲兲2 艋 冑2M储y − z储,
FIG. 1. 共Top兲 a star coupled network; 共bottom兲 a directed ring network.
that is, Assumption 1 is satisfied. So we may take hi = 冑2M for i = 1 , 2 , . . . , N. For complete synchronization, the map i is defined as yi = i共xi兲 = 共xi兲 = xi ,
x˙ =
冢
a共x2 − x1兲 + x4 − x1x3 + cx2 x1x2 − bx3 x1x3 + dx4
冣
then .
共15兲
When a = 36, b = 3, c = 20, system 共15兲 has a periodic orbit for −1.03艋 d 艋 −0.46, a chaotic attractor for −0.46⬍ d 艋 −0.35, and a hyperchaotic attractor for −0.35⬍ d 艋 1.30. In what follows, we will take the unified chaotic system and the hyperchaotic Lü system as node dynamics to illustrate our proposed method for generalized outer synchronization. For brevity, we always take P and Q as identity matrices with proper dimensions. In all the following simulations, we assume the drive network is a star network, while the response network is a directed ring network, as shown in Fig. 1. All the coupling strength is set to be 1 and the network size N is taken as 50.
冢 冣 1 0 0
Di共xi兲 = 0 1 0 . 0 0 1 Furthermore, m共A + A / 2兲 = 31, m共Q + Q / 2兲 = 0, which gives k* = 112.6. With the parameters specified above, the controllers are designed according to the control law 共4兲. The initial values for the ith node in the drive and response networks are set to be xi共0兲 = 共0.1i , −0.2i , 0.3i兲 and y i共0兲 = 共0.2i , −0.3i , 0.4i兲, respectively. Let E共t兲 = 兺Ni=1储y i共t兲 − i共xi共t兲兲储 denote the synchronization error between the two networks. Figure 2 shows the phase diagrams of node 5 in the two networks without control. Figure 3共a兲 displays the evolution of E共t兲 along time t without control. When the control law is imposed, the synchronization error quickly tends to zero, as displayed in
013109-6
Chaos 19, 013109 共2009兲
Wu, Zheng, and Zhou 30
20
25
xi2
10 20 E(t)
0
−10
15
−20 10
xi1 0
10
30 −10
(a)
20 x i3
10
5
(a) 0
5
10
t 15
20
25
30
10
20 10
8
y
i2
0
6 E(t)
−10 −20
4
10 0 y
i1
(b)
−10
10
20
30
40
2
yi3
FIG. 2. 共Color online兲 Phase diagrams of node 5 without control. 共a兲 Node 5 in the drive network with = 兩sin 5兩 = 0.9589; 共b兲 Node 5 in the response network with = 兩sin 52兩 = 0.1324.
Fig. 3共b兲 for k = 5. This further indicates that k 艌 k* is only a sufficient condition, and a small value of k can also render a fast synchronization.
Still take the unified chaotic system as node dynamics, with for the ith node being 兩sin i兩 and 兩sin i2兩 in the drive and response networks, respectively. Define two different types of maps i as i = 1,2, . . . ,
2
4
6
t
8
10
FIG. 3. 共Color online兲 Complete synchronization errors of the drive and response networks, where the ith node in the drive and response networks is a unified chaotic system 共14兲 with = 兩sin i兩 and = 兩sin i2兩, respectively. 共a兲 without control; 共b兲 control imposed with k = 5.
冢
0
1
0
冣
Di共xi兲 = xi2 xi1 0 , 1 0 2
B. Generalized synchronization with uniform node dimension in the drive and response networks
yi = i共xi兲 = 共xi兲 = 共2xi1,xi2 + 1,x2i3兲,
0
(b) 0
i=
N N + 1, + 2, . . . ,N. 2 2
With the expressions above, we design controllers ui 共i = 1 , 2 , . . . , N兲 according to the control scheme 共4兲. The initial conditions are set as the same as those in the previous subsection. The synchronization error with k = 3 is shown in Fig. 4. It is seen from the figure that the generalized outer
N 2
70 60
and
50
yi = i共xi兲 = 共xi兲 = 共xi2,xi1xi2,xi1 + 2xi3兲,
30 20
Then
冢
2 0
0
冣
Di共xi兲 = 0 1 0 , 0 0 2xi3 and
40 E(t)
N N i = + 1, + 2, . . . ,N. 2 2
10
N i = 1,2, . . . , ; 2
0 0
0.5
1
1.5 t
2
2.5
3
FIG. 4. 共Color online兲 Generalized synchronization error between the drive and response networks with k = 3.
013109-7
Chaos 19, 013109 共2009兲
Generalized outer synchronization
180 160 140
10
E(t)
x
i2
120 100
0
−10
80 60 40
10 x
20
i1
0 −10
(a)
0 0
0.5
1
1.5 t
2
2.5
3
FIG. 5. 共Color online兲 Generalized synchronization error between the drive and response networks with k = 15, where each node in the drive network displays a hyperchaotic Lü attractor with a = 36, b = 3, c = 20, d = −0.4.
yi = i共xi兲 = 共2ixi1,xi2 + 0.5xi4,xi3 − i兲,
0
xi1 0
−10
(b)
10
x
10
15 y
冣
The controllers ui 共i = 1 , 2 , . . . , N兲 are then designed according to the control law 共4兲. Select the initial values as xi共0兲 = 共−0.1i , −0.2i , 0.3i , 0.4i兲 in the drive network and yi共0兲 = 共0.2i , −0.3i , 0.4i兲 in the response network. To begin with, let all the nodes in the drive network be the identical hyperchaotic Lü system with a = 36, b = 3, c = 20, d = −0.4, where each node displays a chaotic attractor. Figure 5 plots the generalized outer synchronization error E共t兲 along time t, with k = 15. Figure 6 shows the dynamics of node 5 in the drive and response networks, where projections on different phase space are displayed. Next, we assume that each node in the drive network is distinct. Assume the ith node is a hyperchaotic Lü system with a = 36, b = 3, c = 20 but d = −1 + 共2.3i / N兲. So d varies from about −1 to 1.3, and the nodes transit from a periodic orbit to a hyperchaotic attractor, as discussed previously. Figure 7 displays the generalized outer synchronization error E共t兲 with k = 15, which tends to zero
30
i3
i2
y
0
−20
100 i1
冢
25
20
i = 1,2, . . . ,N.
Di共xi兲 = 0 1 0 0.5 . 0 0 1 0
20
15
y 0
0
i3
10
Therefore, 2i 0 0
20 x
−50
C. Generalized synchronization with different node dimensions in the drive and response networks
In this subsection, we consider the hyperchaotic Lü system as node dynamics in the drive network, and the unified chaotic system as node dynamics in the response network, where for the ith node is still taken as 兩sin i2兩. The maps i are defined variously for different nodes, with
15
30
50
xi4
synchronization is attained after a quite short transient period. Remark 4: In this simulation, though there are two different forms of the maps i共xi兲 which vary for nodes, the generalized outer synchronization is achieved under our proposed scheme. For more different types of the maps i共xi兲, similar work can be generalized easily.
10
25
(c)
−100
5
20
25
i3
FIG. 6. 共Color online兲 Phase diagrams for node 5, where 共y i1 , y i2 , y i3兲 = 共2ixi1 , xi2 + 0.5xi4 , xi3 − i兲 with i = 5. 共a兲 projection in the 共xi3 , xi1 , xi2兲-phase space of node 5 in the drive network; 共b兲 projection in the 共xi3 , xi1 , xi4兲-phase space of node 5 in the drive network; 共c兲 controlled node 5 in the response network with = 兩sin 52兩 = 0.1324.
after a short transient period. To take a clearer view of the relationships between dynamics of nodes in two networks, we also depict some corresponding subvariables of node 25 in the xy plane, as shown in Fig. 8, where transients are discarded. V. CONCLUSIONS
Synchronization within complex networks has been extensively studied in the past decade. However, investigation on synchronization between two networks 共called outer synchronization兲 is still at the initial stage. To the best of our knowledge, there have been only a few papers in the literature that focus on complete outer synchronization between
013109-8
Chaos 19, 013109 共2009兲
Wu, Zheng, and Zhou
200
outer synchronization. When two complex networks have the same topological structures or identical dynamics, the proposed control scheme for achieving generalized outer synchronization reduces to a simpler form. The applicability of the theoretical findings has been validated by the computer simulations.
E(t)
150
100
ACKNOWLEDGMENTS
50
0 0
0.5
1
1.5 t
2
2.5
3
FIG. 7. 共Color online兲 Generalized synchronization error between the drive and response networks with k = 15, where the ith node in the drive network displays a hyperchaotic Lü attractor with a = 36, b = 3, c = 20, and d = −1 + 共2.3i / N兲.
two networks, where it is required that all the nodes should have the same dynamical behaviors. In this paper, we have investigated generalized outer synchronization between two complex dynamical networks with different topologies and diverse node dynamics. We have proposed a nonlinear control scheme which is guaranteed to achieve this generalized
800 600
yi1
400 200 0 −200 −400 −600 −10
(a)
−5
0 x i1
5
10
15
10 5
y
i3
0
−5 −10 −15
(b)
10
15
20 x
i3
25
30
35
FIG. 8. 共Color online兲 Relationships between the subvariables for node 25 in the drive and response networks. 共a兲 y i1 = 2ixi1 for i = 25; 共b兲 y i3 = xi3 − i for i = 25.
The authors wish to thank the anonymous reviewers for their helpful comments and suggestions. This work was supported in part by a research grant from the Australian Research Council and in part by the Chinese National Natural Science Foundation 共Grant Nos. 60804039, 70771084, and 60574045兲. D. J. Watts and S. H. Strogatz, Nature 共London兲 393, 440 共1998兲. S. H. Strogatz, Nature 共London兲 410, 268 共2001兲. 3 A. L. Barabási, R. Albert, and H. Jeong, Physica A 272, 173 共1999兲. 4 H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai, and A.-L. Barabási, Nature 共London兲 407, 651 共2000兲. 5 S. Goto, T. Nishioka, and M. Kanehisa, Bioinformatics 14, 591 共1998兲. 6 A. B. Horne, T. C. Hodgman, H. D. Spence, and A. R. Dalby, Bioinformatics 20, 2050 共2004兲. 7 L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 80, 2109 共1998兲. 8 M. Barahona and L. M. Pecora, Phys. Rev. Lett. 89, 054101 共2002兲. 9 Y. H. Chen, G. Rangarajan, and M. Z. Ding, Phys. Rev. E 67, 026209 共2003兲. 10 C. W. Wu and L. O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 42, 430 共1995兲. 11 C. W. Wu, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 48, 1257 共2001兲. 12 C. W. Wu, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 50, 294 共2003兲. 13 C. W. Wu, IEEE Trans. Circuits Syst., I: Regul. Pap. 52, 282 共2005兲. 14 C. W. Wu, Synchronization in Coupled Chaotic Circuits and Systems 共World Scientific, Singapore, 2002兲. 15 X. F. Wang and G. Chen, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 49, 54 共2002兲. 16 X. F. Wang and G. Chen, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 187 共2002兲. 17 J. Lü and G. Chen, IEEE Trans. Autom. Control 50, 841 共2005兲. 18 J. Zhou, J. Lu, and J. Lü, IEEE Trans. Autom. Control 51, 652 共2006兲. 19 W. Lu, Chaos 17, 023122 共2007兲. 20 C. Li, W. Sun, and J. Kurths, Phys. Rev. E 76, 046204 共2007兲. 21 For predator-prey relationships, visit the website http://en.wikipedia.org/ wiki/Predation and links therein. 22 For intestinal balance, visit the website http://life-enthusiast.com/index/ Education/NutritionPrinciples. 23 H. Tang, L. Chen, J. Lu, and C. K. Tse, Physica A 387, 5623 共2008兲. 24 N. F. Rulkov, M. M. Sushchik, and L. S. Tsimring, Phys. Rev. E 51, 980 共1995兲. 25 L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 共1996兲. 26 J. Meng and X. Wang, Chaos 18, 023108 共2008兲. 27 D. V. Senthikumar, M. Lakshmanan, and J. Kurths, Chaos 18, 023118 共2008兲. 28 P. A. Regalia and M. K. Sanjit, SIAM Rev. 31, 586 共1989兲. 29 H. K. Khalil, Nonlinear Systems, 3rd ed. 共Prentice-Hall, Englewood Cliffs, NJ, 2002兲. 30 E. N. Lorenz, J. Atmos. Sci. 20, 130 共1963兲. 31 H. Richter, Chaos, Solitons Fractals 12, 2375 共2001兲. 32 I. Stewart, Nature 共London兲 406, 948 共2000兲. 33 G. Chen and T. Ueta, Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, 1465 共1999兲. 34 G. Q. Zhong and W. K. S. Tang, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 1423 共2002兲. 35 J. Lü, G. Chen, and S. Zhang, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 1001 共2002兲. 36 J. Lü, G. Chen, D. Cheng, and S. Čelikovsky, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 2917 共2002兲. 1 2
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Y. X. Li, W. K. S. Tang, and G. Chen, Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, 3367 共2005兲. 41 A. Chen, J. Lu, J. Lü, and S. Yu, Physica A 364, 103 共2006兲. 42 D. Li, J. Lu, X. Wu, and G. Chen, J. Math. Anal. Appl. 323, 844 共2006兲.
CHAOS 19, 013110 共2009兲
Regular and chaotic dynamics of magnetization precession in ferrite–garnet films Anatoliy M. Shuty and Dmitriy I. Sementsov Ulyanovsk State University, ul. L. Tolstogo 42, Ulyanovsk 432970, Russia
共Received 16 August 2008; accepted 7 January 2009; published online 10 February 2009兲 By numerically solving equations of motion and constructing the spectrum of Lyapunov exponents, nonlinear dynamics of uniformly precessing magnetization in 共110兲 thin film structures with perpendicular magnetic bias is investigated over a wide frequency range of the alternating field. Bifurcational changes in magnetization precession and the states of dynamical bistability are discovered. Conditions for the realization of high-amplitude regular and chaotic dynamic regimes are revealed. The possibility of controlling those precession regimes by using external magnetic fields is shown. The features of time analogs of the Poincaré section of trajectories in the chaotic regimes are studied. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3076395兴 In this paper, it is shown that the dynamics of magnetization in (110) films differs substantially from the dynamics of magnetization that was studied earlier in films with other orientations of crystallographic axes. The differences consist not only in the form of attractors of the dynamic regimes but also in the existence of two intervals of the magnetizing field corresponding to chaotic precessing regimes. A strong dependence of features of the established regimes on the direction of polarization of the alternating field is revealed. The construction of time sections of attractors in the chaotic regimes has demonstrated a strong dependence on the choice of the initial phases of the alternating field. In some cases, two sections of phase trajectory are not intersected at all. In this work, alongside with construction of bifurcation diagrams, the spectrum of the Lyapunov exponents is calculated and a number of its features for the dynamical systems under study is revealed. The use of rather simple models of magnetic dynamics allows us to conclude that the revealed features are general for a wide range of physical systems. I. INTRODUCTION
Interest in nonlinear dynamics of magnetization in magnetically ordered crystals is caused by the diversity of nonlinear effects appearing under the action of high frequency field in dissipative spin systems,1–3 and also by the possibility of achieving large precession angles and realizing dynamic chaos and different static and dynamic self-organizing structures.4–6 Interest in the investigation of magnetization dynamics for large precession angles is also generated by great potentials of its practical application, in particular for modulation of laser radiation, where efficiency mainly depends on the value of the precession angle.4,7 It is known that, for the perpendicular orientation of high-frequency and static fields 共transverse pumping兲, there exist two mechanisms of energy transfer from uniform precession to spin waves.5,8 These mechanisms restrain the growth of precession amplitude. The first mechanism is re1054-1500/2009/19共1兲/013110/14/$25.00
lated to a three-magnon process, in which one magnon with wave vector k = 0 is destroyed, and there appear two magnons with wave vectors k and −k and frequency k = 0 / 2, where 0 is the resonance frequency of uniform precession. The second mechanism is related to a four-magnon process, where two magnons with k = 0 disappear, and two magnons with wave vectors k and −k and frequency k = 0 appear. When the magnitude of the high-frequency field exceeds a threshold, those processes result in the development of spinwave instabilities. As a result, both regular and chaotic nonlinear dynamic regimes can be realized in the spin system.9–12 To achieve large angles of uniform precession of magnetization, one has to meet such conditions that the Suhl instabilities caused by the three- and four-magnon interactions cannot develop. For this, the frequency of the alternating field should be below the lowest frequency in the spectrum of spin waves.7 For a perpendicularly magnetized thin layer, the frequency of the first 共nonuniform兲 spin-wave mode can be shifted by choosing the layer thickness far from the frequency of the uniform resonance 0 ⬍ infk⫽0 共k兲.14 Exactly because of this, the mentioned mechanisms of energy transfer from the uniform precession to spin waves are not realized in films with perpendicular magnetic bias for frequencies 艋 0. As a result, with an increase in the amplitude of the high-frequency field, there is no saturation of resonance in the uniform mode,13 and the features of nonlinear dynamics of magnetization appear already in the case of its uniform precession.14 As the analysis shows, the symmetry of the magnetic anisotropy field of the material, which is related to the crystallographic symmetry, significantly influences the nonlinear dynamic regimes of magnetization. The uniform magnetization precession in the case that the frequency of alternating field and the magnitude of magnetizing field are related by conditions of linear ferromagnetic resonance8 was investigated for the perpendicularly magnetized 共111兲 monocrystal films15 and for the 共100兲 films.16 However, concerning the realization of complex regular and chaotic precession re-
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gimes, it is promising to study the conditions that exist in the case of weaker magnetizing fields at frequencies below the frequency of linear resonance. Chaotic and quasiperiodic magnetization dynamics was studied for this case in crystals with uniaxial anisotropy14 and in the 共100兲 and 共111兲 films with cubic crystalline anisotropy for the third and fourth order symmetry axes, respectively.17,18 In the present work, by using the construction of bifurcation diagrams, the time analogs of the Poincaré sections, and the spectrum of the Lyapunov exponents, we investigate the features of nonlinear precession dynamics of magnetization that is realized under the indicated conditions in the 共110兲 films with the symmetry axis of the second order, which is one of the most standard orientations of the crystallographic axes. For these films, the expression for the energy of crystalline anisotropy is significantly different from the cases studied earlier. This should strongly affect the features of nonlinear dynamics of magnetization. In addition, due to symmetry of the structure, the precession dynamics would depend to a much greater extent on the polarization of the alternating magnetic field. The one-domain condition, i.e., homogeneous distribution of magnetization over the film, will be considered. As it is known,8 this can be achieved in a rather small film sample, whose size is determined by the saturation magnetization, the anisotropy constants, and the nonuniform exchange constant and is ⬃1 micron for the ferrites with garnet structure. Homogeneous magnetization in the case when the magnetizing field decreases below the values corresponding to saturation of the film can be maintained as a result of dynamic stabilization, which consists of the following. A homogeneously magnetized film 共by means of a rather strong magnetizing field兲 is subject to a resonant alternating magnetic field. Then the magnetizing field decreases down to values that are interesting for us, which, in particular, can correspond to static bistability of the magnetization vector. The precessing magnetization, due to strong exchange interaction, continues to be homogeneous at lower fields than in the static case. This happens mostly when the precession amplitude is large and the attractor of system does not depend on the initial orientation of magnetization 共this situation is realized in the majority of cases corresponding to the considered conditions兲. With the specified method of maintenance of a homogeneous dynamic state of magnetization, no special restrictions on sizes of the film samples are needed. II. BASIC EQUATIONS AND RELATIONS
We investigate nonlinear precession regimes of magnetization in thin ferrite-garnet films, which are widely used as magnetic elements in the integrated technology. The epitaxial ferrite-garnet films are monocrystal layers with the cubic crystal lattice. In the process of liquid-phase epitaxy, conditions for the growth of the 共110兲 films are easily realized.19 Let the 关110兴 crystallographic axis coincide with the x axis ¯ 10兴 and and be normal to the surface of the film and the 关1 关001兴 axes coincide with the y and z axes 共see Fig. 1兲. The dynamic behavior of the magnetization vector M in the external static H and alternating h magnetic fields is described by the Landau–Lifshitz equation,8,20
FIG. 1. Geometry of the system and orientation of crystallographic axes.
M ␣ M = − ␥M ⫻ Hef + M ⫻ , t M t
共1兲
where ␥ is the gyromagnetic ratio, ␣ is the dissipation parameter, and the effective magnetic field in the case of uniform distribution of magnetization over the sample is defined by the following derivative of the free energy density of the magnetic system, i.e., Hef = − ⵜMF,
共2兲
where ⵜMF = 兺eiF / M i in the Cartesian coordinates, and ei are the unit vectors along the coordinate axes 共i = x , y , z兲. The free energy of the investigated system is defined by the expression 1 I M兲 + F , F = − M共H + h兲 + 2 M共N a
共3兲
J is the tensor of demagnetizing coefficients, which where N has only one nonzero component Nxx = 4 for the film sample in the chosen coordinate system; Fa is the term taking into account the energy of the magnetic crystalline anisotropy and the growth-induced anisotropy, and, for the chosen orientation of crystallographic axes, it has the following form: Fa = K1关m2z 共m2x + m2y 兲 + 共m2x − m2y 兲2/4兴 − Kum2x ,
共4兲
where Ku and K1 are the constants of the growth-induced and crystalline anisotropies, respectively; mi = M i / M are the normalized components of the magnetization vector. For the analysis of linear ferromagnetic resonance, the dynamical equations are usually transformed from the Cartesian coordinates to the spherical coordinates, taking into account that the amplitude of magnetization in the Landau– Lifshitz equation is conserved.8 However, when a numerical analysis is needed for solving the initial equations during investigation of nonlinear dynamics of magnetization, it turns out that it is much more efficient 共less time consuming兲 to solve the following system of equations 关which are derived from Eq. 共1兲兴 for the magnetization projections in the Cartesian coordinates: M共1 + ␣2兲 F F ˙ y = 共mz + ␣mxmy兲 m − 共mx − ␣mymz兲 ␥ mx mz − ␣共1 − m2y 兲
F , m y
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FIG. 2. Isoenergetic curves of magnetization orientation for H = 共a兲 390, 共b兲 300, 共c兲 200, and 共d兲 170 Oe.
H⬜ ef = 关共H − 4 M + 2Ku/M − 2K1/M兲
M共1 + ␣2兲 F F ˙ z = 共mx + ␣mymz兲 m − 共my − ␣mxmz兲 ␥ m y mx − ␣共1 − m2z 兲
F , mz
⫻共H − 4 M + 2Ku/M + K1/M兲兴1/2 . 共5兲
m2x + m2y + m2z = 1. For small amplitudes of the microwave field 共h Ⰶ Hef兲 at frequency = r, the linear ferromagnetic resonance has small precession angles and the time dependencies mi共t兲 can be found from the linearized equations of motion 共linearized with respect to the small deviation of magnetization from the equilibrium兲. The further study will be limited to the case when orientation of the static field H is perpendicular to the surface of the film. The alternating field is assumed to be orthogonal to the static field 共H ⬜ h兲 and linearly polarized along the y or z axis. When the field H and the anisotropy constants Ku and K1 correspond to the equilibrium orientation of the vector M along the normal to the film 共mx = 1兲, the frequency of the linear resonance precession is equal to r = ␥H⬜ ef , where the effective field is expressed as
共6兲
With an increase in the amplitude of microwave field and, correspondingly, with an increase in the precession angle, the contribution of highest harmonics of the fundamental precession frequency into the dynamics of magnetization increases and the nutational motion of the vector M becomes substantial. In this case, the linear approximation is no longer sufficient for solving Eq. 共5兲, and detailed analysis of the features of precessional motion taking into account all parameters determining the state of magnetization in the film is possible only by numerical methods. III. REGIMES OF DYNAMIC ORIENTATION JUMPS
To understand the features of precession motion of the vector M it is necessary to know the static spatial relief of the free energy 共for h = 0兲. Figure 2 shows the isoenergetic curves in the projection plane of the normalized magnetic moment 共my , mz兲. The curves correspond to magnitudes of magnetizing field H = 共a兲 390, 共b兲 300, 共c兲 200, and 共d兲
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170 Oe. In curves 1–7, the free energy density F is negative and takes the following magnitudes in the units of 共−103兲 erg/ cm3: 共a兲 6.909, 6.9, 6.85, 6.7, 6.4, 6.0, 5.0; 共b兲 5.45, 5.4, 5.37, 5.3, 5.2, 5.0, 4.0; 共c兲 4.14, 4.1, 4.0, 3.895, 3.8, 3.7, 3.67; and 共d兲 3.825, 3.817, 3.75, 3.66, 3.5, 3.25, 3.155. For obtaining the dependencies shown in this figure and for further analysis, the following parameters corresponding to a real Y2.9La0.1Fe3.9Ga1.1O12 ferrite-garnet film were used: 4 M = 214.6 G, ␥ = 1.755⫻ 107 共Oe s兲−1, ␣ = 10−2, K1 = −103 erg/ cm3, and Ku = −103 erg/ cm3.21 From the presented curves it follows that, when the value of magnetizing field H ⬎ H1, there is a local energy minimum for the direction of the vector M along the film normal. This minimum disappears when the value of H decreases to H1, after which there appear two symmetric minima that are not coincident with the normal and lie along the line my = 0. A further decrease in the magnetizing field results in the fact that the minima move away from the normal 共with an increase in the component mz兲 and become more pronounced. The value of the field H1 is found from the condition H⬜ ef = 0 and takes the form H1 = 4 M −
1 共K1 + 2Ku兲. M
共7兲
For the chosen structure, H1 ⬇ 390.3 Oe. For H = H2, there is a next split in the energy minima, which results in the appearance of two pairs of minima, in each of which an angular divergence of minima 共an increase in 兩my兩兲 is observed with a further decrease in the magnetizing field. For the considered case, the value of field H2 is also found numerically: H2 ⬇ 180.2 Oe. For numerical analysis of the equations of dynamics, we used the Runge–Kutta method of the forecast-correction type. Numerical analysis shows that, in the case H ⬍ H1 for small amplitudes of the microwave field h 艋 hc共H兲 共the dependence of hc on H will be investigated in the discussion of Fig. 11兲, it is realized that a low-amplitude precession of magnetization near one of the two 共for H ⬎ H2兲 or one of the four 共for H ⬍ H2兲 energy minima mentioned above. For sufficiently large amplitudes of the alternating field 共h ⬎ hc兲 linearly polarized along the z axis and having low frequency 共 / 2 艋 1 MHz兲, which is substantially below the frequency of linear resonance, there appear dynamic orientation jumps of the vector M between two equilibrium orientations when the magnetizing field is in the interval H−1 ⬍ H ⬍ H1. In the case when polarization of the alternating field is along the y axis, analogous jumps take place for one of the two pairs of the energy minima in the field interval H−2 ⬍ H ⬍ H2. The specified dynamic regime turns out to be connected with one pair of the energy minima. The pair is determined by the initial equilibrium orientation of magnetization. Thus, when the magnetizing field decreases quasistatically down to magnitudes in the indicated interval, the dynamical bistability appears: the realization of dynamic jumps 共influenced by the alternating field兲 between energy minima of one or other pair is influenced by incidental parameters, including initial phase of the alternating field. When the magnetizing field is close to critical magnitudes and H1 ⬍ H ⬍ H+1 and H2 ⬍ H ⬍ H+2 , precession regimes similar to the dynamic jumps between
the equilibrium orientations take place as well. The feature of these regimes is a complex trajectory consisting of two symmetric multiturn regions, between which dynamic jumps occur. Note that the magnitudes of H1,2 are located significantly closer to the right-side boundaries of the corresponding intervals, i.e., H+1 − H1 ⬍ H1 − H−1 and H+2 − H2 ⬍ H2 − H−2 . Figure 3 shows the projections of the normalized magnetic moment onto the yz plane in different regimes of dynamic jumps. These regimes are established under the action of the alternating magnetic field having frequency / 2 = 1 MHz and amplitudes 共a, b兲 hz = 1 Oe, hy = 0 and 共c, d兲 hy = 1 Oe, hz = 0 for the constant of the growth-induced anisotropy 共a, b, c兲 Ku = −103 erg/ cm3, and 共d兲 Ku = 0 共here and hereinafter where not mentioned, the previous magnitude of K1 is used兲 and the magnetizing field H = 共a兲 391, 共b兲 377, 共c兲 175, and 共d兲 111 Oe. The dynamic regime realized for H = 391 Oe corresponds to the case of H ⬎ H1. A distinctive feature of this and similar regimes occurring for H1 ⬍ H ⬍ H+1 and H2 ⬍ H ⬍ H+2 is a strong elongation of the precession trajectories in the z or y axis, which is caused by substantial elongation of the isoenergetic curves in the corresponding direction. The magnitude of precession amplitude 共i.e., the maximum deviation of magnetization from the precession axis兲 in the case of H−1 ⬍ H ⬍ H1 and H−2 ⬍ H ⬍ H2 共b, c, d兲 is close to the magnitude of deviation of equilibrium states from the normal and, therefore, weakly depends on the changes in the parameters of the alternating field 共in certain limits兲. Thus, the regimes of dynamic jumps should be attributed to the self-oscillating regimes. From the presented dependencies, it is also seen that the trajectories of regimes realized for polarization of the alternating field along the y axis are elongated in the direction of the z axis, in accordance with the shape of isoenergetic curves 关see Fig. 2共d兲兴. In the case of small growth-induced anisotropy 关Fig. 3共d兲兴, there appear a third multiturn region in trajectory close to this direction 共the 关001兴 crystallographic axis兲—the precessing magnetic moment stays in a state with a small absolute magnitude of the component my. In the case when the initial equilibrium orientation of the magnetization vector is with mz ⬍ 0 and polarization of the alternating field is along the y axis, regimes with trajectories symmetric to the trajectories given in the figure are established. As is seen from the figure, the dynamic jumps are accompanied by rapidly damping high-frequency oscillations with period Tq ⬇ 2 / r, which is close to the period of linear resonance with precession axis different from the normal, and the full period of these steadystate regimes, i.e., the period of jumps, corresponds to the period of the alternating magnetic field T = Th = 2 / . IV. BIFURCATION DIAGRAMS
For more detailed investigation of nonlinear dynamic regimes having complex trajectories, it is convenient to construct bifurcation diagrams.22–24 In Fig. 4 such diagrams are presented in the plane 共mym ; H兲, where each value of the magnetizing field H corresponds to extremum values of the y-component of the normalized magnetic moment, mym, which precesses under the action of the alternating field with parameters hz = 0, hy = 共a , c兲 1, 共b, d兲 2 Oe, / 2 = 1 MHz,
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FIG. 3. Projections of normalized magnetic moment making dynamic orientation jumps under the action of alternating magnetic field; 共a, b兲 h = hz = 1 Oe, 共c, d兲 h = hy = 1 Oe; 共a, b, c兲 Ku = −103 erg/ cm3, 共d兲 Ku = 0; H = 共a兲 391, 共b兲 377, 共c兲 175, and 共d兲 111 Oe.
and Ku = 共a , b兲 0, 共c, d兲 −103 erg/ cm3. In this case, if one value of the magnetizing field corresponds to only two points in the bifurcation diagram 共my max and my min兲, then the regular precession regime is realized, whose nonlinear character shows itself only in the nutational motion. If, otherwise, one value of H corresponds to a multitude of points 共more than two兲, then the regime of dynamic jumps is established. For magnitudes of H located in the right side of the interval corresponding to the jumps 共i.e., for large magnetizing fields兲, a precession with axis directed along the normal is realized, and on the left side of the indicated interval, the precession is low-amplitude with the axis different from the normal. It is seen from the figure that a decrease in the value of magnetizing field results in an insignificant growth in the amplitude of the dynamic regimes. A decrease in the growthinduced anisotropy 共in the value of 兩Ku兩兲 and an increase in the amplitude of the alternating field result also in an increase in the precession amplitude, which is accompanied in addition by broadening of the interval of values of H that corresponds to the regimes of dynamic jumps. However, even when hy is doubled, the amplitude of the regimes increases only by 10%–20%. It is also seen that, for a small growth-induced anisotropy 关Figs. 4共a兲 and 4共b兲兴, the interme-
diate multiturn region of trajectory 共i.e., the region with small values of 兩my兩兲 that is shown in Fig. 3共d兲 is realized only for larger magnitudes of the magnetizing field with the field H in the interval corresponding to the regimes of jumps. The bifurcation diagrams for the case of polarization of the alternating field directed along the z axis are similar to the diagrams in Figs. 4共c兲 and 4共d兲 when they are constructed for extreme values of the z-component of the normalized magnetic moment. For the construction of the bifurcation diagram presented here and below, the magnetization vector in the numerical modeling was brought to a steady-state regime, i.e., regime, in which an increase in the time of observation over the system does not change its attractor 共within the limits of accuracy of the computational procedure兲. The time during which the extremum magnitudes of my were registered was taken to be equal to the period of the regular dynamic regime. For the chaotic regimes considered further, this time was chosen so that the shape of the corresponding chaotic attractor was fully revealed 关in most cases this time is t ⬇ 共50– 100兲 / 兴. In this case the shape of the attractor does not change with an increase in t, and only the density of the corresponding points in the bifurcation diagram increases.
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FIG. 4. Bifurcation diagrams for the regimes of the dynamic orientation jumps: dependence of extremum magnitudes of the y-component of the normalized magnetic moment on the value of the magnetizing field; h = hy = 共a , c兲 1, 共b, d兲 2 Oe; / 2 = 1 MHz; Ku = 共a , b兲 0, 共c, d兲 −103 erg/ cm3.
The regimes of dynamic jumps are established, as was mentioned above, only for quite small frequencies of the alternating field. For higher frequencies and the same magnitudes of h and of the magnetizing field, they realized other high-amplitude regimes of precession 共deviation of magnetization from the normal my,z ⬎ 0.1兲, including chaotic ones. In Fig. 5, two points 共for a fixed value of H兲 correspond to a regular oscillating regime with one maximum mi max 共where i = y , z兲 and one minimum mi min; a larger finite number of points corresponds to a more complex oscillation; and a multitude of closely located points corresponds to the chaotic magnetization dynamics. The simulation was carried out with the following parameters: the amplitude of the alternating field 共a, b, c, d兲 h = hy = 1 Oe and 共e, f兲 h = hz = 1 Oe, its frequency / 2 = 10 MHz, and the constant of growthinduced anisotropy Ku = 共a , b兲 0, 共c, d, e, f兲 −103 erg/ cm3. It is seen that when approaching the zone of the chaotic dynamics from the side of large values of the magnetizing field, it is observed first an increase in the amplitude of regular precession, which is accompanied by a complication of the trajectory of precession. This effect becomes stronger with an increase in the growth-induced anisotropy of the film 共c, d兲. After the appearance of chaotic character, a further de-
crease in H results in a continuation of growth of the precession amplitude. In the considered range of the field H 共close to H2兲, the precession amplitude along the y axis significantly exceeds the amplitude along the z axis independently of the polarization of the alternating field. From the side of the small magnitude of the magnetizing field, the zone of chaotic regimes is bounded by low-amplitude regular oscillations. Here, the precession amplitude sharply decreases. In the chaotic zone itself, the regions of chaotic regimes alternate with the regions corresponding to regular regimes. The chaotic dynamics, as seen from the figure, is formed in many cases through “smearing” of attractors corresponding to the regular regimes. For h = hy ⫽ 0, the interval for the value of the magnetizing field in which chaotic and regular dynamic regimes with complex trajectories are realized is approximately equal to the interval corresponding to the regimes of dynamic jumps. In the case of polarization of the field along the z axis, this interval is significantly narrower 共e, f兲. An increase in the amplitude of the alternating field results in broadening of the chaotic zone and its complication, in particular, the number and the width of regions with complex regular precession regimes increase.
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FIG. 5. Bifurcation diagram for the high-amplitude chaotic and regular regimes: field dependence of extrema of two components of precessing magnetization; 共a, b, c, d兲 h = hy = 1 Oe, 共e, f兲 h = hz = 1 Oe; / 2 = 10 MHz; Ku = 共a , b兲 0, 共c, d, e, f兲 −103 erg/ cm3.
Figure 6 shows a bifurcation diagram in the plane 共mym ; 兲 for magnitudes of the magnetizing field H = 共a兲 175, 共b兲 180 Oe, and amplitude of the alternating field h = hy = 1 Oe; the constant of growth-induced anisotropy here and below is Ku = −103 erg/ cm3. As in the previous diagram, a multitude of closely located points for a fixed value of corresponds to the chaotic magnetization dynamics 共except
for the region of / 2 ⬃ 1 MHz, where the regime of dynamic jumps is realized兲. In the case of a comparatively small value of the field H, when the frequency of the alternating field increases up to magnitude / 2 ⬃ 10 MHz, the trajectory corresponding to the regimes of jumps is continuously “smeared” and is transformed into a trajectory of a regime with more pronounced chaotic character 关Fig. 6共a兲兴.
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the attractor into the chaotic regimes, to which narrower frequency regions correspond. It is seen from Fig. 4 that the interval of the field H corresponding to the regimes of dynamic jumps is divided into two parts, one of which corresponds to the case shown in Fig. 6共a兲 and the other corresponds to Fig. 6共b兲. V. REGULAR PRECESSION REGIMES AND BISTABLE STATES
FIG. 6. Bifurcation diagrams: dependence of extremum magnitudes of the y-component of normalized magnetic moment on frequency of alternating field; H = 共a兲 175, 共b兲 180 Oe; h = hy = 1 Oe; Ku = −103 erg/ cm3.
Through growing frequency intervals, narrow 共and also growing兲 zones of corresponding to the regular regimes are observed. A change in frequency up to / 2 ⬵ 170 MHz insignificantly increases the amplitude of the chaotic and regular regimes. The largest increase in precession amplitude accompanies an increase in frequency at the end 共from the side of large values of 兲 of the frequency interval corresponding to the high-amplitude regular regimes. Note that the high-amplitude regular regime, which precedes the chaotic regimes realized for / 2 ⬵ 145– 170 MHz, can also be established at frequencies higher than it is shown in Fig. 6共a兲, but for this the initial state of the magnetization vector should not be in equilibrium. To overcome the zone of the strange attractor and to be trapped by the attractor corresponding to the indicated regular regime, the magnetization vector should be either shifted to the latter attractor or it should be given a sufficient initial acceleration. Thus, there is a dynamic bistability consisting of attractors corresponding to the regular and chaotic regimes. The dynamic bistability with several regular regimes is also observed at higher frequencies corresponding to the low-amplitude precession. In the figure, the numbers mark the points corresponding to different regimes constituting the dynamic bistability. In the given case, realization of one or another regime 共“1,” “2,” or “3”兲 can be influenced by fluctuations in some parameter of the system or in the initial phase of the alternating field. For sufficiently large magnetizing fields 关Fig. 6共b兲兴, the regimes of dynamic jumps are not transformed through relatively continuous “smearing” of
The forms of attractors for high amplitude regular regimes are determined by the isoenergetic surfaces, since the larger part of the trajectory of the magnetization vector approximately follows this surface. Trajectories of the high amplitude regular regimes realized under the considered conditions are determined by the form of isoenergetic surfaces. Near the magnitude H2, large amplitudes of the magnetization precession are achieved in the cases of polarization of the alternating field directed in the y and z axes. In the other interval of magnetizing field, i.e., near H1, the precession amplitudes realized for polarization of the alternating field along the z axis are several times larger than the amplitudes realized for h = hy. The presence of regions where magnitudes of parameters 共in particular, frequency of the alternating field兲 correspond to dynamical bistability is characteristic for both the intervals of the field H. As an example, Fig. 7 shows the projections of trajectories of the regular precession regimes of magnetization vector onto the yz plane. These regimes are established for magnitudes of growth-induced anisotropy Ku = 共a兲 0 and 共b, c, d兲 −103 erg/ cm3, amplitude of the alternating field h = hy = 1 Oe, and various magnitudes of its frequency and the magnetizing field: 共a, 1 and 2兲 / 2 = 85 MHz, H = 110 Oe; 共a, 3 and 4兲 / 2 = 110 MHz, H = 107 Oe; 共b, 1 and 2兲 H = 180 Oe, / 2 = 120 MHz; 共b, 3 and 4兲 H = 180 Oe, / 2 = 170 MHz; and 共c, d兲 / 2 = 10 MHz, H = 176, 179 Oe. The regimes corresponding to curves 1 and 2, and also to curves 3 and 4 are the states of dynamic bistability: realization of one or another regime 共“1” or “2,” “3” or “4”兲 in this case was influenced by the initial phase of the alternating field, which is written in the form h共t兲 = h sin共 + t兲.
共8兲
The considered regimes are obtained for initial phase 共a, 1 and 3; b, 1 and 3兲 = 0 and 共a, 2 and 4; b, 2 and 4兲 / 2. It is seen that the regimes constituting the dynamic bistability may have very different precession amplitudes. In addition, these regime may have different periods, which are either equal to the period of the alternating field 共a, 2, 4; b, 1, 2, 4兲 T = Th or multiples of it: T = nTh, where 共a, 3兲 n = 2 and 共a, 1; b, 3兲 n = 3. The characteristic trajectories of the high-amplitude regular regimes are also trajectories with two close symmetric multiturn regions, i.e., they are similar to the trajectories of the regimes of dynamic jumps but have a significantly smaller number of turns. In Figs. 7共c兲 and 7共d兲, the projections of trajectories for such regimes are given. Note that the period of the regime in case 共c兲 is equal to the period of the alternating field, and there is period doubling in case 共d兲, i.e., the trajectory consists of two closely located parts, and the full period of this regime turns out to be equal to T = 2Th.
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FIG. 7. Projections of trajectories of regular precession regimes of magnetization vector that are realized in the condition of dynamical bistability for various and H 共a, b兲, and are established for / 2 = 10 MHz; H = 共c兲 176, 共d兲 179 Oe; h = hy = 1 Oe; Ku = 共a兲 0, 共b, c, d兲 −103 erg/ cm3.
In the case when polarization of the alternating field is along the z axis, the trajectories of the high-amplitude regular regimes for the value of the magnetizing field close to H2 are similar 共except for somewhat larger asymmetry兲 to the above presented trajectories having small number of turns. For fields close to H1, these trajectories are similar to the trajectories of dynamic jumps 关see Fig. 3共b兲兴. VI. CHAOTIC REGIMES AND SPECTRUM OF LYAPUNOV EXPONENTS
For the analysis of chaotic regimes, the spectrum of the Lyapunov exponents 共LEs兲 i is highly informative.22 These exponents are introduced as follows. At the initial moment of time, a region having small radius with its center at the attractor is chosen. The region moves along the attractor and at the moment of time t has semiaxes li in N basic dimensions, 兵l1,l2, . . . ,lN其 = 关 exp共1t兲, exp共2t兲, . . . , exp共Nt兲兴,
共9兲
where the number of LEs is equal to the number of dimensions of the phase space. A chaotic regime occurs when one of the exponents i is positive; at the same time, the sum of
LEs should be negative 共the attractor exists in the phase space when this condition is satisfied兲. The dynamical system of Eq. 共5兲 has the variables my, mz, and mx. However, only two of them are independent. The calculation was carried out according to the procedure described by Benettin et al.25 In the initial system, the expres˙ x is written in the form that is similar sion for the derivative m to the expressions for the other two components of magnetization. We solve this system jointly with variation of each variable 共the relation between variations ␦mx, ␦my, and ␦mz is not used in an explicit form for this numerical analysis兲. Figure 8 shows the dependencies of the following quantities on the value of the magnetizing field: the three LEs, the sum of the exponents S = 兺i 共dashed curves兲, and the quantity A = 关共my max − my min兲2 + 共mz max − mz min兲2兴1/2 ,
共10兲
which characterizes the amplitude of the precession regime. The alternating field with amplitude h = 1 Oe and frequency / 2 = 10 MHz is polarized along the z axis, and the growth-induced anisotropy Ku = −103 erg/ cm3. It is seen from this figure that the high-amplitude precession regimes for such polarization of the alternating field correspond to
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returning to the regular regimes, opposite changes in the LEs occur with conservation of the value of S. Note that the conservation of the sum of the LEs is rather accurately satisfied in small intervals of parameter controlling the process 共in the case considered here, this parameter is the value of magnetizing field兲, whereas the value of S undergoes an insignificant monotonic change in a large interval. Small jumps in the sum of LEs, as follows from the presented dependencies, occur only during transition between the high-amplitude precession regime and the low-amplitude regime with precession axis different from the normal 共see Fig. 5兲. In the considered case, the Lyapunov dimension of chaotic attractors 共taking into account that 2 = 0兲 is defined by the following expression: r=2+
FIG. 8. Dependencies of the Lyapunov exponents i, their sum 共dashed curves兲, and the value of A, which characterizes the amplitude of precession regime, as functions of magnetizing field; h = hz = 1 Oe; / 2 = 10 MHz; Ku = −103 erg/ cm3.
two intervals of the magnetizing field: intervals near the values H1 and H2. For polarization of the field h along the y axis, large amplitudes of both the regular and chaotic magnetization precession are realized only for magnetizing fields close to the value H2; however, the amplitudes achieved in this case are almost twice larger than the amplitudes of the regimes excited by the field polarized along the z axis. From the analysis of LEs, the following conclusion can be made. One of the exponents is equal to zero. This reflects the fact that the norm of the vector M is a constant. In this case, two close trajectories with slightly different norms ˜ 兩 = 1 + , where Ⰶ 1 共the two trajectories coin兩m兩 = 1 and 兩m cide after appropriate normalization of the perturbed vector ˜ 兲, does not diverge nor converge. If the time is considered m as an independent variable, one more LE can be obtained, which is also equal to zero, since it is related to the perturbation shifting along the attractor trajectory. In the case of regular precession regimes, one of the exponents is equal to zero 共we designate it as 1兲, and the other two exponents are negative and are usually equal to each other 共2 = 3兲 for the simplest attractors. When one of the negative LEs increases 共we designate it as 2兲 the other exponent 共3兲 decreases so that their sum is constant. The transition to the chaotic regimes is accompanied by an increase in the exponent 2 up to zero, an increase in 1, which becomes positive, and a decrease in the exponent 3 such that the sum of all exponents, S, is constant. The positive LE is responsible for the exponential divergence of two points 共in the phase plane兲 that were initially close to each other and belonged to an attractor of the system. Therefore, this LE is a characteristic of the chaotic precession dynamics. When
1 1 =2+ . 兩3兩 兩S − 1兩
共11兲
Since S ⬇ const, the dependence r共H兲 qualitatively repeats the dependence 1共H兲. With an increase in the positive exponent 1, the dimension of attractors increases. “Narrow” attractors have the smallest dimension close to 2, in particular, when the chaotic character shows itself in “smearing” of the trajectory of regular regimes, to which these attractors are close. In many cases such regimes are realized for values of the parameter 共values of H兲 corresponding to the edge of the regions related to chaotic regimes. High fractal dimension is observed in “wide” attractors with a pronounced chaotic character. For example, Fig. 9 shows the projections of trajectories of chaotic regimes corresponding to the given LEs and established for H = 共a兲 179.37, 共b兲 179.6, 共c兲 380.9, and 共d兲 381 Oe. The dimensions of attractors corresponding to the given regimes are r ⬇ 2.004, 2.126, 2.325, 2.135. It is seen that a small change in the field H can result in a significant change in the chaotic attractor along with a change in the value of the Lyapunov dimension. Since the intervals of magnetizing field corresponding to the chaotic regimes and the regime of dynamic jumps in most cases almost coincide, the calculation of the dependence of LEs on various parameters allows obtaining the corresponding dependencies for the values H−1,2 and H+1,2 mentioned above. In Fig. 10, a diagram in the axes 共H , h兲 is presented where points mark the magnitudes of magnetizing field and the amplitudes 共h = hy兲 of alternating field corresponding to the case of the largest positive LE and, therefore, also corresponding to realization of the chaotic dynamic regime. Unmarked points in the diagram correspond to the values of parameters for which the regular magnetization dynamics is established. In the calculation, the previous magnitudes for the frequency of alternating field and the growthinduced anisotropy constant were used. The presented diagram consists of regions similar to each other and having the same fractal geometry but different sizes. It is seen that, in the process of any continuous change in the value of magnetizing field, the system can avoid the chaotic dynamics of magnetization when the amplitude of alternating field is tuned in coordination with the change in magnetizing field. The field magnitude H+2 limiting the high-amplitude precession regimes from the side of larger magnetizing fields 共in the figure, this magnitude is adjoint to the right-side bound-
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FIG. 9. Projections of trajectories of chaotic regimes; h = hz = 1 Oe; / 2 = 10 MHz; H = 共a兲 179.37, 共b兲 179.6, 共c兲 380.9, 共d兲 381 Oe; Ku = −103 erg/ cm3.
ary of the diagram兲 is almost independent of h. The approximate dependence of H−2 共h兲 shown in the diagram by the dashed line 共and, therefore, the interval width of the field H corresponding to the high-amplitude regimes兲 is linear in the
FIG. 10. Bifurcation diagram of chaotic regimes: points mark the magnitudes of magnetizing field and the amplitudes of alternating field corresponding to the largest positive Lyapunov exponent of attractor of precession regimes; h = hy; / 2 = 10 MHz; Ku = −103 erg/ cm3.
case of sufficiently large h, and H−2 ⬃ h1/2 in the case of a small alternating field. For the magnitudes of parameters that are positioned in the diagram below the line, the lowamplitude regular precession regimes are realized, and, above the line, the high-amplitude 共both chaotic and regular兲 dynamics or the regime of dynamic jumps is realized in the case of low frequency of alternating field. Thus, the indicated separating line corresponds 共with small deviations depending on frequency兲 to the dependence hc共H兲 mentioned in Sec. III. It is seen that the dependence hc ⬃ H2 is realized for the magnitudes of magnetizing field close to H2, and this dependence approximates to a linear function with a decrease in H. The diagram 共H , h兲 constructed for the field polarized along the z axis is similar; therefore, the dependencies H+1 共h兲 and H−1 共h兲 and also hc共H兲 are similar to the aforementioned dependencies. In the case of an increase in the growth-induced anisotropy with other conditions unchanged, a shift of the considered field interval proportional to Ku is observed, accompanied by its minor broadening. In other words, the dependencies H+1,2共Ku兲, H−1,2共Ku兲, and H1,2共Ku兲 are linear with similar proportionality factors. In particular, for Ku = −共0.5, 1.0, 1.5兲 ⫻ 102 erg/ cm3, we have H−1 ⬇ 310, 368, 426 Oe and H+1 ⬇ 330, 389, 447 Oe for h = hy
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FIG. 11. Discrete in time 共⌬t = / 兲 representation of projections onto the yz plane of trajectories of magnetization for h = hz = 2 Oe; / 2 = 共a兲 7, 共b兲 40, 共c兲 90, 共d兲 230 MHz; H = 370 Oe; Ku = −103 erg/ cm3.
= 2 Oe and H−2 ⬇ 136, 169, 203 Oe and H+2 = 146, 181, 215 Oe for h = hz = 2 Oe, respectively. The dependencies H+1 共K1兲, H−1 共K1兲, and H1共K1兲 are also linear and similar to the corresponding dependencies on Ku. The dependencies H+2 共K1兲, H−2 共K1兲, and H2共K1兲 have a different character; H2 weakly depends on the crystalline anisotropy constant, and the width of the interval 兩H−2 − H+2 兩 becomes rapidly narrower with an increase in the crystalline anisotropy in the region of small values of 兩K1兩 and becomes slowly narrower in the region with large values of 兩K1兩. In particular for h = hy = 2 Oe, Ku = −103 erg/ cm3, and K1 = −共0.2, 0.4, 0.8, 1.2兲 we have H+2 − H−2 ⬇ 45, 25, 13, 7 Oe, ⫻ 103 erg/ cm3, respectively. VII. TIME SECTIONS OF ATTRACTORS IN THE CHAOTIC REGIMES
For practical purposes and for more detailed investigation of the chaotic precession regimes, the phase trajectories corresponding to these regimes should be represented as a set of points obtained in equal time intervals, in particular, intervals equal to the period or the half-period of the alternating field 共an analog of the Poincaré section18,26,27兲. Figure 11
shows a representation of projection of the trajectories of magnetization onto the yz plane in discrete time 共with a step of ⌬t = / 兲 for h = hz = 2 Oe, / 2 = 共a兲 7, 共b兲 40, 共c兲 90, and 共d兲 230 MHz, H = 370 Oe, and Ku = −103 erg/ cm3. It follows from the figure that, due to a change in frequency for a constant magnitude of magnetizing field, a significant change in the shape of the chaotic attractor can also be achieved with an insignificant change in precession amplitude. An important feature of the chaotic regimes realized in the considered system is a strong dependence of the corresponding time sections on the phase of alternating field, , at which these sections are obtained. This dependence can influence mainly the shape of the section, affecting only weakly the occupied region of phase space; however, it can also strongly change this region and can result in the fact that the sections of the same phase trajectory of chaotic regime taken at different phases of the alternating field have very different sizes and do not even intersect in some cases. In the numerical modeling, different sections 共of the same regime兲 are realized by the choice of different phases of the alternating field. The sections in Fig. 11 are constructed for phase = / 2. For = 0, the sections corresponding to the cases
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FIG. 12. 共a, b兲 Projections of trajectories and 共c, d兲 the corresponding time 共⌬t = / 兲 sections of chaotic attractors taken for phase of alternating field = 0 共Sec. I兲 and = / 2 共Sec. II兲; / 2 = 10 MHz; h = hy = 2 Oe; 共a, c兲 H = 175 Oe, 共b, d兲 H = 389 Oe; Ku = −103 erg/ cm3.
共b兲, 共c兲, and 共d兲 differ mainly from the sections given in Fig. 11 by their shape, and the corresponding section in case 共a兲 turns out to be substantially smaller and almost fits inside the two rings of the section constructed for = / 2. Figure 12 shows 共a, b兲 the projections of the phase trajectories and 共c, d兲 the corresponding time sections 共⌬t = / 兲 of the chaotic attractors realized for / 2 = 10 MHz, h = hy = 2 Oe, 共a, c兲 H = 175 Oe, 共b, d兲 H = 389 Oe, and Ku = −103 erg/ cm3. The construction of the sections was made for phases = 0 共Sec. I兲 and = / 2 共Sec. II兲. To make the appreciation of the precession trajectories in Figs. 12共a兲 and 12共b兲 easier, the corresponding time t was chosen to be significantly shorter than the time used for obtaining the sections. It is seen that one of the sections of the dynamical regime in cases 共a兲 and 共c兲 turns out to be twice as large as the other, and the section corresponding to = 0 in cases 共b兲 and 共d兲 lies between the two parts of the section corresponding to = / 2. When a section is formed, the points corresponding to this section propagate to all parts of the section during a short period of time, and, with increasing time, the new points are placed due to phase diffusion near the earlier obtained points. When the calculation time is large, clearly shaped areas corresponding to the time section are observed,
as it is seen from the figures. Thus, the phase diffusion does not result in the filling of the whole attractor area, and there always exist areas of the attractor unoccupied by the time section corresponding to the chosen value of the phase. This fact indicates that the considered attractors can be attributed to the phase-coherent attractors. This is valid both for the attractors corresponding to the time sections with simple forms and the attractors corresponding to complex sections 共Fig. 11兲, which also have clear shape structure 共i.e., there are areas unoccupied by the section兲. In the case of 共100兲 and 共111兲 films, the attractors are also phase-coherent, but the time sections for different phases differ insignificantly and occupy close areas of the attractors. A feature of the 共110兲 films is that for these films it is possible to obtain attractors with strongest dependence of the time sections on the magnitude of the phase. We note that the calculations made for the precession regimes of interacting magnetic moments in the presence of domain structure has shown that clearly shaped areas occupied by the time sections are absent. In this case, the characteristic sections are sections with fuzzy areas of concentration. With increasing time, these sections fill—as a result of phase diffusion—the whole area of the attractor, which is phase-incoherent.
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VIII. CONCLUSION
We investigated the features of nonlinear regimes of magnetization dynamics in the 共110兲 films by constructing bifurcation diagrams and analyzing dynamical attractors of the system. For the chaotic regimes, the spectrum of Lyapunov exponents is calculated and the time sections of attractors are constructed. We showed that, at low frequencies of the microwave range, the nonlinear character of precession results in the appearance of resonance areas, where different types of bifurcations are realized. These bifurcations result in a change of dynamic regimes and also in the appearance of dynamic bistability. The most characteristic regimes are those related to orientation jumps of magnetic moment from one equilibrium orientation to another under the action of alternating magnetic field and also the highamplitude regular and chaotic regimes caused by the presence of bistable states. The amplitude of the indicated precession regimes is determined mainly by the equilibrium orientation of magnetization and depends weakly on parameters of the alternating field in a rather broad interval of these parameters. The chaotic regimes may strongly differ in the degree of chaotic behavior 共attractor width兲 and in the fractal dimension of attractor, which can be effectively controlled. Also it is possible to make transitions from the regular dynamics to the chaotic one 共and vice versa兲 by means of changing parameters of the external magnetic fields. The calculation of the spectrum of Lyapunov exponents has shown that, during transitions between the high-amplitude chaotic regime and the regular regime, the sum of the exponents is conserved. The time sections of attractors of the considered chaotic regimes depend strongly on the phase of the alternating magnetic field at which they are obtained. A feature of the 共110兲 film structures distinctive from structures having other orientations of crystallographic axes 关in particular, the 共100兲 and 共111兲 films兴 is the presence of two separated regions of the magnetizing field in which the nonlinear dynamical precession regimes mentioned above are realized. In structures of the type considered here, the precession amplitude and the shape of attractors is determined to a great extent also by the direction of polarization of the alternating field, and the time sections of the chaotic attractors depend significantly on the phase of the alternating magnetic field at which they are made. The revealed features of precession dynamics of magnetization in the considered structures allow
one to significantly increase the variety of dynamic regimes and also the methods of controlling these regimes and, correspondingly, to broaden the scope of their practical application. ACKNOWLEDGMENTS
This work was supported by a grant of the President of the Russian Federation 共N MD-3169.2007.2兲. 1
Nonlinear Phenomena and Chaos in Magnetic Materials, edited by P. E. Wigen 共World Scientific, Hackensack, NJ, 1994兲. 2 G. Bertotti, A. Magni, I. D. Mayergoyz, and C. Serpico, J. Appl. Phys. 91, 7559 共2002兲. 3 W. van Saarloos, Phys. Rep. 386, 29 共2003兲. 4 B. Neite and H. Dotsch, J. Appl. Phys. 62, 648 共1987兲. 5 S. M. Rezende and F. M. de Agular, Proc. IEEE 78, 893 共1990兲. 6 Th. Gerrits, M. L. Schneider, A. B. Kos, and T. J. Silva, Phys. Rev. B 73, 094454 共2006兲. 7 A. K. Zvezdin and V. A. Kotov, Modern Magneto-Optics and MagnetoOptical Materials 共Institute of Physics Publishing, London, 1997兲. 8 A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves 共Fizmatlit, Moscow, 1994; CRC Press, Boca Raton, 1996兲. 9 G. Gibson and C. Jeffries, Phys. Rev. A 29, 811 共1984兲. 10 H. Benner, F. Rödelsperger, and G. Wiese, “Chaotic dynamics in spinwave instabilities,” in Nonlinear Dynamics in Solids, edited by H. Thomas 共Springer, Heidelberg, 1992兲, pp. 129–155. 11 S. M. Rezende and A. Azevedo, Phys. Rev. B 45, 10387 共1992兲. 12 V. V. Tikhonov and A. V. Tolmachev, Phys. Solid State 36, 101 共1994兲. 13 P. E. Zil’berman, A. G. Temiryazev, and M. P. Tikhomirova, JETP 108, 151 共1995兲. 14 L. F. Álvarez, O. Pla, and O. Chubykalo, Phys. Rev. B 61, 11613 共2000兲. 15 A. M. Shuty and D. I. Sementsov, JETP 91, 531 共2000兲. 16 A. M. Shuty and D. I. Sementsov, Crystallogr. Rep. 51, 303 共2006兲. 17 A. M. Shuty and D. I. Sementsov, JETP 104, 758 共2007兲. 18 D. I. Sementsov and A. M. Shuty, Phys. Usp. 50, 793 共2007兲. 19 A. H. Bobeck and E. Della Torre, Magnetic Bubbles 共North-Holland, Amsterdam, 1975兲. 20 T. H. O’Dell, Ferromagnetodynamics. The Dynamics of Magnetic Bubbles, Domains and Domain Walls 共MacMillan, New York, 1981兲. 21 B. Neite and H. Dotsch, SPIE Electro-Optic and Magneto-Optic Materials 1018, 115 共1988兲. 22 P. Bergé, Y. Pomeau, and C. Vidal, L’ordre dans le Chaos. Vers une Approche Deterministe de la Turbulence 共Hermann, Paris, 1988; Mir, Moscow, 1991兲. 23 G. Bertotti, A. Magni, I. D. Mayergoyz, and C. Serpico, J. Appl. Phys. 89, 6710 共2001兲. 24 A. M. Shuty and D. I. Sementsov, JETP Lett. 78, 480 共2003兲. 25 G. Benettin, L. Galgani, A. Giorgilli, and J. M. Strelcin, Meccanica 15, 9 共1980兲. 26 L. Keefe, P. Moin, and J. Kim, J. Fluid Mech. 242, 1 共1992兲. 27 Yu. I. Neimark and P. S. Landa, Stochastic and Chaotic Oscillations 共Kluwer Academic, Dordrecht, 1992兲.
CHAOS 19, 013111 共2009兲
Distinguished trajectories in time dependent vector fields J. A. Jiménez Madrid and A. M. Manchoa兲 Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Serrano 121, 28006 Madrid, Spain
共Received 6 June 2008; accepted 4 December 2008; published online 10 February 2009兲 We introduce a new definition of distinguished trajectory that generalizes the concepts of fixed point and periodic orbit to aperiodic dynamical systems. This new definition is valid for identifying distinguished trajectories with hyperbolic and nonhyperbolic types of stability. The definition is implemented numerically and the procedure consists of determining a path of limit coordinates. It has been successfully applied to known examples of distinguished trajectories. In the context of highly aperiodic realistic flows our definition characterizes distinguished trajectories in finite time intervals, and states that outside these intervals trajectories are no longer distinguished. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3056050兴 This paper attempts to generalize the concepts of fixed point and periodic orbit to time dependent aperiodic dynamical systems. Fixed points and periodic orbits are keystones for describing solutions of autonomous and time periodic dynamical systems, as the stable and unstable manifolds of these hyperbolic objects form the basis of the geometrical template organizing the description of the dynamical system. The mathematical theory of aperiodic dynamical systems is far from complete. In this context, this work deals with a general definition that encompasses the concepts of fixed point and periodic orbit and which when applied to finite time and aperiodic dynamical systems identifies special trajectories that play an organizing role in the geometry of the flow. I. INTRODUCTION
In recent years the theory of dynamical systems has provided a useful framework for describing transport in fluid flows. Since the seminal work by Aref1 on chaotic advection much progress has been made both in theory and applications. Dynamical systems techniques were first applied to Lagrangian transport in the context of two-dimensional, time-periodic flows2 and stationary 3D flows such as the ABC flow.3 More recently these techniques have been extended to describe aperiodic flows4–6 and finite timedependent flows, such as, those rising in geophysical applications.7,8 However, the mathematical theory for both aperiodic time-dependent flows and finite time aperiodic flows is far from being completely developed. For stationary flows the idea of fixed point is a key for describing geometrically the solutions. Fixed points may be classified as hyperbolic or nonhyperbolic depending on their stability properties. Stable and unstable manifolds of hyperbolic fixed points organize the phase portraits of the flow away from the region close to the fixed points.9,10 These manifolds comprise, respectively, the trajectories that apa兲
Author to whom correspondence should be addressed. Telephone: ⫹34 91 5616800, ext. 2408. Fax: ⫹34 91 5854894. Electronic address:
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proach the fixed points as time tends to plus or minus infinite. As they are formed of trajectories they act as barriers to transport as particles cannot cross them without violating the uniqueness of the solution. They are useful because they allow qualitative predictions for the evolution of sets of initial conditions avoiding explicit integration of initial conditions on the whole domain. Hyperbolic fixed points and their stable and unstable manifolds are the basic notions used for the geometrical description of flows in autonomous dynamical system. The concept of the fixed point is extended to time periodic flows by means of the Poincaré map, as periodic orbits with period T become fixed points of the Poincaré map. For hyperbolic periodic orbits there also exists stable and unstable manifolds that are geometric objects that organize the global dynamics. Again they are, respectively, the sets of orbits asymptotically approaching the periodic orbit as time tends to plus or minus infinity. Aperiodic flows are still poorly understood, as theory that is well established for autonomous or periodic flows do not apply to them directly. For instance, there exists efforts in the mathematical community to extend the well known concept of bifurcation for stationary flows to nonautonomous systems.11,12 To gain insight on the geometrical structure of aperiodic flows, concepts, such as, Lyapunov exponents are used, however these are defined strictly on infinite time systems. Realistic flows, like those arising in geophysics or oceanography, are not infinite time systems and for their description, finite time versions of the definition of Lyapunov exponents, such as, finite size Lyapunov exponents 共FSLE兲 共Ref. 13兲 and finite time Lyapunov exponents 共FTLE兲 共Refs. 14 and 15兲 are used. Special trajectories, such as, detachment and reattachment points,16 are observed in highly aperiodic or turbulent flows. In particular these separation trajectories occur on the boundaries in simplified ocean models17 and also in technological applications in air foil design.18 Recent articles by Ide et al. and Ju et al.19,20 referring to these special trajectories introduce the concept of distinguished hyperbolic trajectory 共DHT兲 which encompasses not only trajectories on the boundaries, but also special trajectories in the
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interior of the flow. DHT are hyperbolic trajectories that, like hyperbolic fixed points and periodic orbits, have stable and unstable manifolds that are key for describing geometrically the solutions on the phase space. This generalization is an important step forward in the study of aperiodic flows, as it is a powerful tool for describing transport in realistic oceanographic flows.7,8,21,22 Distinguished hyperbolic trajectories as defined in Refs. 19 and 20 are computed from hyperbolic instantaneous stagnation points 共ISPs兲 by means of an iterative procedure. If instantaneous stagnation points bifurcate and do not persist for all times the technique developed in Refs. 19 and 20 cannot be applied in those time intervals, leaving many questions unanswered, such as, what happens to the distinguished trajectories at those times, for distinguished hyperbolic trajectories are trajectories, and as trajectories exist at all times. In fact, Ref. 7 provides examples of vector fields with exact distinguished hyperbolic trajectories that exist on time intervals without hyperbolic ISP. References 7 and 8 discuss the impossibility of this technique for tracking DHTs after ISP bifurcations and as a consequence the difficulties in establishing whether DHTs obtained at different times are part of the same trajectory or not. In this paper, following ideas discussed in Refs. 7, 19, and 20, we propose a new definition of distinguished trajectory 共DT兲 which generalizes the concepts of fixed point and periodic orbit to aperiodic flows. We have taken the liberty of calling them distinguished as in Refs. 7, 19, and 20, since although the definitions are not strictly equivalent, it is found that the studied hyperbolic trajectories are encompassed by both definitions. We remark that our notion has the advantage over the method proposed in Refs. 19 and 20 that the DTs may be computed without the presence of hyperbolic instantaneous stagnation points. Our definition does not depend on the dimension n of the space on which the vector field is defined and is valid both for hyperbolic and nonhyperbolic types of stabilities. Nonhyperbolic DTs have not been studied in Refs. 19 and 20 and in this sense our definition is broader than that proposed there. In particular, we will show that exact nonhyperbolic periodic orbits fall within the category of distinguished trajectories. Trajectories of this type could be of special interest for their applications in oceanography, as they are related to eddies and vortices. Ocean eddies are well studied.23 Frequently they are long lived, and water trapped inside can maintain its biogeochemical properties for a long time, being transported with the vortex. In steady horizontal velocity fields, the presence of closed streamlines is the mathematical reason for the isolation of the vortex core from the exterior fluid. In the two-dimensional, incompressible, time-periodic velocity fields the KAM tori enclose the core, a region of bounded fluid particle motions that do not mix with the surrounding region.4 But how can one define an eddy from the Lagrangian point of view in aperiodic flows? This is still an open question22,24 for which we will discuss new possibilities suggested by the definitions given in this paper. The structure of the paper is as follows: Section II introduces the definition of distinguished trajectory and explains its motivation in the context of 1D examples. Section III explains the algorithm used to verify the applicability of our
definition of distinguished trajectories to the solutions of the periodically forced Duffing equation. Details about technical issues arising from implementation of the definition are given. Section IV reports the results obtained in several other 2D and 3D examples, both periodic and nonperiodic, hyperbolic and nonhyperbolic. Section V discusses results on realistic flows. Attention is paid to open questions on distinguished trajectories, such as, those mentioned above and pointed out in Refs. 7 and 8. Finally, Sec. VI presents the conclusions. II. DISTINGUISHED TRAJECTORIES: A DEFINITION
We start by recalling the definition of distinguished hyperbolic trajectory provided in Ref. 19. Given the system dx = Dx + gNL共x,t兲, dt
x 苸 Rn .
共1兲
Let x共t兲 be a trajectory of Eq. 共1兲 that remains in a bounded region for all time. Then x共t兲 is said to be a distinguished hyperbolic trajectory if 1. 2.
3.
it is hyperbolic, there exists a neighborhood B in the flow domain having the property that the DHT remains in B for all time, and all other trajectories starting in B leave B in finite time, as time evolves in either a positive or negative sense, it is not a hyperbolic trajectory contained in the chaotic invariant set created by the intersection of the stable and unstable manifolds of another hyperbolic trajectory.
Remark 1: If the data span only a finite time interval, then the DHT cannot be determined uniquely. Instead, there is a small region in B where the DHT can exist. In Ref. 19 this setup is extended to general vector fields as follows. Coordinate transformations are sought which put the system in the form of Eq. 共1兲 and then the previous definition is applied. We give now our definition of distinguished trajectory for a general vector field, dx = v共x,t兲, dt
x 苸 Rn,t 苸 R.
共2兲
We assume that v共x , t兲 is Cr 共r 艌 1兲 in x and continuous in t. This will allow for unique solutions to exist, and also permit linearization, although linearization will not be used in our construction. Before giving our definition of DT, we first need to introduce some notation and to make some definitions. Let x共t兲 denote a trajectory of the system 共2兲 and denote its components in Rn by 共x1 , x2 , . . . xn兲. For any initial condition x* in an open set B 傺 Rn, consider the function M : B → R, M共x*兲t*, =
冕 冑兺 冋 册 t*+
t*−
n
i=1
dxi共t兲 dt
2
dt,
共3兲
M is the function that associates to each initial condition x* in B the arc length of the trajectory that passes through x* at time t*. The arc length of the trajectory is considered over its projection in the phase space 共x1 , x2 , . . . xn兲 and depends on
Chaos 19, 013111 共2009兲
Distinguished trajectories
4
4
3
3
2
2
1
1
0
0
t
t
013111-3
−1
−1
−2
−2
−3
−3
−4 −3
(a)
−2
−1
0
x
1
2
3
−4 −3
(b)
FIG. 1. Solutions 共6兲 for different initial conditions x*. 共a兲 Solutions for positive times t ⬎ 0. 共b兲 Solutions for positive and negative times.
−2
−1
t* and . As the function M is defined over an open set it does not necessarily attain a minimum, but if it does, the minimum is denoted by min共M共x*兲t*,兲. Definition 2: (-distinguished trajectory). A trajectory ␥共t兲 of Eq. (2) is -distinguished at time t* if there exists an open set B around ␥共t*兲 on which the defined function M共x*兲t*, has a minimum and min共M共x*兲t*,兲 = M共␥共t*兲兲t*, .
The elements of the above definition deserve a detailed justification. We illustrate our explanations with examples in 1D. First we consider an example taken from Refs. 19 and 25. It is the linear one-dimensional nonautonomous dynamical system given by 共5兲
For this example we consider the DHT reported in Ref. 19, which is given by x = t − 1. This is the particular solution of the linear equation 共5兲 towards which all trajectories decay. The solution through the point x* at t = 0 is given by
3
−0.2
16
−0.4
14
FIG. 2. 共a兲 Function M共x*兲t=0, evaluated over the solutions 共6兲. Dashed line = 3, solid line = 4. 共b兲 Position of the x*-coordinate at the minimum of the function M t=0, as a function of . The horizontal dashed line marks the position of the DHT.
−0.6
12 10
−0.8
8
−1 6
−1.2
4 2
−1.4
0 −2,0
共6兲
b) min(M)
20 18
M
2
Figure 1共a兲 displays several trajectories starting at times ranging from t = 0 to t = 4 and Fig. 1共b兲 displays the same, but starting at time t = −4. Note that in this case part of the trajectories are out of the displayed domain. For each initial condition the function M provides the length of the projection of the trajectory over the x-axis in the range of times 关− , 兴. Geometrically it is clear that in this example the function M should have a minimum for a certain x value and that this value depends on . Ideally the minimum of M should coincide with the position of the DHT at t = 0, however this would not be possible if in the definition of M only positive times were considered, i.e., if the limits of the integration were 共0 , 兲 the dashed trajectory in Fig. 1共a兲 would have a lower projection in positive times than the particular solution. An analogous problem would be encountered where only negative times are considered, that is, if the limits of the integration would have been 共− , 0兲. To determine precisely the position of the DHT at t = 0, both positive and negative times must be considered in the definition of M. Figure 1共b兲 confirms that with this choice the dashed trajectory cannot be distinguished as it increases its projection in negative times. Figure 2共a兲 displays the function M共x*兲t=0, evaluated along the trajectories 共6兲, for several values. Figure 2共b兲 displays
A. A discussion of the definition
a)
1
x共t兲 = t − 1 + e−t共x* + 1兲.
共4兲
dx = − x + t. dt
0
x
−1,5
−1,0 x*
−0,5
0,0
0
5
τ
10
15
013111-4
Chaos 19, 013111 共2009兲
J. A. Jiménez and A. M. Mancho
min(M)
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−1.4 0
5
τ
10
15
FIG. 3. Position of the x*-coordinate at the minimum of the function M t=0, as a function of . The function M t=0, is considered for the solutions in Eq. 共8兲. The horizontal dashed line marks the position of the DHT.
the position of the minimum of the function M t=0, as a function of . These minima correspond to the positions of the -distinguished trajectories at t = 0 and as increases they approximate the coordinate of the DHT at this time, which is at x* = −1. The pair 共tl , xl兲 formed by the time at which M is computed and the value of the coordinate xl to which the minimum of the function M tl, converges for increasing is called the limit coordinates. Figure 2共b兲 illustrates the idea of approaching a point 共t0 , x0兲 of the distinguished trajectory by means of the limit coordinates. In practice the convergence to the limit coordinates cannot be examined in the limit → ⬁, either because it is impracticable in a numerical implementation, or because in the large limit errors accumulate, or simply because the dynamical system is defined by a finite time data set. For these reasons the convergence to the limit coordinates will be tested up to a finite . Figure 2共b兲 raises the question: What controls the rate of the convergence of the minima of M to the coordinates of the DHT? It is hard to answer this question rigorously for a vector field as general as in Eq. 共2兲. However, some insight may be provided by particular examples. For instance, the system dx = − 2x + 2t − 1, dt
共7兲
has the same DHT as Eq. 共5兲. Its solution through the point x* at t = 0 is given by x共t兲 = t − 1 + e−2t共x* + 1兲.
共8兲
Here the decay of the solution towards the DHT is faster due to the presence of the exponential term e−2t. Figure 3 shows that in this case the rate of the convergence of the minima of M towards the coordinates of the DHT at time t = 0 is also faster than before. However there exist systems in which the exponential decay of the solution is not a determining factor affecting the rate of the convergence of the minima of M to the coordinates of the DHT. For instance, in autonomous systems fixed points are the DTs, and clearly they are mini-
mizers of M for any ⬎ 0 whatever is the exponential rate of growth or decay of the nearby solution. In these examples the function M has a unique minimum, but as we will see the situation will not always be so simple when nonlinearities are involved in the vector field. Also it is important to notice that the function M obtained at different values has been used to obtain the limit coordinates 共tl0 , xl0兲 and that these approach the x0 coordinate of DHT at a given time t0 共here t0 = tl0兲. Once this is obtained, approaching the DHT at later times tk = t0 + k⌬t would require applying the same procedure to get the limit coordinates 共tlk , xlk兲. We remark here that the proposed algorithm does not ensure that the set of limit coordinates 共tlk , xlk兲 are in fact part of a trajectory. Later we will see that in practice, in many examples these points approach a true trajectory, however in realistic aperiodic flows this has to be verified a posteriori. These considerations lead us to the definition of a distinguished trajectory. Definition 3: (Distinguished trajectory). A trajectory ␥共t兲 is said to be distinguished with accuracy ⑀ 共0 艋 ⑀兲 in a time interval 关t0 , tN兴 if there exists a continuous path of limit coordinates 共tl , xl兲, where tl 苸 关t0 , tN兴, such that, 储␥共tl兲 − xl共tl兲储 艋 ⑀,
∀ tl 苸 关t0,tN兴.
共9兲
Here 储·储 represents the distance defined by
冑兺 n
储a − b储 =
共ai − bi兲2
with a,b 苸 Rn .
i=1
In the numerical exploration of this definition we will replace the continuous path of limit coordinates 共tl , xl兲 and the continuous trajectory ␥共t兲 by discrete representations 共tlk , xlk兲 and ␥共tlk兲, where t0 艋 tlk 艋 tN. By definition 3 any trajectory is distinguished for sufficiently large ⑀, however the interesting distinguished trajectories are those for which ⑀ is close to zero, which means it is of the order of the accuracy in which ␥共tlk兲 and xl共tlk兲 are numerically determined, or zero, if an exact expression is known for both. Underlying definitions 2 and 3 is the geometrical idea that distinguished trajectories, which act as organizing centers of the flow in phase space, are those that “move less” 共in a certain sense兲 than other nearby trajectories. This property of “moving less” is satisfied by minima of the function M as it measures the length of the displacement in phase space of a trajectory forwards and backwards in time. In fact this property is related somehow to property 共2兲 of the definition provided in Ref. 19 and presented at the beginning of Sec. II, as the trajectory that “moves least” is not expected to leave the neighborhood B. Definitions 2 and 3 are made for a general dynamical system in any dimension n. The purpose of this paper is the exploration of these definitions, but more in an illustrative than demonstrative way, as it is impossible to provide examples for every possible n, and one cannot deal with every possible example at a given n. Even if one wants to provide a rigorous formal proof that the definition recovers specific trajectories, such as, periodic orbits 共it is not obvious that in general they have to satisfy our definition兲, this has to be done with some further hypotheses on the vector field and
Chaos 19, 013111 共2009兲
Distinguished trajectories
proofs will not be valid beyond the assumed hypotheses. Therefore we restrict the discussion to dimensions up to 3, as these are the dimensions important for geophysical flows, which are what originally motivated the definition. However it is sensible to make the same definition for any dimension n, as it is clear that it works for autonomous systems of any dimension. Fixed points are the kind of trajectory expected to be recovered by the definition and they do not move at all in the phase space. For these M = 0, while M ⬎ 0 for any other trajectory in the neighborhood which is not a fixed point. We conclude this section with some remarks. First, it is not guaranteed a priori that for an arbitrary vector field, satisfying only some rather general hypotheses such as those of Eq. 共2兲, the function M will have a minimum, however this is not a problem from the point of view of the definition. For instance the same thing happens for general nonlinear autonomous systems. In these systems fixed points are perfectly defined although one does not know a priori if such points exist for arbitrary examples. If they exist, it is possible to find them by either solving the nonlinear equation v共x兲 = 0, or by applying definitions 2 and 3. In the same way one does not know a priori if distinguished trajectories exist for a general vector field, however if they exist they can been found with the tools proposed in this article. Second, even if a path of limit coordinates is found, it is not guaranteed that it will be a trajectory, although if that is the case then from definition 3 follows that this trajectory is distinguished. Third, one might think that if limit coordinates are found at t0 that approach with great accuracy a point of an existing DT, then the iterative procedure described above for finding a set of limit coordinates 共tlk , xlk兲 approaching the DT at later times is an unnecessary computational effort, as those coordinates could have been equally well obtained by integrating forwards the initial data. However there exist examples such as a hyperbolic DT in dimension greater than one with at least 1D unstable manifold, that cannot be integrated like this, as the integrated trajectory will eventually leave the neighborhood of the DT through the unstable manifold no matter how small the initial error is. In summary, the proposed methodology based on limit coordinates provides a systematic way of finding DT, which can be elusive and difficult to obtain. We will discuss these issues in detail in later sections.
III. A NUMERICAL ALGORITHM
In this section we propose an algorithm for computing a path of limit coordinates in a time interval, and we verify that it is close to a DT of a known example. For this purpose we calculate, at increasing values, the minimum of the function M t=0,共x兲 for x in an open set in Rn. The method is illustrated in a 2D case, the periodically forced Duffing equation, x˙ = y,
y˙ = x − x3 + sin共t兲,
共10兲
where is a small parameter. The hyperbolic fixed point of the unperturbed autonomous system 共i.e., = 0兲 is at the origin x = 共0 , 0兲. For small , it is possible to compute by per-
0.2 0.15 0.1 0.05
Y
013111-5
0 −0.05 −0.1 −0.15 −0.2 −0.2
−0.15
−0.1
−0.05
0
X
0.05
0.1
0.15
0.2
FIG. 4. Contour plot of the function M t=0,=2共x兲 in the open set D = 共−0.2, 0.2兲 ⫻ 共−0.2, 0.2兲. The minimum corresponds to the black tone.
turbation theory 共see Ref. 26兲, the following periodic trajectory which stays close to the origin: xDHT共t兲 = −
冉 冊 冉
冊
3 3 2 sin t 3 2 sin t + 2 sin t cos t − + O共5兲. 40 23 cos3 t + 3 sin2 t cos t 2 cos t
共11兲 For = 0.1, Eq. 共11兲 is accurate up to the fifth digit. This trajectory is identified as distinguished in Ref. 19, for this reason we have labelled it a DHT. Substituting the expression, x = 共x,y兲 = xDHT共t兲 + 共1, 2兲
共12兲
into Eq. 共10兲 and by dropping the nonlinear terms, one finds that the linearized equations have two linearly independent solutions in terms of which the time evolution of the components 共1 , 2兲 is 共 1, 2兲 = ␣ e t
冉 冑冑 冊 冉 冑冑 冊 1/ 2
1/ 2
+ e−t
− 1/ 2 1/ 2
+ O共2兲.
共13兲
Equation 共13兲 confirms the hyperbolicity of the solution 共11兲. This explicit expression for the distinguished hyperbolic trajectory is a benchmark for testing the utility of our definition. The procedure starts by determining the coordinates of xDHT at time t = 0. We consider the open set D 傺 R2, defined by D = 共−0.2, 0.2兲 ⫻ 共−0.2, 0.2兲 and in the function M t=0,共x兲 we take to be 2. Figure 4 displays a contour plot of M t=0,=2共x兲 which has a minimum at x = 共0 , −5.7057⫻ 10−2兲. M t=0,=2共x兲 quantifies displacements of particles in phase space, and its minimum corresponds to the initial condition that “moves less” over the interval 关−2 , 2兴. As noted in the previous section, when the value of is increased, the position of the minimum gets closer and closer to the coordinates of the DHT at t = 0. Figure 5 shows contour plots of the function M for several values. Figure 5共a兲 displays a typical hyperbolic structure for M for = 5 where the directions of the stable and unstable manifolds are easily recognized. In Fig. 5共a兲 the function M has a unique minimum at x = 共0 , −4.979⫻ 10−2兲 while in Fig. 5共b兲 there appear
013111-6 0.2
b)
0.2
c)
0.2
0.15
0.15
0.15 0.1
0.05
0.05
0
0
0
Y
0.1
Y
0.1 0.05
Y
a)
Chaos 19, 013111 共2009兲
J. A. Jiménez and A. M. Mancho
−0.05
−0.05
−0.1
−0.1
−0.1
−0.15
−0.15
−0.15
−0.2 −0.2
−0.15
−0.1
−0.05
0
X
0.05
0.1
0.15
0.2
−0.2 −0.2
−0.05
−0.15
−0.1
−0.05
0
X
0.05
0.1
0.15
0.2
−0.2 −0.2
−0.15
−0.1
−0.05
0
X
0.05
0.1
0.15
0.2
FIG. 5. Contour plot of the function M in the open set x 苸 共−0.2, 0.2兲 ⫻ 共−0.2, 0.2兲. 共a兲 M t=0,=5共x兲; 共b兲 M t=0,=10共x兲; 共c兲 M t=0,=50共x兲.
several minima for = 10. The global minimum in this picture corresponds to x = 共0 , −5.004256⫻ 10−2兲. Figure 6共a兲 compares the x-coordinate of xDHT as a function of time with trajectories having initial conditions at the global minima of M t=0,=2 and M t=0,=10. Taking as the initial condition the global minimum of M t=0, for = 10 provides a trajectory that stays close to xDHT for a longer time interval than for = 2, which confirms that larger values more closely approach the coordinates of the DHT. Figure 5共c兲 displays the contour plot of M t=0,=50共x兲. Its global minimum is at x = 共0 , −5.003760⫻ 10−2兲. The associated trajectory depicted in Fig. 6共b兲 shows that this initial condition tracks the DHT for a longer time interval than those obtained for = 2 and 10, however the figure shows that the integration of the DHT in 共−50, 50兲 is not possible. In fact the associated trajectory stays close to the DHT only in the time interval 共−20, 20兲. This confirms that results obtained for = 50 are the same as those obtained for = 20. In practice for a finite precision numerical scheme, such as, a fifth order Runge–Kutta used here, the approach to the DHT has an upper bound depending on . This occurs because the stable and unstable manifolds of the hyperbolic trajectory magnify any initial error in
x
a)
0.5
0
−0.5 −50
−40
−30
−20
−10
0
10
20
30
40
50
−40
−30
−20
−10
0
10
20
30
40
50
t
0.5
x
b)
0
−0.5 −50
t
FIG. 6. 共a兲 x-coordinate vs time for the DHT 共thick solid line兲 and those trajectories integrated with initial conditions at the global minima of Fig. 5共a兲 共solid line兲 and Fig. 5共b兲 共dashed line兲; 共b兲 x-coordinate vs time for the DHT 共thick solid line兲, a trajectory integrated with initial condition at the global minimum of Fig. 5共c兲 共solid line兲 and a trajectory integrated at a nonglobal but relative minimum of the same figure 共dashed line兲.
either negative or positive time and beyond this -limit numerical errors dominate. The convergence towards the DHT is confirmed in Fig. 7 which displays the evolution of the coordinates x and y of the global minimum of M as a function of the parameter . New minima appearing in Figs. 5共b兲 and 5共c兲 relate to the existence of different -distinguished trajectories. As illustrated in Fig. 6共b兲, they correspond to trajectories which stay close to xDHT in a small time range contained in the interval − ⬍ t ⬍ , but which later fly apart from the DHT. We now describe a numerical scheme to compute a path of limit coordinates. The algorithm has the following steps: Step 1. Discretize the domain D at the initial time t = t0 at which one wishes to compute a DT. For instance, the grid size of this domain in Fig. 4 is 101⫻ 101. The function M is evaluated at each grid point for a given 0. Step 2. Search for the local minima of M t0,0 in the interior of the grid. These minima approach the coordinates of 0-distinguished trajectories within the accuracy of the grid. In what follows we restrict our description to the case of a unique minimum, as this simplifies the description; the procedure is easily generalized to the case of multiple minima. Step 3. Improve the approach of the coordinates of the 0-distinguished trajectory up to precision ␦. For this purpose build up a 3n grid centered on the candidate point provided by step 2 共for the 2D case this is a 3 ⫻ 3 grid as Fig. 8 illustrates兲, setting the distance between nodes equal to ␦. Then evaluate M t0,0 at the points of the ␦-grid. If the minimum of M t0,0 is in the interior of the grid, then the coordinates of the 0-distinguished trajectory are known to within ␦ accuracy. Otherwise the ␦-grid must be rebuilt centered on the boundary point where the minimum has been located, and M t0,0 must be re-evaluated in the new ␦-grid. This procedure stops when the minimum of M t0,0 is in the interior of the grid. Step 4. Computing the limit coordinates at time t0. Define a sequence of increasing -values as follows: 1 = 0 + ⌬ and 2 = 0 + 2⌬. Then evaluate M t0,0, M t0,1 and M t0,2 on the ␦-grid. If the minimum is at an interior position for the three cases, then we consider that limit coordinates have been found within ␦ accuracy. We note that this is a necessary but not sufficient condition as one does not know a priori the convergence rate to the distinguished trajectory. Although this criterion could be strengthened, it has been
013111-7 a)
Chaos 19, 013111 共2009兲
Distinguished trajectories
1e-10
b)
0
5e-11 -0.01
0
-5e-11 -0.02
Coordinate Y
Coordinate X
-1e-10
-1.5e-10
FIG. 7. Evolution of the coordinates of the global minimum of M vs . 共a兲 The x coordinate; 共b兲 the y coordinate. These plots show the convergence to the DHT whose position is marked with a dashed horizontal line.
-0.03
-2e-10 -0.04
-2.5e-10
-3e-10 -0.05 -3.5e-10
-4e-10
0
10
20
30
40
50
60
70
80
90 100
-0.06
0
10
20
Integration time τ
tested and found to be adequate for the examples explained in subsequent sections. If the condition defined above of having a minimum at an interior position for the sequence of -values is not satisfied, then after replacing 0 by 1, we return to step 3 and then to step 4. The loop between steps 3 and 4 is stopped when the condition of step 4 is satisfied for some k. Step 5. Compute the limit coordinates at time t1 = t0 + ⌬t. Once the limit coordinates have been approached at time t0, they are integrated forward numerically up to time t1. If the limit coordinates converge to a hyperbolic DT with an unstable manifold, the position x共t1兲 obtained should deviate from the position of the DT at time t1. In order to correct this, the procedure described above is repeated from step 3 onwards. For that purpose in the definition of M, t0 is replaced by t1 and the -value is reset to 0. The coordinates x共t1兲 are the first approximation to the 0-distinguished trajectory at time t1. Once the limit coordinates are found for time t1 it is possible to repeat the procedure to locate them at successive times t2 , t3 , . . . , tN.
30
40 50 60 70 Integration time τ
80
90
100
The algorithm requires as inputs: an explicit expression for the dynamical system 共2兲; the definition of the domain D 傺 Rn; the initial and final times t0, tN at which DTs are required, and the time step ⌬t for intermediate times; the initial 0 and the increment ⌬; the precision ␦. As an output the algorithm gives a path of limit coordinates at the selected times tk. Next we discuss in more detail some technical aspects related to the implementation of the above algorithm. Steps 1 and 3 require evaluating M t0,0 as defined in Eq. 共3兲. We explain how this is done for the contour plots displayed in Figs. 4 and 5, which refer to the system 共10兲 at t0 = 0. Figure 9 shows a schematic projection onto the R2 plane of a possible trajectory x共t兲 of the system from − to . As it was obtained numerically, only a finite number of points 共L兲 appear. This picture suggests the following discrete version of Eq. 共3兲 for M:
xL τ
δ
xj (p)
x j+1 xj x 1 −τ
FIG. 8. A ␦-grid in R2. The center or interior point is marked with the white dot.
FIG. 9. A schematic projection onto the R2 plane of a possible trajectory from − to with L points.
013111-8
Chaos 19, 013111 共2009兲
J. A. Jiménez and A. M. Mancho
TABLE I. Relative errors for several ellipse lengths, computed with a linear interpolation over L points on the curve. Linear interpolation Ratio between axes L 10 102 103 104 105 106 107 108 109
L−1
M共x兲0, = 兺 j=1
冉冕 冑冋 pf
pi
1
2
5
10
100
1000
8.16 2.63 0.83 0.26 8.34⫻ 10−2 2.64⫻ 10−2 8.34⫻ 10−3 2.64⫻ 10−3 8.34⫻ 10−4
8.80 3.36 1.08 0.34 0.11 3.42⫻ 10−2 1.08⫻ 10−2 3.42⫻ 10−3 1.08⫻ 10−3
9.48 3.86 1.24 0.39 0.12 3.94⫻ 10−2 1.25⫻ 10−2 3.95⫻ 10−3 1.25⫻ 10−3
9.73 3.99 1.29 0.41 0.13 4.08⫻ 10−2 1.29⫻ 10−2 4.08⫻ 10−3 1.29⫻ 10−3
9.99 4.05 1.31 0.41 0.13 4.14⫻ 10−2 1.31⫻ 10−2 4.14⫻ 10−3 1.31⫻ 10−3
10.00 4.05 1.31 0.41 0.13 4.14⫻ 10−2 1.31⫻ 10−2 4.14⫻ 10−3 1.31⫻ 10−3
dx j共p兲 dp
册 冋 册 冊 2
+
dy j共p兲 dp
2
dp ,
共14兲
where the functions x j共p兲 and y j共p兲 represent a curve interpolation parametrized by p, and the integral
冕 冑冋 pf
pi
dx j共p兲 dp
册 冋 册 2
+
dy j共p兲 dp
x j共p兲 = x j + pt j + j共p兲n j
共16兲
for pi = 0 艋 p 艋 p f = 1 with x j共0兲 = x j and x j共1兲 = x j+1, where t j = 共a j,b j兲 = x j+1 − x j, n j = 共− b j,a j兲,
t j 苸 R2 ,
共17兲
n j 苸 R2 ,
共18兲
2
共15兲
dp
is computed numerically. In our case we use the Romberges method 共see Ref. 27兲 of the order 2K with K = 5. It is clear that the accuracy of the evaluation of M will depend on the number of points on the trajectory L, which is controlled by the size of the time step, h, of the integrator 共a fifth order Runge–Kutta method兲 and on the interpolation scheme between points. Two interpolation methods are compared in Tables I and II. Results in Table I are obtained with linear interpolation between nodes. Results in Table II correspond to the interpolation method used by Dritschel28 in the context of contour dynamics, which has been successfully applied in Ref. 26 to the computation of invariant manifolds for aperiodic flows. This method interpolates a piece of the curve in Fig. 9 between consecutive nodes as follows:
j共p兲 = j p +  j p2 + ␥ j p3,
j 苸 R.
The cubic interpolation coefficients j,  j, and ␥ j are
j = − 31 d j j − 61 d j j+1,
 j = 2d j j,
␥ j = 61 d j共 j+1 − j兲,
where d j = 兩x j+1 − x j兩 and
j = 2
a j−1b j − b j−1a j 兩d2j−1t j + d2j t j − 1兩
is the local curvature defined by the circle through the three points, x j−1, x j, and x j+1. Tables I and II show the errors in the computed lengths of the ellipses for different ratios of major to minor axis. The reference length is that obtained with GNU Octave version 2.1.73, as it provides 16 correct digits for the known circumference. The tables confirm that the Dritschel’s method is
TABLE II. Relative errors for several ellipse lengths, computed with Dritschel interpolation over L points on the curve. Dritschel interpolation Ratio between axes L 10 102 103 104 105 106 107 108 109
共19兲
1
2
5
10
100
1000
0.99 3.49⫻ 10−2 1.11⫻ 10−3 3.53⫻ 10−5 1.12⫻ 10−6 3.51⫻ 10−8 1.11⫻ 10−9 2.18⫻ 10−11 3.72⫻ 10−10
0.67 1.37⫻ 10−2 3.83⫻ 10−4 1.17⫻ 10−5 3.64⫻ 10−7 1.14⫻ 10−8 3.68⫻ 10−10 1.30⫻ 10−11 3.37⫻ 10−10
0.33 4.47⫻ 10−3 9.20⫻ 10−5 4.44⫻ 10−6 2.17⫻ 10−6 2.10⫻ 10−6 2.10⫻ 10−6 2.10⫻ 10−6 2.10⫻ 10−6
0.25 2.81⫻ 10−3 4.43⫻ 10−5 5.23⫻ 10−6 4.48⫻ 10−6 4.46⫻ 10−6 4.46⫻ 10−6 4.46⫻ 10−6 4.46⫻ 10−6
0.26 2.08⫻ 10−3 2.23⫻ 10−5 2.80⫻ 10−7 5.45⫻ 10−8 5.222⫻ 10−8 5.21⫻ 10−8 5.22⫻ 10−8 5.22⫻ 10−8
0.27 2.61⫻ 10−3 1.98⫻ 10−5 7.01⫻ 10−7 9.20⫻ 10−7 9.22⫻ 10−7 9.22⫻ 10−7 9.22⫻ 10−7 9.25⫻ 10−7
013111-9 0.06
7e-07
b)
0.04
6.5e-07
0.02
6e-07
Error
Y
a)
Chaos 19, 013111 共2009兲
Distinguished trajectories
0
5.5e-07
-0.02
5e-07
-0.04
4.5e-07
-0.06 -0.06
-0.04
-0.02
0 X
0.02
0.04
4e-07
0.06
0
1
2
3 Time t
4
5
6
FIG. 10. 共a兲 Representation of both the distinguished hyperbolic trajectory 共11兲 and its approximation obtained with the proposed numerical algorithm for = 0.1; 共b兲 distance between the exact and the numerical approach.
IV. APPLICATIONS TO EXACT EXAMPLES
In this section we apply the algorithm explained in the previous section to selected examples. A. A nonhyperbolic distinguished trajectory
The unperturbed autonomous system 共10兲 obtained with = 0 has nonhyperbolic fixed points at 共−1 , 0兲 and 共1,0兲. 15
14.5
14 Integration time τ
superior to linear interpolation and it is the one used to compute the function M. In the trajectory from − to the number of points L is determined by the time step size of the Runge–Kutta method which is set to 10−2. Another important element of the algorithm needing discussion is the value of the input parameters, in particular of 0 and ⌬. It is clear from Fig. 5 that large values are not convenient as they increase the roughness of the function M and several local minima may appear in the neighborhood of a DHT that correspond to trajectories that stay close to it for some time. On the other hand it is clear that sufficiently large values are required to fix the coordinates of the DHT to within the prescribed accuracy. Combining these observations suggests the use of relatively small values for the initial 0. In the example above 0 = 2, provides, as a starting point, a smooth M as that of Fig. 4. The increments should not be large. In practice we have chosen ⌬ = 0 / 2. This prevents us from stepping to a too rough M before getting close enough to the sought after DHT. Some of the local minima appearing in Fig. 5共b兲 are just apparent and disappear with a more refined grid. However, as already observed, others belong to true -distinguished trajectories, which are secondary and can be avoided if the increment of the -values is conveniently small. These choices are found to be appropriate for determining with great accuracy the DHT in Eq. 共11兲 by means of a path of limit coordinates. Figure 10共a兲 represents both the analytical DHT and the numerical limit coordinates and Fig. 10共b兲 displays the distance between the exact and the numerical approach, confirming that the DHT in Eq. 共11兲 is also a DT in the sense of our definition 3 with accuracy ⑀ = 10−6. Other parameters in the algorithm are ␦ = 10−6, step size in the Runge–Kutta method h = 10−2, t0 = 0, tN = 6, and ⌬t = 0.01. To locate the DHT with accuracy ␦ = 10−6 requires increasing values of up to 15, which is near the limit of the integration method. Figure 11 shows the maximum required at each tk.
13.5
13
12.5
12 0
1
2
3
4
5
6
Time t
FIG. 11. Representation of the maximum required to approach the DT to within accuracy ␦ = 10−6 vs time.
013111-10
b)
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
Y
Y
a)
Chaos 19, 013111 共2009兲
J. A. Jiménez and A. M. Mancho
−0.05
−0.05
−0.1
−0.1
−0.15
−0.15
−0.2
−0.2
−0.25
−0.25
−1.3
−1.2
−1.1
−1
−0.9
X
−0.8
−0.7
−1.3
−1.2
−1.1
−1
X
−0.9
−0.8
−0.7
FIG. 12. Contour plot of the function M in the open set x 苸 共−1.2, −0.8兲 ⫻ 共−0.2, 0.2兲兴. 共a兲 M t=0,=10共x兲; 共b兲 M t=0,=300共x兲.
Obviously these fixed points correspond to DTs which are also -distinguished trajectories for all ⬎ 0. For the periodically forced system 共10兲 with small it is possible using perturbation theory to find periodic solutions close to these fixed points in a manner similar to the analysis of the hyperbolic example made in the previous section. For instance close to the point 共1,0兲 we find the periodic trajectory, xDET共t兲 = −
冉冊 冉 冊 冉 1 0
+
sin t
cos t
+ 3
1 2
2
cos2 t
− sin t cos t
冊
+ O共3兲. 共20兲
This solution has not been considered distinguished in previous works,19,20 as these have been focused on hyperbolic trajectories and this solution, as is proved next, is not hyperbolic. However, in anticipation of its having the distinguished property, we have labelled it DET for two reasons. One is that it is periodic, and we expect periodic orbits to be distinguished, and second is that it is in clear correspondence to the elliptic fixed point 共−1 , 0兲 in the case = 0, and fixed points are DTs. To determine the stability of Eq. 共20兲 we proceed as before, by substituting the expression x = 共x,y兲 = xDET共t兲 + 共1, 2兲
共21兲
into Eq. 共10兲. We find that the linearized system at order 0 is d1 = 2 , dt
共22兲
d2 = − 21 . dt
共23兲
Therefore the linearized flow around xDET共t兲 evolves according to 共 1, 2兲 = ␣ e i
冉冑 冊
冑 冑2t 1/ 3
i 2/3
+ ␣*e−i
冑2t
冉
1/冑3
− i冑2/3
冊
+ O共兲,
共24兲
which clearly is not hyperbolic. Here ␣ and ␣* are complex conjugate numbers.
We apply our algorithm to determine the limit coordinates approaching Eq. 共20兲, as we want to verify whether definition 10 also works for time-dependent nonhyperbolic solutions. The following input is considered: D = 共−1.2, −0.8兲 ⫻ 共−0.2, 0.2兲, 0 = 2, ⌬ = 1, ␦ = 10−4, t0 = 0, tN = 6, and time step 10−2 for the Runge–Kutta integrator. We note that the accuracy ␦ is not as demanding as before, since now the exact xDET for = 0.1 is only accurate up to the third digit. Figure 12 shows a rather different structure for the function M. An important feature is the smoothness of M close to the DET even for large . In Fig. 12共b兲 the differences between the rather flat region around the position of the DT given by Eq. 共20兲, which appears in the dark tone, and the roughness of the outer part are remarkable. The irregularity of this region suggests that inside it nearby trajectories follow rather different paths as happens for chaotic motions, while the regularity of the central core suggests the existence of trapped trajectories circling around the DET. From this perspective the function M for large seems a useful tool for fixing the boundaries of a Lagrangian eddy, different from the methods proposed in Refs. 22 and 24. Figure 13 shows the rate of convergence to the global minimum of M in the domain D as a function of . The convergence towards the coordinates of the DT is oscillatory and rather slow since values up to 600 are required. A slight difference between the exact coordinates of the DT and the numerically computed limit coordinates is evident, however we note that these differences are consistent with the precision to which the exact DT is known, which is only to the third digit. Figure 14, and more specifically Fig. 14共b兲, confirms that the exact expression in Eq. 共20兲 is in fact a distinguished trajectory according to our definition 3 with accuracy ⑀ = 4 ⫻ 10−3. Figure 15共a兲 shows a forward and backward integration along the time interval 共−50, 50兲 taking as initial data the limit coordinates supplied by our algorithm at time t0 = 0, and compares it with the exact solution of the DT. From Fig. 15共b兲 it can be seen that this trajectory evolves close to the exact solution in the entire time range. This result shows that
013111-11
Chaos 19, 013111 共2009兲
Distinguished trajectories a)
-0.975
b)
-0.98
0.14
0.12
-0.985
-0.99
Coordinate Y
Coordinate X
0.1
-0.995
0.08
0.06
-1 0.04 -1.005 0.02 -1.01
0
200
400
600
800
0
1000
200
400
600
800
1000
Integration time τ
Integration time τ
FIG. 13. Evolution of the coordinates of the global minimum of M vs at t0 = 0. 共a兲 The x coordinate; 共b兲 the y coordinate. These plots show the convergence of the minima to the coordinates of the DET whose position is marked with a continuous horizontal line.
contrary to what happens near hyperbolic trajectories, near nonhyperbolic trajectories, a small error does not amplify and as a consequence, once a DT is known to exist it could have been computed simply by integrating forwards and backwards the limit coordinates found at a given time tk. However one needs to be careful here, as a trajectory is not necessarily distinguished at all times, and for it to be properly called distinguished, it should be verified that it stays close to the limit coordinates in the whole time interval, and
a)
therefore one cannot avoid computing limit coordinates along the time interval in this case either. We will return to this point in the next section. B. The rotating Duffing equation
Next we analyze the aperiodic hyperbolic distinguished trajectory of a system already studied in Ref. 26 the rotating Duffing equation,
b)
0.1
0.0035
0.003 0.05
0.0025
Error
Y
0.002 0
0.0015
0.001
-0.05
0.0005 -0.1 -1.1
-1.05
-1 X
-0.95
-0.9
0
0
1
2
3 Time t
4
5
6
FIG. 14. 共a兲 Dotted line represents the exact nonhyperbolic distinguished trajectory and the solid line stands for the numerically computed limit coordinates; 共b兲 distance between the exact nonhyperbolic trajectory 共20兲 and the limit coordinates.
013111-12 a)
Chaos 19, 013111 共2009兲
J. A. Jiménez and A. M. Mancho
C. A 3D extension of the Duffing equation
−0.9
x
−0.95
In this section we apply our definitions to an example in higher dimension. In particular we consider a 3D extension of the Duffing equation,
−1 −1.05 −1.1 −50
b)
4
−40
x 10
−30
−20
−10
0
t
10
20
30
40
50
x˙ = y,
y˙ = x − x3 + sin共t兲,
x
xDHT共t兲 = − 2
0 −2 −40
−30
−20
−10
0
t
10
20
30
40
冉冊冉 ˙2
=
sin 2t
cos 2t +
cos 2t −
− sin 2t
冊冉 冊 1 2
+ 共 sin t − 关cos t1 − sin t2兴3兲
冉 冊 sin t
cos t
.
共25兲
This Duffing equation is quasiperiodic in time when the rotation rate is irrational. It is obtained from the system 共10兲 by applying the rotation x = R共t兲, where R共t兲 =
冉
cos t − sin t sin t
cos t
冊
共26兲
.
The DHT can also be obtained through the coordinate transformation,
DHT共t兲 = R共t兲−1xDHT共t兲.
共27兲
Figure 16, in particular Fig. 16共b兲, confirms that the DHT 共27兲 is also a DHT according to our definition 10 with accuracy ⑀ = 4 ⫻ 10−6.
a)
冢
50
FIG. 15. 共a兲 x-coordinate vs time for the DET 共solid line兲 and the trajectory integrated taking as initial data the limit coordinates located at time t0 = 0 共dashed line兲; 共b兲 time evolution of the differences between these trajectories.
˙1
共28兲
The hyperbolic fixed point of the unperturbed autonomous system 共i.e., = 0兲 is at the origin x = 共0 , 0 , 0兲. The solution for small becomes
3
2
−50
z˙ = z + sin共t兲.
0.06
−
3 40
sin t cos t cos t − sin t
冢
2 sin3 t +
冣 3 sin t cos2 t 2
3 cos3 t + 3 sin2 t cos t 2 0
冣
+ O共5兲.
共29兲
The numerical scheme explained in Sec. III is easily adapted to higher dimensions. However some changes must be made. The computation of M requires approximating lengths of trajectories which in 3D needs an interpolation scheme different to that of Eq. 共16兲, which is only valid in R2. We consider the linear interpolation instead. This interpolation evaluates the function M satisfactorily if trajectories are represented by a large number of points. This is achieved by using a Runge–Kutta method with time step h = 10−4. Figure 17 indicates the evolution of coordinates associated with the minimum of M as a function of 共solid line兲. The dashed line corresponds to the exact perturbative solution. There is evidence of a clear convergence towards the exact position although there is a significant jump in the asymptotic behavior beyond ⬃ 50. This jump is due to round off errors in the determination of M for large . The third equation in Eq. 共28兲 is just a linear equation and for this reason solutions which are in the neighborhood of the DHT have z-coordinate growing exponentially in backwards time. Thus for large values, the evaluation of M is made along very long trajectories in
b)
3.5e-06
3e-06
0.04
2.5e-06
0.02
Error
Y
2e-06 0
1.5e-06 -0.02 1e-06 -0.04
-0.06 -0.06
5e-07
-0.04
-0.02
0 X
0.02
0.04
0.06
0
0
2
4
6
8
10
Time t
FIG. 16. 共a兲 Dashed line represents the exact distinguished hyperbolic trajectory of the rotating Duffing equation and the solid line stands for the numerically computed one; 共b兲 distance between the exact and the numerical distinguished hyperbolic trajectories.
013111-13
0.0006
Coordinate X
Coordinate Y
0.0004
0.0002
0
-0.0002
c)
b)
0.0008
0
10
20
30
40
-0.0497
-0.0497
-0.0498
-0.0498
-0.0499
-0.0499
Coordinate Z
a)
Chaos 19, 013111 共2009兲
Distinguished trajectories
-0.05
-0.05
-0.0501
-0.0501
-0.0502
-0.0502 0
50
10
Integration time τ
20 30 Integration time τ
40
0
50
10
20
30
40
50
Integration time τ
FIG. 17. Evolution of the coordinates of the global minimum of M for the 3D example vs at t0 = 0. 共a兲 The x coordinate; 共b兲 the y coordinate; 共c兲 the z coordinate. These plots show the convergence of the minima to the coordinates of the DHT whose position is marked with a dashed horizontal line.
on the ␦-grid displayed in Fig. 8. The number of neighbors of the interior point grows with the dimension n as 3n, therefore when the problem increases its dimension from n to n + 1, the computational demands are multiplied by 3. Another factor that contributes to increased computational time is the decrease of the Runge–Kutta time step h in the evaluation of trajectories on the ␦-grid. This increases the number of points in the trajectory 共and therefore the number of operations兲 with respect to the previous Dristchel approach by a factor 100. This factor is partially balanced by the fact that for the same number of points the arc length is computed more rapidly with the linear than with the Dristchel interpolation.
the z-coordinate, which are underrepresented by points sampled every h = 10−4 共see Table I兲 and where lengths are badly calculated by adding up very small and very large 共and inaccurate兲 numbers. In spite of this, Fig. 18 confirms that the exact distinguished trajectory can be accurately obtained with our methodology and that for ⬍ 50 errors are within the expected margin. The remaining input parameters used in Figs. 17 and 18 are D = 共−0.2, 0.2兲 ⫻ 共−0.2, 0.2兲 ⫻ 共−0.2, 0.2兲, 0 = 2, ⌬ = 1, ␦ = 10−6, step size h = 10−4 in the Runge–Kutta method, t0 = 0, tN = 6, and ⌬t = 10−2. As we explain next, the computational demands made by this example are considerably larger than they were for the previously considered 2D examples. When determining a DT, most of the CPU time is spent computing the value of M
a)
b) 0.1
1.2e-06 1e-06
0.05
0
Error
Z
8e-07
−0.05
6e-07 4e-07 2e-07
−0.1 0.1 0.1
0.05 0.05
0
0
−0.05
Y
−0.1
0
0
1
2
3
4
5
6
Time t
−0.05 −0.1
X
FIG. 18. 共a兲 The solid line represents the exact distinguished hyperbolic trajectory of the 3D equation and circles stand for numerically computed coordinates; 共b兲 distance between the exact and the numerical distinguished hyperbolic trajectories.
013111-14
Chaos 19, 013111 共2009兲
J. A. Jiménez and A. M. Mancho
2000
day = 300
1800 1600 1400 1200 1000 800 600 400 200 0 0
500
1000
FIG. 19. Contour plot of the stream function produced by the quasigeostrophic model at day 300.
V. APPLICATION TO VECTOR FIELDS DEFINED AS FINITE TIME DATA SETS
In this section we explore definition 3 for a highly aperiodic 2D flow in which the vector field is defined as a finite time data set. In particular we consider the output of a quasigeostrophic wind-driven double gyre model in a regime already studied in Refs. 7 and 8. Details of this model may be found in Refs. 8 and 17. Figure 19 shows a typical output for the stream function provided by this model. The velocity data set is obtained on a 1000 km⫻ 2000 km rectangular domain and spans 4000 days. This interval is considered for a fluid started from rest and allowed to spin for 25 000 days. Free slip conditions are considered for the velocities on the boundaries and the wind stress curl is 0.32 dyn/cm2. The equations of motion for this system are given by x˙ = vx共x,y,t兲 = −
y˙ = vy共x,y,t兲 =
, y
, x
共30兲
共31兲
and the variables x and y are in the rescaled domain 关0 , 1兴 ⫻ 关0 , 2兴. Here the velocity fields vx and vy are provided as a finite time data set and are interpolated using bicubic interpolation in space and third order Lagrange polynomials in time. This method has been reported to be good enough for integrating trajectories in Ref. 29. We will focus our analysis in the time interval 关0, 900兴 in the area marked by a rectangle in Fig. 19 for which Ref. 8 reports on the computations of several DHTs. In Ref. 8 distinguished trajectories are computed by means of an iterative algorithm which is
initialized on a hyperbolic instantaneous stagnation point 共ISP兲. In particular two paths of such ISPs are chosen in the Northern gyre in the time intervals 关0, 339兴 and 关446, 880兴. From each of these paths, a DHT is computed which is in the same geographical area although its coordinates are determined for a different time range. In Fig. 20 we show the x and y evolutions for these trajectories. These coordinates have been computed with a different algorithm to that proposed in Ref. 8. Instead each corresponds to a trajectory which is in the intersection of a piece of a stable manifold and a piece of an unstable manifold which are evolved in backwards and forwards time, respectively. In this procedure, in order to avoid the numerous intersections between stable and unstable manifolds, which make difficult the tracking of the trajectory which is distinguished, manifolds are trimmed at each time step following the ideas in Ref. 7, where a method is described to compute a piece of single branch of the stable or of the unstable manifold. This method takes advantage of the fact that a DHT must be in the intersection of both manifolds at all times, as it is a trajectory. However, it does not improve the method explained in Ref. 8 in the sense that it does not allow either to extend the computation of the DHT beyond the time interval in which the ISP exists. Many questions have been raised for these trajectories as has been discussed in Refs. 7 and 8. For instance, as they have been computed only in finite time intervals on which the ISP exists, one can ask how to pursue its computation beyond that interval. Another open issue in Ref. 8 concerns deciding if the two DHT in Fig. 20 computed at different times are part of the same trajectory. In Ref. 7, the question is raised of whether it can happen that a DHT ceases to be distinguished or hyperbolic. In this section we apply our algorithm to compute limit coordinates and verify whether trajectories in Fig. 20 are distinguished or not following our definition 3. Also we will describe how this definition helps address the questions raised in Refs. 7 and 8. We have applied our algorithm to compute limit coordinates in the domain in which the DHT shown in Fig. 20共a兲 exists. In particular we have applied it with the input D = 共55, 75兲 ⫻ 共1325, 1375兲 km2, t0 = 120, tN = 300, ⌬t = 5 days, 0 = 2 days, ⌬ = 5 days, and ␦ = 10−3 km. The time step of the Runge–Kutta method is 0.1 days. Figure 21共a兲 indicates with a solid line the projection onto the x − y plane of the trajectory depicted in Fig. 20共a兲 in the interval 共120, 300兲, and with circles the path of limit coordinates. Figure 21共b兲 shows the evolution of the distances between these trajectories. This confirms that the trajectory displayed in Fig. 20共a兲 is also distinguished in the sense of definition 3 in the time interval 关120, 330兴 with accuracy ⑀ = 8 ⫻ 10−1 km. Thus in this time interval, limit coordinates give a method for computing DT different from those proposed in Refs. 8 and 19. Circles in Fig. 22 show the location versus time of the x limit coordinates computed with our algorithm. The solid line represents a trajectory obtained after integrating with a fifth order Runge–Kutta method forwards and backwards in time the initial condition of the circle at day 285. The dashed line represents the same, but with the initial condition slightly perturbed. It is evident that in both cases the trajectories are aligned with the path of limit coordinates. The distinguished trajectory is highly hy-
013111-15
a)
Chaos 19, 013111 共2009兲
Distinguished trajectories
110
1550
x (km)
y (km) 1500
100
1450 90 1400 80 1350 70
60 0
b)
1300
50
100
150
day200
250
300
350
100
1250 0
100
150
day 200
250
300
350
1500
x (km)
y (km)
90
1450
80
1400
70
1350
60
1300
50
1250
40 450
50
500
550
600
650
700
day
750
800
850
900
1200 450
500
550
600
650
700
day
750
800
850
900
FIG. 20. Distinguished hyperbolic trajectories in the Northern gyre of the quasigeostrophic model reported in Ref. 8. 共a兲 Evolution of the x and y coordinates in the time interval 关5,338兴; 共b兲 evolution of the x and y coordinates in the time interval 关450,880兴.
a) 1500 y (km)
b)
0.8 0.7
1460
0.6
1440
0.5
Error (km)
1480
1420 1400
0.4 0.3
1380
0.2
1360
0.1
1340 65
70
75
80
x (km)
85
90
95
0 120
140
160
180
200
220
240
260
280
300
days
FIG. 21. 共a兲 Solid line represents the projection on the phase space of the distinguished hyperbolic trajectory depicted in Fig. 20共a兲 and the circles stand for the numerically computed limit coordinates; 共b兲 distance between the trajectories represented in 共a兲.
013111-16
Chaos 19, 013111 共2009兲
J. A. Jiménez and A. M. Mancho
FIG. 22. Circles stand for the x component of the limit coordinates in the time range where they approach a DT. The solid line represents a trajectory integrated with a fifth order RungeKutta method passing the through limit coordinates at day 285. The dashed line is a trajectory integrated from the same condition plus a small perturbation.
perbolic backwards in time as in that direction a small perturbation amplifies greatly, while it does not do so forwards in time, suggesting that it has a nonhyperbolic type of stability in that direction 共see comments to Figs. 6 and 15兲. Beyond day 300 it is possible to continue the path of limit coordinates. Figure 23 shows a diagram at day 330; showing the convergence of the x component of the minimum of M versus . This type of convergent diagram is not found in this neighborhood for day 337. On the other hand, although it is possible to continue the path of limit coordinates beyond day 300, Fig. 24 proves that this path is not a trajectory. There can be seen the existence of different trajectories crossing the path, confirming that it is not a trajectory as otherwise it would violate the uniqueness of the solution. Therefore, following our construction it is possible to say that beyond day 300 the trajectory is no longer distinguished. Figure 25 confirms that the trajectory in Fig. 20共b兲 is also distinguished in the sense of definition 3 in the time interval 共470, 860兲 with accuracy ⑀ = 3 km. In particular to compute the path in Fig. 25 we have applied the algorithm of Sec. III with the input D = 共50, 65兲 ⫻ 共1255, 1270兲 km2, t0 = 470, tN = 860, ⌬t = 5 days, 0 = 2 days, ⌬ = 7 days, and
␦ = 10−3 km. The Runge–Kutta time step is 0.1 days. In the time interval from day 600 to day 650 some of the input parameters were modified as follows: D = 共73.5, 75.5兲 ⫻ 共1384, 1392兲 km2, 0 = 40 days, and ⌬t = 1 day. This was due to the presence of nearby elliptic-type minima in the function M, that made it difficult to track the path of the limit coordinates with the previous input. Finally, we discuss the existence of nonhyperbolic distinguished trajectories in this data set. The presence of this type of trajectory has not been addressed before, and we do not have any benchmark solution. We have looked for this type of trajectory in areas of the flow where Eulerian eddies seemed to persist for long times. Figure 26 represents the function M at day 370 for = 150 and = 250. In these figures there can be seen the structure of an eddy at the center even for rather long values. However, Fig. 27 does not confirm the convergence of the minimum of M towards a constant value. On the other hand, the slow convergence in diagram 13 towards the nonhyperbolic trajectory, already suggested that long time intervals were required for that purpose, and those intervals might be difficult to find in realistic flows such like the one analyzed here, in which one is provided just with a finite time data set. VI. CONCLUSIONS
b)
130
1550
125
1545
120
1540
115
1535
Coordinate y
Coordinate x
a)
110
105
1530
1525
100
1520
95
1515
90 0
50
100
150
Integration time τ
200
1510 0
50
100
150
Integration time τ
200
FIG. 23. 共a兲 x component of the minimum of M vs at day 330; 共b兲 y component of the minimum of M vs at the same day.
In this paper we have proposed a new definition of distinguished trajectory that attempts to extend the concept of fixed point and periodic orbit to aperiodic dynamical systems. The concept of fixed point is trivially contained in the definition. Regarding other especially useful trajectories in dynamical systems, for instance, periodic orbits, we have not proven that they fall within the definition in a general way, but we have numerically verified it for selected 2D and 3D examples. The definition can be implemented numerically and the procedure consists of determining a path of limit coordinates. We have analyzed exact examples for the Duffing equation with known distinguished trajectories, both periodic and aperiodic, and we have found that the path of limit coordinates coincides, to within numerical accuracy, with the distinguished trajectories and therefore those trajectories are identified also as distinguished in the framework of our definition. Our definition is novel with respect to previous works dealing with distinguished trajectories, because it is appli-
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Chaos 19, 013111 共2009兲
Distinguished trajectories
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FIG. 25. 共a兲 Solid line represents the projection on the phase space of the distinguished hyperbolic trajectory depicted in Fig. 20共b兲 and circles stand for the numerically computed limit coordinates; 共b兲 distance between the trajectories represented in 共a兲.
FIG. 26. 共a兲 Contour plot of M t=370,=150, the elliptic minimum is in the dark area almost at the center; 共b兲 contour plot of M t=370,=250.
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tional part of this work was done using the CESGA computers SVGD and FINIS TERRAE and using the SIMUMATCSIC cluster ODISEA. The authors have been supported by CSIC Grant No. PI-200650I224 and OCEANTECH 共No. PIF06-059兲, Consolider I-MATH 共C3-0104兲, and the Comunidad de Madrid Project No. SIMUMAT S-0505-ESP-0158.
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H. Aref, J. Fluid Mech. 143, 1 共1984兲. D. V. Khakhar and J. Ottino, Phys. Fluids 29, 3503 共1986兲. 3 T. Dombre U. Frisch J. M. Greene M. Henon, A. Mehr, and A. M. Soward, J. Fluid Mech. 167, 353 共1986兲. 4 S. Wiggins, Chaotic Transport in Dynamical Systems 共Springer-Verlag, New York, 1992兲. 5 N. Malhotra and S. Wiggins, J. Nonlinear Sci. 8, 401 共1998兲. 6 G. Haller and A. Poje, Physica D 119, 352 共1998兲. 7 A. M. Mancho, D. Small, and S. Wiggins, Phys. Rep. 437, 55 共2006兲. 8 A. M. Mancho, D. Small, and S. Wiggins, Nonlinear Processes Geophys. 11, 17 共2004兲. 9 S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos 共Springer-Verlag, New York, 2003兲. 10 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 共Springer-Verlag, New York, 2002兲. 11 J. A. Langa, J. C. Robinson, and A. Suarez, Nonlinearity 15, 887 共2002兲. 12 J. A. Langa, J. C. Robinson, and A. Suarez, J. Differ. Equations 221, 1 共2006兲. 13 E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, and A. Vulpiani, J. Phys. A 30, 1 共1997兲. 14 G. Haller, Physica D 149, 248 共2001兲. 15 J. M. Nese, Physica D 35, 237 共1989兲. 16 G. Haller, J. Fluid Mech. 512, 257 共2004兲. 17 C. Coulliette and S. Wiggins, Nonlinear Processes Geophys. 8, 69 共2001兲. 18 S. Eisenbach and R. Friedrich, Theor. Comput. Fluid Dyn. 22, 213 共2008兲. 19 K. Ide, D. Small, and S. Wiggins, Nonlinear Processes Geophys. 9, 237 共2002兲. 20 N. Ju, D. Small, and S. Wiggins, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 1449 共2003兲. 21 A. M. Mancho E. Hernández-García, D. Small, and S. Wiggins, J. Phys. Oceanogr. 38,1222 共2008兲. 22 M. Branicki, A. M. Mancho, and S. Wiggins, “A Lagrangian description of transport associated with a front-eddy interaction: Application to data from the North-Western Mediterranean Sea,” reprint 共submitted兲. 23 D. B. Chelton, M. G. Schlax, R. M. Samelson, and R. A. de Szoeke, Geophys. Res. Lett. 34, L15606, DOI: 10.1029/2007GL030812 共2007兲. 24 G. Haller, J. Fluid Mech. 525, 1 共2005兲. 25 A. Szeri, L. G. Leal, and S. Wiggins, J. Fluid Mech. 228, 207 共1991兲. 26 A. M. Mancho, D. Small, S. Wiggins, and K. Ide, Physica D 182, 188 共2003兲. 27 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in Fortran 77,” The Art of Scientific Computing, 2nd ed. 共Cambridge University Press, Cambridge, 1999兲. 28 D. G. Dritschel, Comput. Phys. Rep. 10, 77 共1989兲. 29 A. M. Mancho, D. Small, S. Wiggins, and K. Ide, Comput. Fluids 35, 416 共2006兲. 1 2
1215 185 1210
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FIG. 27. 共a兲 x component of the minimum of M vs at day 370; 共b兲 y component of the minimum of M vs at the same day.
cable to nonhyperbolic trajectories. In particular, we have studied a periodic orbit of the Duffing equation with nonhyperbolic stability and it is also recognized as distinguished by our definition. In this case the function M from which the limit coordinates are computed seems to be a suggestive tool for characterizing Lagrangian eddies. We have tested our definition in the context of realistic aperiodic flows where distinguished hyperbolic trajectories had been found.7,8 Again we have identified these trajectories by paths of limit coordinates in certain time intervals. Beyond these time intervals the trajectories are no longer distinguished according to our definition. Thus in the context of the definitions provided in this paper, the property of a trajectory of being distinguished may be lost in time. Also we have found evidence that the hyperbolicity of these trajectories is not constant in time. These two statements provide answers to the open questions mentioned in the text that have been addressed in Refs. 7 and 8. ACKNOWLEDGMENTS
It is a pleasure to acknowledge many useful conversations with Steve Wiggins, Des Small, Peter Haynes, Emilio Hernández-García, Cristóbal López, Antonio Turiel, Emilio García-Ladona, Michal Branicki, Wenbo Tang, and J. J. L. Velázquez on numerous issues related to this project. We are also thankful for the very valuable suggestions from Daniel Fox, Andrew Thompson, and Jodie Holdway. The computa-
CHAOS 19, 013112 共2009兲
Pinning control of fractional-order weighted complex networks Yang Tang,1,a兲 Zidong Wang,1,2 and Jian-an Fang1,b兲 1
College of Information Science Technology, Donghua University, Shanghai 201620, China Department of Information Systems and Computing, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
2
共Received 18 October 2008; accepted 17 December 2008; published online 10 February 2009兲 In this paper, we consider the pinning control problem of fractional-order weighted complex dynamical networks. The well-studied integer-order complex networks are the special cases of the fractional-order ones. The network model considered can represent both directed and undirected weighted networks. First, based on the eigenvalue analysis and fractional-order stability theory, some local stability properties of such pinned fractional-order networks are derived and the valid stability regions are estimated. A surprising finding is that the fractional-order complex networks can stabilize itself by reducing the fractional-order q without pinning any node. Second, numerical algorithms for fractional-order complex networks are introduced in detail. Finally, numerical simulations in scale-free complex networks are provided to show that the smaller fractional-order q, the larger control gain matrix D, the larger tunable weight parameter , the larger overall coupling strength c, the more capacity that the pinning scheme may possess to enhance the control performance of fractional-order complex networks. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3068350兴 Recently, fractional-order differential systems have been widely investigated due to their potential applications in viscoelasticity, dielectric polarization, quantum evolution of complex systems, and many other fields. On the other hand, research of complex networks has triggered tremendous interest during the past decade. Most studies to date have concerned integer-order complex networks. In this paper, we consider the pinning control problem of fractional-order weighted complex dynamical networks. The fractional-order complex networks generalize wellstudied integer-order complex networks. Some local stability properties of such pinned fractional-order networks are derived and the valid stability regions are estimated by utilizing the eigenvalue analysis and fractional-order stability theory. A surprising finding that the fractional-order networks can stabilize itself by reducing the fractional-order q without pinning any node is presented. The numerical algorithms for fractional-order networks are also presented. In the end, computer simulations in scale-free networks are given to show that the smaller fractional-order q, the larger control gain matrix D, the larger tunable weight parameter , the larger coupling strength c, the more capacity that the pinning strategy can possess to accelerate the control rate of fractional-order networks. I. INTRODUCTION
Many large-scale systems in nature and human societies, such as, genetic regulatory networks, food webs, the internet, scientific citation web, etc., can be modeled by networks, where nodes are individuals of the system and the edges a兲
Electronic mail:
[email protected]. Electronic mail:
[email protected].
b兲
1054-1500/2009/19共1兲/013112/9/$25.00
represent interactions between them.1–4 One decade ago, to describe the transition from regular networks to random ones, Watts and Strogatz 共WS兲 共Ref. 1兲 introduced the concept of small-world networks. After that, a scale-free network model was proposed by Barabási and Albert 共BA兲,2 in which the degree distribution of the nodes follows a powerlaw form. Thereafter, the small-world features and the scalefree properties of complex networks have attracted increasing attention from researchers in their studies.5–8 On the other hand, in recent years, fractional calculus has drawn much attention due to its application in physics and engineering.9–12 It has been revealed that, in interdisciplinary fields, various systems have been found to exhibit fractional dynamics. For example, viscoelasticity, dielectric polarization, quantum evolution of complex system, fractional kinetics, and anomalous attenuation can be described by fractional differential equations.9–12 Meanwhile, it has been shown that many chaotic systems display the fractionalorder chaotic dynamics, such as, fractional-order Lorenz system,13 fractional-order Chua’s system,14 fractional-order cellular neural networks,15 fractional-order Rössler system,16 and so on. Recently, the collective dynamics analysis of complex networks has led to a host of interesting effects.6–8,17–34 In particular, the study on controlling the dynamics of a network and guiding it to a desired state, such as, an equilibrium point or a periodic orbit of the network has become an interesting and important direction in this research field.7,18–20,22–24,26,27 It has been revealed that, in the process of controlling various networks, feedback control serves as a simple and effective approach for stabilization and synchronization. However, it is widely believed that it is impossible to add controllers to all nodes. To reduce the control cost, some feedback injections may be added to a
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fraction of network nodes, which is known as pinning control.17–20,22–24,26,27,33 In Ref. 17, pinning control of spatiotemporal chaos was discussed. In Ref. 7, both specific and random pinning schemes were studied. Li et al.18 investigated the pinning control of complex dynamical networks to their equilibriums including both the random networks and the scale-free networks. However, to the best of the authors’ knowledge, the research on the dynamical analysis of complex networks, such as, control and synchronization of complex networks, has mainly focused on integer-order complex networks, and the corresponding research on fractional-order complex networks has received very little attention despite its practical significance. Motivated by the above observations, four seemingly natural questions arise as follows: 共1兲 Does the fractionalorder complex networks exist which can be pinned? 共2兲 What kind of pinning controllers may be designed to ensure a fractional-order complex network is stabilized to its equilibriums? 共3兲 How large of the coupling strength should be used for a fractional-order network with a fixed structure to achieve local stability? 共4兲 What factors effectively affect the convergence rate of controlling the fractional-order networks? The overall aim of this paper is to provide convincing answers to the above four questions by providing a rather general framework. The rest of this paper is organized as follows: In Sec. II, some preliminaries of fractional calculus and fractional-order complex networks are briefly outlined. The main results for pinning control on fractional-order complex networks are given in Sec. III. In Sec. IV, the algorithms for simulating fractional-order networks and numerical examples are provided. Concluding remarks are presented in Sec. V.
Consider a fractional-order weighted complex dynamical network consisting of N identical nodes, described by N
D*xi共t兲 = f关xi共t兲兴 + c q
dmJm−q f共t兲 , dtm
q ⬎ 0,
共1兲
where m = q, i.e., m is the first integer which is not less than q. J␥ is the ␥-order Riemann–Liouville integral operator which can be described by the following expression: J␥g共t兲 =
兰t0共t − 兲␥−1g共兲d , ⌫共␥兲
␥ ⬎ 0,
共2兲
where ⌫共·兲 is a gamma function given by ⌫共z兲 =
冕
⬁
共3兲
tz−1e−tdt.
0
In this paper, we adopt the following definition: Dq*x共t兲 = Jm−qx共m兲共t兲,
q ⬎ 0,
共4兲
where m = q. The operator Dq* is generally called the Caputo differential operator of order q.35,36
共5兲
where 0 ⬍ q 艋 1 is the fractional order; f共·兲 苸 Rn is a given nonlinear continuously differentiable function describing the local dynamics of the nodes; xi = 共xi1 , xi2 , . . . , xin兲T 苸 Rn represents the state vector of the ith node; c ⬎ 0 is the overall coupling strength; ⌫ ⬎ 0 苸 Rn⫻n is a constant matrix indicating inner-coupling between the elements of the node itself; matrix G = 共Gij兲N⫻N is the coupling configuration matrix representing the topological structure of the network, if there is a connection between node i and node j共i ⫽ j兲, Gij is positive if there is a direct influence from node j, where Gij gives a measure of the strength of the interaction. The diagonal elements of matrix G are defined as N
Gii = −
兺
共6兲
Gij ,
j=1,j⫽i
which ensures the diffusion satisfying 兺Nj=1Gij = 0. Suppose that network 共5兲 is connected, then G is an irreducible real matrix, and the network 共5兲 can be rewritten as N
D*xi共t兲 = f关xi共t兲兴 + c 兺 Gij⌫x j共t兲, q
i = 1,2, . . . ,N.
共7兲
j=1
In this paper, we focus on a class of weighted fractionalorder networks where G is diagonalizable and has real eigenvalues. In particular, G can be written as
II. PRELIMINARIES
Dq f共t兲 =
Gij⌫关x j共t兲 − xi共t兲兴,
i = 1,2, . . . ,N,
Gij = − There are many definitions of fractional derivative. The most well-known definition is the Riemann–Liouville fractional derivative, defined by
兺
j=1,j⫽i
Lij ki
共8兲
,
where ki is the degree of node i, and  is a tunable parameter; the real matrix L = 共Lij兲N⫻N is the usual 共symmetric兲 Laplacian matrix with diagonal entries Lii = ki and offdiagonal entries Lij = −1 if node i and j are connected by a link, and Lij = 0 otherwise. In this model, we assume that network 共7兲 is strong connected in the sense of having no isolated clusters, which means that the coupling matrix G is irreducible. In sum, G is irreducible and negative semidefine. Our aim is to stabilize network 共7兲 onto a homogeneous stationary state if lim 储xi共t兲 − ¯x储 = 0,
t→⬁
i = 1,2, . . . ,N,
共9兲
¯ 兲; the where ¯x is an equilibrium point satisfying Dq*¯x = f共x notation 储·储 represents for the Euclidean vector norm. Our task is to stabilize the networks by using pinning strategy. We apply the pinning control on a small fraction 共0 ⬍ ⬍ 1兲 of the nodes in the network 共7兲. Without loss of generality, let the first l nodes be controlled as identified by the set C = 兵c1 , c2 , . . . , cl其, where l = N is the integer part of the real number N. To achieve the controlling fractional-order network to its equilibriums, one should choose the number of controlled nodes l. The decision is influenced by both controllability of
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the network 共7兲 and by selection strategies. A better controllability will probably lead to less cost in terms of control. Moreover, it should be pointed out that applying different strategies for a given network will result in different values of l. A smaller l also means less cost. Assume that the set C contains l nodes where l is a fixed number. As has been shown in Refs. 18–20, 23, and 24, the coupling matrix G and the coupling strength c directly affect the controllability of the network; these parameters also affect the detailed selection, as follows from the above description. Thus, it is difficult to decide on the selection strategy. In the following section, we will show that the fractional-order q, the control gain matrix D, and the tunable parameter  and different pinning techniques also affect the controllability of the network. III. STABILITY ANALYSIS
Before beginning our main results, the following lemmas are needed to derive the main results. Lemma 1 共see Ref. 37兲. Consider the following autonomous system: Dq*X = AX,
共10兲
X共0兲 = X0 ,
with 0 ⬍ q 艋 1 , X 苸 Rn. System 共10兲 is asymptotically stable if and only if 兩arg关i共A兲兴兩 ⬎ q / 2, i = 1 , 2 , . . . , n, where arg关i共A兲兴 denotes the argument of the eigenvalue i of A. In this case, the component of the state decay towards 0 like t−q; system 共10兲 is stable if and only if either it is asymptotically stable or those critical eigenvalues which satisfy 兩arg关i共A兲兴兩 = q / 2 have a geometric multiplicity one. Lemma 2 共see Ref. 22兲. If G = 共Gij兲 苸 RN⫻N is a real irreducible matrix satisfying the diffusive coupling condition, i.e., Gij 艌 0共j ⫽ i兲 and Gii = −兺Nj=1,j⫽iGij, then 共i兲 0 is the largest eigenvalue of G with multiplicity 1 and 关1 , 1 , . . . , 1兴T is the corresponding eigenvector with identical nonzero elements; 共ii兲 there exists a nonsingular matrix ⌽ = 关1 , 2 , . . . , N兴 苸 RN⫻N, such that G k = k k,
k = 1,2, . . . ,N,
where 0 = 1 ⬎ 2 艌 ¯ 艌 N are the eigenvalues of G. Lemma 2 can be easily proved by the Gerschgorin’s disk theorem and the Perron–Frobenius theory. Lemma 3 共see Ref. 22兲. If the matrix G is defined as in Lemma 2, and the nonzero diagonal matrix D is defined as D = diag共d1 , d2 , . . . , dN兲 with di 艌 0共i = 1 , 2 , . . . , N兲, then R = G − D is negative definite. Then, the controlled network 共7兲 can be described as N
Dq*xi共t兲 = f关xi共t兲兴 + c 兺 Gij⌫x j共t兲 + ui,
i = 1,2, . . . ,N,
共11兲
j=1
Let the error states be ei共t兲 = xi共t兲 − ¯x,
Then, one can linearize the controlled networks 共11兲 at state ¯x to obtain ¯ 兲兴 + cRE共t兲⌫, Dq*E共t兲 = E共t兲JT关f共x
i = 1,2, . . . ,N,
where the feedback gain satisfies di =
再
d,i = 1,2, . . . ,l 0,i = l + 1,l + 2, . . . ,N.
冎
共12兲
共13兲
i = 1,2, . . . ,N,
共15兲
¯ 兲兴 is the Jacobian matrix of f evaluated at ¯x; where JT关f共x E共t兲 = 关e1共t兲 , e2共t兲 , . . . , eN共t兲兴T 苸 Rn⫻N. From Lemmas 2 and 3, it follows that R is negative definite, which means that all of its eigenvalues are strictly negative, denoted in an decreasing order as 0 ⬎ 1 艌 2 艌 ¯ 艌 N = min共R兲, with their corresponding 共generalized兲 eigenvectors ⌽ satisfying R k = k k, k = 关1 , 2 , . . . , N兴 苸 RN⫻N, = 1 , 2 , . . . , N. By expressing each column E on the basis of ⌽ basis, one has E = ⌽ .
共16兲
Thus, Eq. 共15兲 can be expanded into the following equations: ¯ 兲兴 + ck⌫其k共t兲, Dq*k共t兲 = 兵J关f共x
k = 1,2, . . . ,N,
共17兲
where k共t兲 = 关k1 , k2 , . . . , kn兴T. Now, the local stability problem of the 共N ⫻ n兲-dimensional system 共15兲 is converted into the stability problem of the much simpler N independent n-dimensional linear systems 共17兲. One can see that in Eq. 共17兲 only k and k are related to k. The importance of Eq. 共17兲 is that the stability problem of the controlled fractionalorder network 共11兲 can be divided into three independent problems: one is to tune the fractional-order q to make the eigenvalue of matrix R into the stable region; the second is to analyze the stable regions of the fractional-order networks 共17兲, depending on the local dynamics of the isolated node, ¯兲 such as, the equilibrium point ¯x, the Jacobian matrix Jf共x and inner coupling matrix ⌫; the last is to analyze the eigenvalue distribution of the matrix R, which depends on the network topological structure and control scheme, i.e., the tunable parameter , the overall coupling strength c, the network size N, the coupling matrix G and the control gain matrix D. The following theorem characterizes a necessary and sufficient condition for system 共7兲 to be locally exponentially stable in the homogenous state ¯x. Theorem 1: For a certain fractional-order q, the controlled network 共11兲 is locally exponentially stable about ¯x if ¯ 兲 + ck⌫兴 and only if all the eigenvalues of the matrix 关Jf共x satisfy ¯ 兲 + ck⌫兴其兩 ⬎ 兩arg兵⌳ki关Jf共x
q , 2
with the local negative feedback controllers are given by ui = − cdi⌫关xi共t兲 − ¯x兴,
共14兲
i = 1,2, . . . ,N.
共18兲 i = 1,2, . . . ,n,
k = 1,2, . . . ,N,
where ⌳ki is the eigenvalue of the kth node matrix ¯ 兲 + ck⌫兴. 关Jf共x Proof: By using Lemma 1, Theorem 1 can be easily proved.
013112-4
Chaos 19, 013112 共2009兲
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Corollary 1: For certain k, inner coupling matrix ⌫, overall coupling strength c, the control gain D, and the coupling matrix G, the controlled network 共11兲 is locally exponentially stable about ¯x if and only if all the eigenvalues of ¯ 兲 + ck⌫兴 satisfy the matrix 关Jf共x
冏
冏
¯ 兲 + ck⌫兴其兲 2 共Im兵⌳ki关Jf共x q ⬍ tan−1 , ¯ 兲 + ck⌫兴其兲 共Re兵⌳ki关Jf共x 共19兲 i = 1,2, . . . ,n,
k = 1,2, . . . ,N.
Remark 1: One can easily see that Corollary 1 is essentially equivalent to Theorem 1. However, the significance of Corollary 1 is to show us that reducing the value of fractional-order q can stabilize the homogenous stationary state when the coupling matrix G, the control gain matrix D, the coupling strength c, the inner matrix ⌫, the pinning schemes are fixed all the time. It can be seen that the stable region is becoming larger, while the unstable region is becoming smaller. Corollary 1 gives us a new idea that we can adjust the fractional-order q to achieve stabilization. It should be noted that even if the networks may not be pinned by any node, the networks can also be stabilized by tuning fractional-order q. One can see the numerical simulation in Sec. IV C which supports this point. Hereafter, we give a more useful theorem to characterizes a necessary and sufficient condition for system 共11兲 to be locally exponentially stable in the homogenous state ¯x. Theorem 2: For certain k, inner coupling matrix ⌫, overall coupling strength c, the control gain D, and the coupling matrix G, the controlled network 共11兲 is locally exponentially stable about ¯x if and only if all the eigenvalues of ¯ 兲 + c1⌫兴 satisfy 兩arg兵⌳1i关Jf共x ¯ 兲 + c1⌫兴其兩 the matrix 关Jf共x ⬎ q / 2, i = 1 , 2 , . . . , n. Proof: Since 1 is the largest eigenvalue of the negative definite matrix R 关see Eq. 共15兲兴, only if the eigenvalues of the ¯ 兲 + c1⌫兴 satisfy 兩arg兵⌳1i关Jf共x ¯ 兲 + c1⌫兴其兩 ⬎ q / 2, matrix 关Jf共x ¯ 兲 + ck⌫兴, k = 2 , . . . , N must satisfy all the eigenvalues of 关Jf共x the Lemma 1. Thus, according to the stability theory of fractional-order system, the controlled network 共11兲 is locally exponentially stable about ¯x. The necessity is obvious. This completes the proof. Corollary 2: For certain k, inner coupling matrix ⌫, overall coupling strength c, the control gain D, and the coupling matrix G, the controlled network 共11兲 is locally exponentially stable about ¯x if and only if all the eigenvalues of ¯ 兲 + c1⌫兴 satisfy the matrix 关Jf共x q⬍
冏
冏
¯ 兲 + c1⌫兴其兲 2 共Im兵⌳1i关Jf共x tan−1 , ¯ 兲 + c1⌫兴其兲 共Re兵⌳1i关Jf共x
i = 1,2, . . . ,n. 共20兲
Suppose that there exists a constant ⬎ 0 such that its ¯ 兲 − ⌫兴 satisfy 兩arg兵⌳1i关Jf共x ¯兲 eigenvalues of the matrix 关Jf共x − ⌫兴其兩 ⬎ q / 2, i = 1 , 2 , . . . , n. If c1 艋 −, then ck 艋 − for ¯ 兲 + ck⌫兴 k = 1 , 2 , . . . , N. Thus, all the eigenvalues of 关Jf共x have satisfied Lemma 1. Then, the following corollary can be obtained without any difficulty. Corollary 3: Assume that there is a constant ⬎ 0 such ¯ 兲 − ⌫兴 satisfy that all the eigenvalues of the matrix 关Jf共x
¯ 兲 − ⌫兴其兩 ⬎ q / 2, i = 1 , 2 , . . . , n. Then, the con兩arg兵⌳1i关Jf共x trolled network 共11兲 is locally exponentially stable about ¯x, provided that c1 艋 − .
共21兲
By the idea of pinning control, the number of controllers is preferred to be very small compared with the entire network size N, namely, l Ⰶ N. According to Lemma 2, R is negative if D ⫽ 0. Considering the best possible case of l = 1, we can easily obtain the following result from the above analysis. Corollary 4: Suppose that the feedback gain matrix D = diag共d , 0 , . . . , 0兲, which means that only one node is to be pinned. Meanwhile, we assume that there exists a constant ⬎ 0 as in Corollary 1 such that all the eigenval¯ 兲 − ⌫兴 satisfy 兩arg兵⌳1i关Jf共x ¯ 兲 − ⌫兴其兩 ⬎ q / 2, ues of 关Jf共x i = 1 , 2 , . . . , n. Then the controlled network 共11兲 is locally exponentially stable about ¯x, provided that c1 艋 − .
共22兲
Remark 2: Obviously, one can see that Corollary 4 is a special case of Corollary 3. The importance of Corollary 4 is to tell us that pinning any one node in the network can stabilize the homogenous stationary state when the network is controllable and connected all the time. From the conditions 共21兲 and 共22兲, it follows that fixed coupling matrix G, control gain matrix D, and the overall coupling strength c guarantee the fractional-order network stabilization. We can easily obtain the following result from the above analysis. Corollary 5: Suppose that the feedback gain matrix D = diag共d , 0 , . . . , 0兲, which means that only one node is to be pinned. Moreover, we assume that there exists a constant ⬎ 0 as in Corollary 1 such that all the eigenval¯ 兲 − ⌫兴 satisfy 兩arg兵⌳1i关Jf共x ¯ 兲 − ⌫兴其兩 ⬎ q / 2, ues of 关Jf共x i = 1 , 2 , . . . , n. Then the controlled network 共11兲 is locally exponentially stable about ¯x, provided that c 艌 − /1 ⬎ 0.
共23兲
The Corollary 5 shows that sufficiently strong overall coupling strength will lead a network to stabilized. On the other hand, the following result can be obtained. Corollary 6: Suppose that the feedback gain matrix D = diag共d , 0 , . . . , 0兲, which means that only one node is to be pinned. Moreover, we assume that there exists a constant ⬎ 0 as in Corollary 1 such that all the eigenval¯ 兲 − ⌫兴 satisfy 兩arg兵⌳1i关Jf共x ¯ 兲 − ⌫兴其兩 ⬎ q / 2, ues of 关Jf共x i = 1 , 2 , . . . , n. Then the controlled network 共11兲 is locally exponentially stable about ¯x, provided that 1 艋 − /c ⬍ 0.
共24兲
Remark 3: It is can be revealed from Corollary 6 that, for a given coupling strength c and the coupling matrix G, we can choose control gain matrix D 共i.e., choose sufficiently large l or large d兲 to stabilize the network to its equilibriums. IV. EXAMPLES
In this section, in order to show the effectiveness of the proposed results, some numerical examples in scale-free
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complex networks are provided by employing different pinning schemes. Moreover, it is worth mentioning that some properties of fractional-order complex networks, such as, the fractional-order q, the tunable parameter , the overall coupling strength c, the coupling matrix G, the control gain matrix D, and different pinning schemes, will affect the controllability of weighted fractional-order networks. A. Numerical algorithms
In order to simulate the fractional-order complex networks, we have to make some modifications of the algorithms proposed in Ref. 38. The algorithms to simulate the fractional-order complex networks without control can be given as follows: The fractional-order differential equations with coupling terms are given by
xih共tn+1兲 =
q−1 k tn+1 共k兲 xi0 k=0 k!
兺
N
D*xi共t兲 = f关t,xi共t兲兴 + 兺 Gij⌫x j共t兲,0 艋 t 艋 T, q
j=1
共k兲 x共k兲 i 共0兲 = xi0 ,
共25兲 k = 0,1, . . . ,m − 1,
i = 1,2, . . . ,N,
where q , m are defined in Eq. 共1兲 and N is the size of complex networks. This differential equation with coupling terms is equivalent to Volterra integral equation q−1 k
xi共t兲 =
兺 k=0
t 共k兲 兰t0共t − 兲q−1兵f关,xi共兲兴 + 兺Nj=1Gij⌫x j共兲其d x + . k! i0 ⌫共q兲 共26兲
Set h = T / W, W 苸 Z, tn = nh, 共n = 0 , 1 , . . . , W兲, where h is the step size, T is simulation time, and W is the number of sample points. Then, Eq. 共26兲 can be discretized as follows:
n
p 共tn+1兲 + 兺Nj=1Gij⌫x jh共tn+1兲兴其 hq兵f关tn+1,xih hq + + 兺 al,n+1 ⌫共q + 2兲 ⌫共q + 2兲 l=0
冉再
N
f关tl,xih共tl兲兴 + 兺 Gij⌫x jh共tl兲 j=1
冎冊
, 共27兲
i = 1,2, . . . ,N, where
冦
nq+1 − 共n − q兲共n + 1兲q ,
al,n+1 = 共n − l + 2兲 1, p 共tn+1兲 = xih
q+1
q−1 k tn+1 共k兲 xi0 k=0 k!
兺
+ 共n − l兲
q+1
if l = 0 − 2共n − l + 1兲
n
+
再冋
q+1
冧
, if 1 艋 l 艋 n , if l = n + 1 N
1 Gij⌫x jh共tl兲 兺 bl,n+1 f tl,xih共tl兲 + 兺 ⌫共q兲 l=0 j=1
where bl,n+1 = hq关共n + 1 − l兲q − 共n − l兲q兴 / q. Therefore, the error estimate is max 兩x共ts兲 − xh共ts兲兩 = O共h p兲,
s=0,1,. . .,N
册冎
,
d qx = 共25␣ + 10兲共y − x兲, dtq 共28兲
where p = min共2 , 1 + q兲. Remark 4:. In order to simulate the fractional-order system, the simulation algorithm proposed in this paper needs to be effective and solvable. However, if the networks size N and sampled point W are very large, the procedure usually needs large-scale computation and data processing. Therefore, it is still a challenging problem to solve the optimization problem with low computational complexity.
B. Eigenvalues distribution analysis
In 2002, Lü et al.39 introduced a new chaotic system which can unify chaotic systems, such as, the Lorenz, Chen, Lü systems. The fractional-order differential equation of the unified system can be given by
dqy = 共28 − 35␣兲x − xz + 共29␣ − 1兲y, dtq ␣+8 d qz z, q = xy − 3 dt
共29兲
where 0 ⬍ q 艋 1 , ␣ 苸 关0 , 1兴. When ␣ = 0, ␣ = 0.8, ␣ = 1, and q = 1, the unified integer-order chaotic system is the Lorenz, Lü, and Chen system, respectively. The chaotic behaviors can be seen from Figs. 1 and 2 the chaotic behaviors when q = 0.99, ␣ = 0, and q = 0.9, ␣ = 1, respectively. For the chaotic system in the continuous time domain, ¯ 兲, which does not there exists at least one eigenvalue of Jf共x satisfy Lemma 1. The goal is to ensure all the eigenvalues of ¯ 兲 + c1⌫兴 are well located in the stable region ⌳1i关Jf共x through designing the feedback gain matrix in the case of fixed topology. Here, we aim at investigating that the Lorenz system is taken as the isolated node to illustrate the theoretical result. With this set of system parameters, the unstable equilibrium points are ¯x = 关⫾6冑2 , ⫾ 6冑2 , ⫾ 27兴T and
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Chaos 19, 013112 共2009兲
Tang, Wang, and Fang
45
15
40
0.094+10.1945i 10
35
5
Stable region
25
Im(λ)
z
30
20
5 −20
−qπ/2 Stable region
−5 50 −10 x
0
0
10
20
−50
−15 −1
FIG. 1. 共Color online兲 Chaotic attractor of Lorenz oscillator with q = 0.99, ␣ = 0, and initial vector 关x共0兲 , y共0兲 , z共0兲兴 = 共10, 3 , 12兲.
UnStable region
0.094−10.1945i
−10
y
UnStable region
0
15 10
qπ/2
−0.5
0 Re(λ)
0.5
1
¯ 兲. One of the FIG. 3. Eigenvalue distribution of the Jacobian matrix Jf共x eigenvalues located in the stable region is omitted.
¯x = 关0 , 0 , 0兴T. Here, we stabilize the controlled networks to ¯x = 关6冑2 , 6冑2 , 27兴T. It is an easy matter to obtain the eigen¯ 兲 as shown in Fig. 3. The objective value distribution of Jf共x is to make two of the eigenvalues in the unstable region cross the critical line and enter the stable region by adding an appropriate control matrix D. In the simulation, the size of the scale-free network is N = 20. The model can be described as follows. The growth starts from with three nodes and no edges. At each step, a new node with three edges is added to the existing network. Repeating this rule will produce a scale-free networks. In the following, we choose Gij = Lij / ki , where  is a tunable parameter, L is the Laplacian matrix of the network. Consider a scale-free network composed of N = 20 identical Lorenz oscillators in the case of c = 10 and ⌫ = diag共1 , 1 , 1兲. We select only one of the nodes to be pinned and set the feedback gain d as 10. The fractional-order q is chosen as q = 0.99 and the tunable parameter  = 0.1. As can be seen in Fig. 4, two eigenvalues are in the unstable region 共as in Fig. 3兲, and then they move across the line and enter the stable region. This suggests that the feedback gain designed is efficient for stability.
C. Pinning by adjusting fractional-order q
Here, we consider a scale-free network and Lorenz system discussed in Sec. IV B. The pinning scheme is employed to pin only one node of the networks. We set c = 10,  = 0.1, d = 10 and only tuning fractional-order q. Note that we still select the same single node to control the network while adjusting the fractional-order q. For the sake of pinning scheme, we introduce a quantity E共t兲 =
冑
兺Ni=1储xi共t兲 − ¯x储2 , N
共30兲
which is used to measure the quality of the pinning process. From Fig. 5, one can easily see that the control performance is better with the decreasing of fractional-order q. As seen in Lemma 1, Figs. 3 and 4, when the fractional-order q decreases, the stable region expands, while the unstable region becomes smaller. Remark 5: Surprisingly, from Corollary 1, if any node of 15
10
40 35
5
30
−0.2407+10.1945i
Unstable region
Stable region
Im(λ)
qπ/2 −qπ/2
z
25
0
20
−5
15
−10
Unstable region
Stable region −0.2407−10.1945i
10 5 −30
50 −20
−10
x
0
10
20
−50
0 y
FIG. 2. 共Color online兲 Chaotic attractor of Chen oscillator with q = 0.90, ␣ = 1, and initial vector 关x共0兲 , y共0兲 , z共0兲兴 = 共10, 3 , 12兲.
−15 −1
−0.5
0 Re(λ)
0.5
1
¯ 兲 + c1⌫. One of the eigenvalFIG. 4. Eigenvalue distribution of matrix Jf共x ues located in the stable region is omitted.
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Chaos 19, 013112 共2009兲
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q=1 q=0.975 q=0.95 q=0.9
1
d=7 d=15 d=25 d=50
1.2
1
0.8
E(t)
E(t)
0.8
0.6
0.6
0.4
0.4
0.2
0
0.2
0
0.5
1 t
1.5
0
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
t
FIG. 5. 共Color online兲 The quantity E共t兲 of pinning a certain node in scalefree network under different fractional-order q.
FIG. 7. 共Color online兲 The quantity E共t兲 of pinning a certain node in the scale-free network under different control gain d.
the networks is not pinned, the fractional-order networks can stabilize itself without any controllers, while q should satisfy q ⬍ 0.9941. It is can be seen from Fig. 6 that, after 2.5 s, the fractional-order networks is stable without control at its equilibrium, where q = 0.9. On the other hand, the results in this paper can be easily extended to other chaotic systems, such as, the Chua system, hyperchaotic systems, and so on.
E. Pinning by tuning overall coupling strength c
D. Pinning by adjusting control gain matrix D
We consider the same scale-free network discussed in Sec. IV B and Chen system as each node. The pinning scheme is employed to pin one node of the networks. Let c = 10,  = 0.5, q = 0.90, and only tuning control gain matrix D = diag共d , 0 , 0 , . . . , 0兲. Note that we still select the same single node to control the network. We also use Eq. 共30兲 to evaluate the control performance. Therefore, only the control gain of one node is changed. From Fig. 7, one can easily see that the control performance is better with the increase of control gain d.
Consider the same scale-free network discussed in Sec. IV B and Chen system as each node. The pinning scheme is applied to pin the nodes of the networks. Let  = 0.5, q = 0.90, d = 10 and only tuning the overall coupling strength c. Notice that we still select the same single node to control the network. Clearly, from Fig. 8, we can easily see that the control performance is better with the increase of overall coupling strength c. F. Pinning by adjusting tunable parameter 
Consider the same scale-free network discussed in Sec. IV B and Chen system as each node. The pinning scheme is used to pin one node of the networks. Let c = 10, q = 0.95, d = 10, and only change the tunable parameter . It should be emphasized that we still select the same single node to control the network.  = 0 indicates that the coupling matrix G is an unweighted symmetric matrix. Obviously, from Fig. 9, we can easily see that the control performance is better, while
40 35
1.2
30
zi(t)
1
25
0.8 E(t)
xi(t),yi(t),zi(t)
c=7 c=10 c=20 c=50
20
0.6 15
0.4
xi(t),yi(t)
10
0.2
5 0
0
0.5
1
t
1.5
2
2.5
FIG. 6. 共Color online兲 Convergence of states in the scale-free network under q = 0.9 without pinning any node.
0
0
0.1
0.2
0.3
0.4
t
0.5
0.6
0.7
0.8
0.9
FIG. 8. 共Color online兲 The quantity E共t兲 of pinning a certain node in the scale-free network under different coupling strength c.
013112-8
Chaos 19, 013112 共2009兲
Tang, Wang, and Fang
nodes are pinned, the quicker the network is stabilized. On the other hand, in a scale-free network, to reach the same control efficiency, much larger coupling strength and pinning fraction are required in the randomly pinning control than that in the specifically pinning control. That is to say, the randomly pinning scheme is less efficient than the specifically pinning scheme, as compared in Fig. 10.
β=0 β=0.1 β=0.5 β=1
1.2 1
E(t)
0.8 0.6
V. CONCLUSIONS
0.4 0.2 0
0
0.1
0.2
0.3
0.4
t
0.5
0.6
0.7
0.8
0.9
FIG. 9. 共Color online兲 The quantity E共t兲 of pinning a certain node in the scale-free network under different parameters .
parameter  is increasing. The simulation shows that a fractional-order complex network-weighted or unweighted, symmetric or asymmetric-can be stabilized through pinning. G. Randomly pinning and specially pinning
Consider the same scale-free network discussed in Sec. IV B and Chen system as each node. Two different pinning schemes are used to pin the nodes of the networks. Let c = 10, q = 0.95, d = 10,  = 0.5 and keep constant. First, only one largest degree node is pinned which has degree 13. Second, the two largest degree nodes are pinned which have degree 13 and 11, respectively. Finally, the randomly pinning scheme is used to pin only one node and two nodes, respectively. Obviously, we can easily see that the control performance is better by pinning a larger degree node and more forceful nodes from Fig. 10. Remark 6: Clearly, increasing the pinning fraction will accelerate the convergence of the network stabilization, as shown in Fig. 10. In other words, the more forcefully the
Randomly pin one node Specially pin one node Randomly pin two nodes Specially pin two nodes
1.2
1
E(t)
0.8
ACKNOWLEDGMENTS
The authors are grateful to the editor and reviewers for their kind help and constructive comments which helped to improve the presentation of the paper. This research was supported in part by the National Natural Science Foundation of P. R. China under Grant No. 60874113, the Key Creative Project of Shanghai Education Community 共09ZZ66兲, the Research Fund for the Doctoral Program of Higher Education 共200802550007兲, and Shanghai Natural Science Foundation 共08ZR1400400兲, the Engineering and Physical Sciences Research Council 共EPSRC兲 of the U.K. under Grant No. GR/S27658/01, an International Joint Project sponsored by the Royal Society of the U.K., and the Alexander von Humboldt Foundation of Germany. D. J. Watts and S. H. Strogatz, Nature 共London兲 393, 440 共1998兲. A. L. Barabási and R. Albert, Science 286, 509 共1999兲. 3 S. H. Strogatz, Nature 共London兲 410, 268 共2001兲. 4 S. Boccalettia, V. Latorab, Y. Morenod, M. Chavezf, and D. U. Hwanga, Phys. Rep. 424, 175 共2006兲. 5 V. Latora and M. Marchiori, Phys. Rev. Lett. 87, 198701 共2001兲. 6 C. Zhou, A. E. Motter, and J. Kurths, Phys. Rev. Lett. 96, 034101 共2006兲. 7 X. F. Wang and G. Chen, Physica A 310, 521 共2002兲. 8 M. Barahona and L. M. Pecora, Phys. Rev. Lett. 89, 054101 共2002兲. 9 R. Hilfer, Applications of Fractional Calculus in Physics 共World Scientific, Singapore, 2000兲. 10 I. Podlubny, Fractional Differential Equations 共Academic, New York, 1999兲. 1
0.6
2
0.4
0.2
0
The pinning control problem of a class of fractionalorder weighted complex dynamical networks has been investigated in detail. The general strategy is to apply a feedback control scheme to a small fraction of the network nodes. By utilizing eigenvalue analysis approach and fractional-order stability theory, we employ the numerical algorithms for fractional-order complex networks addressed in this paper to establish some local stability criteria and valid stability regions of such pinned fractional-order networks. The analytical analysis show that the largest eigenvalue of matrix R determines the control of the weighted fractional-order complex networks. By seeking an appropriate R, overall coupling strength c, and fractional-order q, we are able to achieve our goal. It is surprising to find that a network can realize stabilization under a suitable q, even without controller. In addition, it is interesting to find that the fractional-order q, the control gain matrix D, the tunable weight parameter , the overall coupling strength c, the specially pinning of largest nodes will effectively affect the convergence rate of controlling fractional-order complex dynamical networks. In the end of the paper, some simulation examples in scale-free networks are exploited to demonstrate the applicability of the proposed results.
0
0.2
0.4
0.6
0.8
1
t
FIG. 10. 共Color online兲 The quantity E共t兲 of pinning one or two certain nodes in the scale-free network under different pinning schemes.
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Chaos 19, 013112 共2009兲
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S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications 共Gordon and Breach, Yverdon, 1993兲. 12 W. Deng and J. Lü, Chaos 16, 043120 共2006兲. 13 I. Grigorenko and E. Grigorenko, Phys. Rev. Lett. 91, 034101 共2003兲. 14 T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 42, 485 共1995兲. 15 P. Arena, L. Fortuna, and D. Porto, Phys. Rev. E 61, 776 共2000兲. 16 C. Li and G. Chen, Physica A 341, 55 共2004兲. 17 R. O. Grigoriev, M. C. Cross, and H. G. Schuster, Phys. Rev. Lett. 79, 2795 共1997兲. 18 X. Li, X. F. Wang, and G. Chen, IEEE Trans. Circuits Syst., I: Regul. Pap. 51, 2074 共2004兲. 19 Q. Miao, Z. H. Rong, Y. Tang, and J. A. Fang, Physica A 387, 6225 共2008兲. 20 J. Zhou, J. Lu, and J. Lü, Automatica 44, 996 共2008兲. 21 C. Zhou and J. Kurths, Phys. Rev. Lett. 96, 164102 共2006兲. 22 L. Y. Xiang, Z. X. Liu, Z. Q. Chen, F. Chen, and Z. Z. Yuan, Physica A 379, 298 共2007兲. 23 F. Sorrentino, M. di Bernardo, F. Garofalo, and G. Chen, Phys. Rev. E 75, 046103 共2007兲. 24 T. P. Chen, X. W. Liu, and W. L. Lu, IEEE Trans. Circuits Syst., I: Regul. Pap. 54, 1317 共2007兲. 25 M. Chavez, D.-U. Hwang, A. Amann, H. G. E. Hentschel, and S. Boccaletti, Phys. Rev. Lett. 94, 218701 共2005兲.
W. Yu, G. Chen, and J. Lü, Automatica 45, 429 共2009兲. L. Y. Xiang, Z. X. Liu, Z. Q. Chen et al., J. Control Theory Appl. 6, 2 共2008兲. 28 C. W. Wu and L. O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 42, 430 共1995兲. 29 Y. Liu, Z. Wang, and X. Liu, Phys. Lett. A 372, 3986 共2008兲. 30 Y. Liu, Z. Wang, J. Liang, and X. Liu, IEEE Trans. Syst., Man, Cybern., Part B: Cybern. 38, 1314 共2008兲. 31 Y. Tang, R. Qiu, J. A. Fang, Q. Miao, and M. Xia, Phys. Lett. A 372, 4425 共2008兲. 32 Y. Tang and J. A. Fang, Phys. Lett. A 372, 1816 共2008兲. 33 L. Wang, H. P. Dai, H. Dong, Y. Y. Cao, and Y. X. Sun, Eur. Phys. J. B 61, 335 共2008兲. 34 L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 80, 2109 共1998兲. 35 M. Caputo, Geophys. J. R. Astron. Soc. 13, 529 共1967兲. 36 F. Keil, W. Mackens, and J. Werther, Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties 共Springer-Verlag, Heidelberg, 1999兲. 37 K. Diethelm K, N. J. Ford, and A. D. Freed, Nonlinear Dyn. 29, 3 共2002兲. 38 D. Matignon, Stability results of fractional differential equations with applications to control processing, in: IMACS, IEEE-SMC, Lille, France, 1996. 39 J. Lü and G. Chen, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 659 共2002兲. 26 27
CHAOS 19, 013113 共2009兲
Chimera states in heterogeneous networks Carlo R. Lainga兲 Institute of Information and Mathematical Sciences, Massey University, Private Bag 102-904 NSMC, Auckland, New Zealand
共Received 9 November 2008; accepted 17 December 2008; published online 10 February 2009兲 Chimera states in networks of coupled oscillators occur when some fraction of the oscillators synchronize with one another, while the remaining oscillators are incoherent. Several groups have studied chimerae in networks of identical oscillators, but here we study these states in heterogeneous models for which the natural frequencies of the oscillators are chosen from a distribution. For a model consisting of two subnetworks, we obtain exact results by reduction to a finite set of differential equations, and for a network of oscillators in a ring, we generalize known results. We find that heterogeneity can destroy chimerae, destroy all states except chimerae, or destabilize chimerae in Hopf bifurcations, depending on the form of the heterogeneity. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3068353兴 Synchronization of interacting oscillators is a problem of fundamental importance, with applications from Josephson junction circuits to neuroscience.19,24,22,27 Since oscillators are unlikely to be identical, the effects of heterogeneity on their collective behavior is of interest. One wellstudied system of heterogeneous phase oscillators is the Kuramoto model,4,9,23 for which there is global coupling. Generalizations of this model with nonlocal coupling,3,21,2,10,16 or several populations of oscillators,1 have shown interesting types of behavior referred to as “chimera” states, in which some oscillators are synchronized with one another while the remainder are incoherent. Even though the effects of heterogeneity on synchronization have been emphasized in the past,4,13,9,23 all chimerae have so far been studied in networks of identical oscillators. This raises the obvious question: Do chimerae exist in networks of nonidentical oscillators? Here we address the question analytically, first using recent results to exactly derive a finite set of differential equations governing the dynamics of chimerae in two coupled networks of heterogeneous phase oscillators, and then using a similar idea to extend the results of Abrams and Strogatz3 for chimera states in a ring of coupled oscillators.
I. INTRODUCTION
Networks of coupled oscillators have been studied for many years.24,13,1,3,19,20,5 One well-known system is the Kuramoto model4,9,6,10,23 of phase oscillators. In the last few years, several authors have studied chimera states in networks of identical Kuramoto oscillators, in which some oscillators are synchronized with one another, while the remainder are incoherent. Much analytical progress has been made in the study of these states.1–3,21,10,16 It is very unlikely that any physical system being modeled by a network of coupled oscillators will have identical units, so the robusta兲
Electronic mail: c
[email protected].
1054-1500/2009/19共1兲/013113/8/$25.00
ness of chimerae to network heterogeneity is naturally of interest. Certain networks of coupled oscillators are known to have nongeneric properties,25,26 and it is of interest to know whether chimera states are generic and stable 共and thus expected to be generally observed兲 or not. Here we conduct an analytical investigation into the robustness of previously studied chimerae with respect to heterogeneity in the intrinsic frequencies of oscillators. We find that chimerae are robust with respect to this type of heterogeneity, and show some of the bifurcations that chimera and other states undergo as the oscillators in the network are made more dissimilar. Our results provide more evidence that the Ott–Antonsen ansatz17 correctly describes attracting states in Kuramoto-type networks when the oscillators are not identical. Some of the ideas here have recently been used by others to study a single population of oscillators with a bimodal frequency distribution14 and the periodically forced Kuramoto model.7 Note that the term “chimera” has been used in the past to refer to certain states in networks of identical oscillators,16,21,1,3 but here we also use the term to describe similar states in heterogeneous networks in the obvious way. A state found by numerically continuing from a chimera state in a network of identical oscillators is also referred to as a chimera. In Sec. II we present the first model of two coupled networks, consider its continuum limit, and use the remarkable recent result of Ott and Antonsen17 to derive three ordinary differential equations 共ODEs兲 which exactly describe some of its behavior. In Sec. III we perform a limited bifurcation analysis of these ODEs and interpret the results. In Secs. IV and V we consider other distributions of the intrinsic oscillator frequencies, and generalizations, respectively. In Sec. VI we consider oscillators on a ring. II. TWO COUPLED NETWORKS
We first consider two networks of coupled oscillators with uniform coupling between oscillators within each net19, 013113-1
© 2009 American Institute of Physics
013113-2
Chaos 19, 013113 共2009兲
Carlo R. Laing
work, and a weaker coupling to those in the other network. Our model equations are 共1兲
for i = 1 , . . . , N and k = 1 , 2, where the natural frequencies ki are chosen from a distribution gk共k兲. Our system is the same as that of Montbrió et al.;15 the system of Abrams et al.1 is a special case of that studied here. A similar system was studied by Barreto et al.,6 but their focus was the onset of synchrony, as was Montbrió et al.’s. Like Abrams et al.,1 we choose K11 = K22 = and K12 = K21 = , set + = 1 共by rescaling time if necessary兲, and choose ⬎ . We define A = − and  = / 2 − ␣. Abrams et al. found that for  and A sufficiently small and positive 共and all ki equal兲, both the completely synchronized state 共mj = ki for all j , i , m , k兲 and the chimera state 共all oscillators in one population perfectly synchronized, all oscillators in the other population incoherent兲 were stable. We take the continuum limit of Eq. 共1兲, letting N → ⬁. The system is then described by the probability density function 共PDF兲 f k共k , k , t兲 for each population k. We define two order parameters,
冕冕 ⬁
−⬁
2
exp共ik兲f k共k, k,t兲dkdk ,
共2兲
0
for k = 1 , 2. Each f k satisfies a continuity equation
fk + k 共f kvk兲 = 0, t
共3兲
where the velocity vk = k + 兵exp关− i共k + ␣兲兴共zk + zk⬘兲
− exp关i共k + ␣兲兴共¯zk + ¯zk⬘兲其/共2i兲,
共4兲
k⬘ = 3 − k, and an overbar denotes the complex conjugate. Writing f k as a Fourier series in k, we have f k共k, k,t兲 =
冋 再
g k共 k兲 1+ 2
⬁
冎册
兺 hn共k,t兲exp共ink兲 + c.c.
n=1
, 共5兲
where “c.c.” denotes the complex conjugate of the previous term. Substituting Eq. 共5兲 into Eqs. 共3兲 and 共2兲, one can derive an infinite set of integro-differential equations for the hn.15 However, Ott and Antonsen17 noticed that for the special choice hn共k,t兲 = 关ak共k,t兲兴n ,
⬁
¯ak共k,t兲gk共k兲dk .
共8兲
−⬁
The ansatz 共6兲 is not trivial, and the reduction from an infinite set of differential equations to one is remarkable. Ott and Antonsen17 give more detail on the circumstances under which this ansatz is valid, and we discuss its usefulness in describing attracting states below. As is well known,15,17,9,14 if gk is a Lorentzian distribution the integral 共8兲 can be evaluated analytically. Suppose that g k共 k兲 =
D k/ 共k − ⍀k兲2 + D2k
共9兲
,
i.e., the ki are from a distribution centered at ⍀k with half width at half maximum Dk. Then, zk共t兲 = ¯ak共⍀k − iDk , t兲 and evaluating Eq. 共7兲 at k = ⍀k − iDk, we obtain ¯k dz ¯k − 共ei␣/2兲共¯zk + ¯zk⬘兲 + 共Dk + i⍀k兲z t ¯2k = 0, + 共e−i␣/2兲共zk + zk⬘兲z
共10兲
i.e., a complex ODE for each k. Writing z1 = r1e−i1 and z2 = r2e−i2, and defining = 1 − 2, we obtain the three real ODEs:
冉 冊 冉 冊
1 − r21 dr1 = − D 1r 1 + 关r1 cos ␣ + r2 cos共 − ␣兲兴, 2 dt
共11兲
1 − r22 dr2 = − D 2r 2 + 关r2 cos ␣ + r1 cos共 + ␣兲兴, 2 dt
共12兲
冉 冊 冉 冊
r2 + 1 d = 1 关r1 sin ␣ − r2 sin共 − ␣兲兴 dt 2r1 + ⍀2 −
r22 + 1 关r2 sin ␣ + r1 sin共 + ␣兲兴, 2r2
共13兲
where, without loss of generality, we have set ⍀1 = 0. When D1 = D2 = ⍀2 = 0, we recover the results of Abrams et al.1 In particular, r1 = 1 共1i all equal兲 is invariant. If r1 = 1, there also exists the perfect synchrony state 共r2 = 1, = 0兲 and, depending on parameters 共see Fig. 4 in Ref. 1兲 two other fixed points with r2 ⫽ 1 共the chimerae兲. When they exist, one of these fixed points is a saddle, while the other is either stable or unstable, changing stability via a supercritical Hopf bifurcation. We now proceed with a limited analysis of Eqs. 共11兲–共13兲.
共6兲
i.e., hn is ak raised to the nth power, all of these differential equations are actually the same, and we are left with a single partial differential equation governing the dynamics of ak共k , t兲:
ak + ikak − 共ei␣/2兲共¯zk + ¯zk⬘兲 + 共e−i␣/2兲共zk + zk⬘兲a2k = 0, t 共7兲 where
冕
N
2
dki Kkm = ki + 兺 兺 sin共mj − ki − ␣兲 dt m=1 N j=1
zk共t兲 =
zk共t兲 =
III. RESULTS A. Varying distribution widths, no frequency offset
First consider the case when D1 = D2 = D and ⍀2 = 0. The system 共11兲–共13兲 possesses Z2 symmetry: 共r1 , r2 , 兲 → 共r2 , r1 , −兲. We choose A = 0.2 and  = 0.07, so that for D = 0, there exist five fixed points of Eqs. 共11兲–共13兲: the perfect synchrony state 共r1 , r2 , 兲 = 共1 , 1 , 0兲 and two chimerae 共one stable and one a saddle兲 for which r1 = 1 and r2 ⫽ 1 and ⫽ 0, and their symmetrically related states. The fixed
013113-3
Chaos 19, 013113 共2009兲
Chimera states in heterogeneous networks 1
(a)
1
(a)
0.9
r2
r
0.9
0.8
0.8
0.7
0.7 0
0.005
D
0.01
0.6 −0.01
0.015
−0.005
0 D
0.005
0.01
2
1
0.2
(b)
(b)
0.9 r1,r2
φ
0.1
0
0.7
−0.1
−0.2 0
0.8
0.005
D
0.01
0.015
FIG. 1. 共Color online兲 Fixed points of Eqs. 共11兲–共13兲 when D1 = D2 = D, ⍀2 = 0. 共a兲 r1 and r2 as a function of D. 共b兲 as a function of D. Blue curve: stable chimera. Red curve: saddle chimera. Black curve: symmetric state 共r1 = r2兲. Solid lines indicate stable solutions; dashed lines unstable. Other parameters: A = 0.2,  = 0.07.
points of Eqs. 共11兲–共13兲 and their stability as a function of D are shown in Fig. 1. There are several interesting observations to be made here. First 共for these parameter values兲, increasing the heterogeneity of the network first destabilizes the symmetric state 共r1 = r2兲, then restabilizes it. Second, increasing the heterogeneity actually decreases the width of the angular distribution of the unsynchronized population in the chimera state 关lower blue branch in panel 共a兲 of Fig. 1兴. B. Varying one distribution width, no frequency offset
Now consider varying D2, while D1 = ⍀2 = 0. The system no longer possesses any symmetry, so the effect of increasing D2 from zero on the chimerae with r1 = 1, r2 ⫽ 1 will be different from its effect on the chimerae with r1 ⫽ 1, r2 = 1. We choose A = 0.2 and  = 0.1, so that as before, when D2 = 0 there exist five fixed points of Eqs. 共11兲–共13兲. Results are shown in Fig. 2. With five solutions to track, we do not show . Also, even though D2 ⬍ 0 is not physically meaningful, we plot fixed points for D2 ⬍ 0 to show how branches of solutions are connected. Panel 共a兲 in Fig. 2 shows fixed points for which r1 = 1 共recall that we are making population 2 heterogeneous兲. As D2 is increased, the perfectly synchronous solution that exists at D2 = 0 is destroyed in a saddle-node bifurcation, while the chimera with population 2 desynchronized persists. Panel 共b兲 shows the fate of the two chimerae 共one stable and one a saddle兲 for which r2 = 1 when D2 = 0. We see that they are both destroyed in a saddle-node bifurcation as D2 is increased. From this figure, we see that if one population is
0.6 −0.01
−0.005
0 D2
0.005
0.01
FIG. 2. 共Color online兲 Fixed points of Eqs. 共11兲–共13兲 when D1 = ⍀2 = 0. 共a兲 r2 as a function of D2 when r1 = 1. 共b兲 r1 共red, lower curve兲 and r2 共blue, upper curve兲 as a function of D2. Solid lines indicate stable solutions; dashed lines unstable. Other parameters: A = 0.2,  = 0.1.
made sufficiently heterogeneous, the only solution that persists is the chimera for which the oscillators in that population are desynchronized. Interestingly, if D2 is increased to larger values 共D2 ⬇ 0.1兲, the state where both populations are in the “splay” state, with uniform angular density, i.e., r1 = r2 = 0 and is no longer meaningful, becomes stable 共not shown兲. C. Varying frequency offset ⍀2
We now consider varying ⍀2 with D1 = D2 = 0. Since we are interested in states for which at least one of the populations is in complete synchrony, we set r1 = 1 and only consider Eqs. 共12兲 and 共13兲. The completely synchronized state, i.e., 共r2 , 兲 = 共1 , 0兲, exists when ⍀2 = 0, and as ⍀2 is increased it persists as the fixed point 共r2 , 兲 = 共1 , 兲, where is the solution closest to zero of 2 cos ␣ sin = ⍀2. Numerical results are shown in Fig. 3. We see that as ⍀2 is increased from zero, the synchronized state for which r2 = 1 is destroyed in a transcritical bifurcation involving the saddle chimera, while the stable chimera undergoes a supercritical Hopf bifurcation, leading to oscillations in r2 and . However, decreasing ⍀2 from zero causes destruction of the stable chimera in a saddle-node bifurcation with the saddle chimera. For these parameter values, one can see that the stable chimera is much more robust to speeding up the asynchronous oscillators, as opposed to slowing them down. D. Discussion
Our bifurcation analysis in this section has found all four co-dimension-1 bifurcations of ODEs 共saddle-node, pitch-
013113-4
Chaos 19, 013113 共2009兲
Carlo R. Laing 1 0.95
1
0.9 0.85
0.6
r
r
2
0.8
0.8
0.4
0.75
0.2 0
0.7
0
0.02
0.04 Ω
0.06
0.08
0.1
0.65 0
0.005
0.01 σ
2
FIG. 3. 共Color online兲 r2 as a function of ⍀2, when r1 = 1. Solid lines indicate stable fixed points of Eqs. 共11兲–共13兲; dashed lines unstable. Circles are the maximum and minimum of r2 during stable periodic oscillations. Note that the branch with r2 ⬎ 1 is not physically meaningful. Other parameters: A = 0.2,  = 0.1.
fork, transcritical, and Hopf兲 and a two-parameter study is likely to find higher co-dimension bifurcations. Pikovsky and Rosenblum18 recently studied Eq. 共1兲 with identical ki 共i.e., the system of Abrams et al.,1 and our system when D1 = D2 = ⍀2 = 0兲 as a special case and found that the ansatz 共6兲 did not completely describe the possible dynamics of this system. However, Pikovsky and Rosenblum18 and other authors14 found that when the oscillators are nonidentical, this ansatz does successfully allow one to describe attracting states. We also find this behavior here: extensive numerical simulations show that the stable states shown in Figs. 1 and 2 for D , D2 ⬎ 0 are attracting, and that the angular distributions of these stable states are given by Eqs. 共5兲 and 共6兲, even if the initial distributions are not. However, the same cannot be said for the results in Fig. 3, for which oscillators within each of the two networks are identical. This figure correctly predicts the dynamics if the initial angular distribution is given by Eqs. 共5兲 and 共6兲 but other initial conditions give solutions not described by Fig. 3 共not shown兲. This relationship between initial conditions and dynamics when oscillators within each network are identical was also noticed by Montbrió et al.15 The results in Fig. 3 are likely to be a subset of those that could be found using the approach in Ref. 18. In related results, Montbrió et al.15 fixed ⍀2 ⫽ 0 and varied both ␣ and and found chimera states, both for homogeneous and heterogeneous networks.
A. Gaussian distribution: Numerical simulations
Figure 4 shows the results of fitting the time-dependent PDF
0.02
FIG. 4. 共Color online兲 r1 and r2 fitted to simulations of Eq. 共1兲, where all ki are chosen from a normal distribution of mean zero and standard deviation . Blue circles joined by a line: stable chimera. Black crosses joined by a line: stable symmetric state 共r1 = r2兲. Red dashed line: presumed unstable symmetric state. Compare with Fig. 1. See the text for details on the fitting. Other parameters: A = 0.2,  = 0.07, N = 1000.
再冋
1 1+ f k共,t兲 = 2 =
⬁
册冎
兺 共rkei 兲nein + c.c. k
n=1
1 − r2k 2关1 − 2rk cos共k − 兲 + r2k 兴
共14兲
to each population in simulations of Eq. 共1兲 after transients, where all ki are chosen from a normal distribution of mean zero and standard deviation . We found that both r1 and r2 tended to constant values, as did 2 − 1 共not shown兲. Only stable states are shown in Fig. 4, but the results are compatible with those shown in Fig. 1, suggesting that there is nothing special about the Lorentzian distribution, as has been noted by others.14,7 The unstable states could presumably be found using the “equation-free” method13,8,12 of analyzing low-dimensional descriptions of high-dimensional systems, under the assumption that these states are also exactly described by the variables r , for each population. B. Another distribution
As an alternative,17 we suppose that the ki are chosen from the distribution g k共 兲 =
冑2s3k
冉
1
+ s4k 4
冊
,
共15兲
which has mean zero 共for simplicity兲 and variance s2k . This gk共兲 has poles at = sk共⫾1 ⫾ i兲 / 冑2, and the integral 共8兲 gives
IV. OTHER DISTRIBUTIONS
Now we consider the effects of choosing the ki from distributions other than the Lorentzian, first numerically and then analytically.
0.015
zk =
冉 冊 冉 冊
1+i − 1−i + zk + zk , 2 2
共16兲
where z⫾ k satisfy dz⫾ k −i␣ − 关sk共1 ⫾ i兲/冑2兴z⫾ /2兲共zk + zk⬘兲 k − 共e dt 2 + 共ei␣/2兲共¯zk + ¯zk⬘兲共z⫾ k 兲 = 0.
共17兲
Thus, we have four coupled complex ODEs, rather than the
013113-5
Chaos 19, 013113 共2009兲
Chimera states in heterogeneous networks
two 关Eq. 共10兲兴. This can clearly be generalized to other distributions which are rational functions of . V. GENERALIZATIONS
We now briefly mention several generalizations of the results above. Suppose that the system 共1兲 is periodically ˆ t − k兲 to Eq. 共1兲, as forced, i.e., we add the term ⌳k sin共⍀ i 7 done recently for a single population. Going to a coordinate ˆ , we find that Eq. frame rotating with angular frequency ⍀ 共10兲 is replaced by
冋
⌳k + ei␣共¯zk + ¯zk⬘兲 ¯k dz ˆ 兲兴z ¯k − + 关Dk + i共⍀k − ⍀ 2 t +
冋
⌳k + e−i␣共zk + zk⬘兲 2
册
¯z2k
册
= 0.
The system is no longer invariant under a translation of time, so during the derivation of ODEs like Eqs. 共11兲–共13兲, we find that we need both 1 and 2, not just their difference. Another possibility is that there is a uniform delay of between the two populations, but zero delay within them, i.e., we replace mj in Eq. 共1兲 by mj 共t − 兲 when m = k⬘. The effect of this is to replace zk⬘共t兲 in Eq. 共10兲 by zk⬘共t − 兲, and the equivalent of Eqs. 共11兲–共13兲 is now the four delay differential equations 共DDEs兲:
冉 冊
r2 − 1 dr1 + D 1r 1 + 1 2 dt
⫻关r1 cos ␣ + r2共t − 兲cos兵1 − 2共t − 兲 − ␣其兴 = 0,
冉 冊
r2 + 1 d1 − 1 dt 2r1
⫻关r1 sin ␣ − r2共t − 兲sin兵1 − 2共t − 兲 − ␣其兴 = 0,
冉 冊
r2 − 1 dr2 + D 2r 2 + 2 2 dt
⫻关r2 cos ␣ + r1共t − 兲cos兵2 − 1共t − 兲 − ␣其兴 = 0,
冉 冊
r2 + 1 d2 + ⍀0 − 2 dt 2r2
⫻关r2 sin ␣ − r1共t − 兲sin兵2 − 1共t − 兲 − ␣其兴 = 0. If all of the variables on the right hand side of Eq. 共1兲 were delayed by , we could define = 1 − 2 as before, and derive three coupled DDEs rather than four above. The analysis of the equations in this section remains an open problem.
perturbations of the oscillators’ natural frequencies, and we now show how to use the ideas above to investigate this analytically. Consider the model consisting of oscillators on a ring studied in Refs. 10 and 3, but include heterogeneity in the intrinsic frequencies of the oscillators. The system is
We now consider a ring of oscillators, with nonlocal coupling between them. The original presentation of chimerae was in such a system, with identical oscillators.3,2,10 The chimera state for this system consists of oscillators on one part of the ring being synchronized, while over the remainder of the ring they are incoherent. Simulations reported in Ref. 3 indicate that these states are also robust with respect to
冊
共18兲
for i = 1 , . . . , N, where the natural frequencies i are chosen from a distribution g共兲. The coupling function G is periodic with period 2. Equation 共18兲 is the discrete version of
=− t
冕
2
G共x − y兲cos关共x,t兲 − 共y,t兲 − 兴dy, 共19兲
0
which for constant is the same as that studied in Refs. 10 and 3. The analysis for a heterogeneous network is very similar to that for a network of identical oscillators, so we skip many of the details here and refer the reader to Ref. 3. The main difference is that is now a variable, and certain quantities now have to be replaced by integrals over , weighted by g共兲. A. Analysis
First we go to a rotating reference frame with angular speed ⍀, i.e., let = − ⍀t. We then define an order parameter R共x,t兲ei⌰共x,t兲 =
冕
2
G共x − y兲ei共y,t兲dy,
0
so that Eq. 共19兲 can be written
= − ⍀ − R cos共 − ⌰ − 兲. t
共20兲
We now look for stationary states, so that R and ⌰ are independent of t. At position y, if R共y兲 ⬎ 兩 − ⍀兩, then the oscillators will move to the stable fixed point *, which is given by the solution of
− ⍀ = R cos关* − ⌰ − 兴. For those drifting oscillators at y with R共y兲 ⬍ 兩 − ⍀兩, we replace ei共y兲 in the order parameter definition with its average over ,3,10 but now weighted by g共兲 共over the appropriate range of 兲. After some calculation the result is that at stationarity, we have
冕 冕冉
R共x兲ei⌰共x兲 = ei
2
G共x − y兲ei⌰共y兲
0
VI. OSCILLATORS ON A RING
冉
N
2兩i − j兩 2 di = i − G cos共i − j − 兲 兺 dt N j=1 N
⫻
⬁
−⬁
冊
− ⍀ − 冑共 − ⍀兲2 − R2共y兲 g共兲d dy. R共y兲 共21兲
In some cases, this double integral can be exactly evaluated. We follow Ref. 3 and suppose that G共x兲 = 共1 + A cos x兲 / 共2兲, so that G共x − y兲 = 共1 + A cos x cos y + A sin x sin y兲 / 共2兲. Let us define
013113-6
Chaos 19, 013113 共2009兲
Carlo R. Laing
冕冉 ⬁
h共y兲 =
−⬁
冊
− ⍀ − 冑共 − ⍀兲2 − R2共y兲 g共兲d . R共y兲
where the overbar indicates complex conjugate. Thus, we have
Thus, under the assumption that R and ⌰ are even 共which can be shown to be self-consistent兲, 共22兲
R共x兲ei⌰共x兲 = c + a cos x, where c=
ei 2
冕
and Aei a= 2
c=
共23兲
ei⌰共y兲h共y兲dy
0
冕
Aei 2
冕
⬁
f共,y兲g共兲d dy
共25兲
−⬁
冕
2
0
cos y c + ¯a cos y
冕
⬁
f共,y兲g共兲d dy,
共26兲
−⬁
− 冑共 − ⍀兲2 − c2 − 2c Re共a兲cos y − 兩a兩2 cos2 y.
共24兲
ei⌰共y兲h共y兲cos y dy.
0
Taking the real and imaginary parts of Eqs. 共25兲 and 共26兲, we obtain four real equations for the four real unknowns: c, Re共a兲, Im共a兲, and ⍀. As in Sec. II, suppose that g共兲 =
= c2 + 2c Re共a兲cos y + 兩a兩2 cos2 y and h共y兲 ei⌰共y兲h共y兲 = R共y兲ei⌰共y兲 R共y兲 = 共c + a cos y兲 ⬁ − ⍀ − 冑共 − ⍀兲2 − R2共y兲 g共兲d ⫻ R2共y兲 −⬁
冕冉
1 = c + ¯a cos y ⫻g共兲d ,
冕
and a=
0
1 c + ¯a cos y
f共,y兲 ⬅ − ⍀
2
R2共y兲 = 关R共y兲ei⌰共y兲兴关R共y兲e−i⌰共y兲兴
− e i 2
2
where
Since Eq. 共21兲 is unchanged by the shift ⌰共x兲 → ⌰共x兲 + ⌰0, we can take c to be real. To write the right hand sides of Eqs. 共23兲 and 共24兲 in terms of a and c, note that
c=
冕
and a=
2
e i 2
− Aei 2
2
0
冕
⬁
册
共27兲
i.e., the i are from a distribution centered at zero with half width at half maximum D. 共There is no loss of generality by assuming that the distribution is centered at zero, since if this was not so, the effect would just be to add a constant to ⍀.兲 Then for any function F共兲 analytic in the lower half of the complex plane,
冕
共 − ⍀ − 冑共 − ⍀兲2 − R2共y兲兲
⬁
F共兲g共兲d = F共− iD兲,
−⬁
−⬁
and thus Eqs. 共25兲 and 共26兲 become
⍀ + iD + 冑共⍀ + iD兲2 − c2 − 2c Re共a兲cos y − 兩a兩2 cos2 y dy c + ¯a cos y
2
0
冕
冊
冋
1 D/ 1 1 − , 2 = +D 2i − iD + iD 2
共28兲
共⍀ + iD + 冑共⍀ + iD兲2 − c2 − 2c Re共a兲cos y − 兩a兩2 cos2 y兲cos y dy. c + ¯a cos y
共29兲
Note that by setting D = 0 in Eqs. 共28兲 and 共29兲, we recover the results of Abrams and Strogatz.3
We now show results of following solutions of Eqs. 共28兲 and 共29兲, using D and  as bifurcation parameters. It is known for identical oscillators 共i.e., D = 0兲 that for fixed A ⬎ 0 chimerae exist for a range 0 ⬍  艋 *, and that * is an increasing function of A 共see Fig. 8 in Ref. 3兲. Here we set A = 0.95. Results for D = 0 are shown in Fig. 5; four types of solution are shown. Blue crosses indicate the modulated drift state which occurs for  = 0, Im共a兲 = 0. In this state, none of the oscillators have synchronized with one another. The green solid line
Re(a)
B. Results 0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
−0.05
−0.05
0
0.1
β
0.2
0.3
0.7
0.8 −Ω
0.9
1
FIG. 5. 共Color online兲 Solutions of Eqs. 共28兲 and 共29兲 when D = 0. Left: Re共a兲 vs . Right: Re共a兲 vs −⍀, for A = 0 95.
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0 −0.05
Chaos 19, 013113 共2009兲
Chimera states in heterogeneous networks
200 Index
Re(a)
013113-7
0.1
β
0.7
0.8 −Ω
0.9
1
FIG. 6. 共Color online兲 Solutions of Eqs. 共28兲 and 共29兲 when D = 0.01. Left: Re共a兲 vs . Right: Re共a兲 vs −⍀, for A = 0.95.
indicates the stable chimera, for which some of the oscillators are synchronized with one another while the remainder drift. The red dashed curve is the unstable chimera. The black dash-dotted line represents the uniform drift state for which  = a = 0. Note that this line has collapsed to a point in the left panel of Fig. 5. If A was decreased, the saddle-node bifurcation seen in the left panel of Fig. 5 would move to a lower value of . 关Note that stability of solutions is inferred, as it was by Abrams and Strogatz.3 All we have in Eqs. 共28兲 and 共29兲 are algebraic equations governing the steady states, with no dynamics.兴 A similar picture when D = 0.01 is shown in Fig. 6, with the same conventions. We see that the modulated drift state 关which no longer has Im共a兲 = 0兴 has moved away from  = 0, as has the uniform drift state 共a = 0兲. Although we have not indicated it in Fig. 5, for D = 0 and 0 ⬍ , the synchronized state 共with a = 0兲 is stable. However, for 0 ⬍ D there is now a range of  共approximately 0.03⬍  ⬍ 0.165 when D = 0.01兲 for which the synchronized state is unstable. 共This is the extent of the dot-dashed line in the left panel of Fig. 6.兲 For  outside this range, i.e., for  small enough or large enough, the synchronized state remains stable. Comparing the left panels of Figs. 5 and 6, we see that the case of identical oscillators 共D = 0兲 is degenerate in the sense that both the modulated and uniform drift states are “hidden” at  = 0, but both occur over finite intervals of  when 0 ⬍ D. Figure 7 shows the results of following the pitchfork and saddle-node bifurcations seen in Fig. 6 as D is varied. Note that the two pitchfork bifurcations emanate from 共D , 兲 0.07
100
200 Time
−0.988
0.5
−0.9885
0
−0.989
−0.5
300
400 1
0
−0.9895 0
−1 1000 0
500 Index
500 Index
−1 1000 0
500 Index
1000
FIG. 8. 共Color online兲 Results of a simulation of Eq. 共18兲 for which D is switched from 0 to 0.02 at t = 50, and then increased to 0.06 at t = 300. Top: sin i. Bottom row: average of di / dt as a function of index i over the time intervals 关0,50兴 共left兲, 关150,300兴 共middle兲, and 关300,450兴 共right兲. N = 1000,  = 0.15, A = 0.95.
= 共0 , 0兲. The right-most pitchfork bifurcation changes from sub- to supercritical at the termination of the curve of saddlenode bifurcations. For approximately 0.047⬍ D ⬍ 0.058 there is no bistability; instead, there is a range of  values for which only the chimera state is stable. Outside this range, only the synchronous state is stable. For D greater than about 0.058, there do not exist any chimera states. From Fig. 7 we see that with  small and fixed, increasing D first destabilizes and then restabilizes the synchronous state. This is the same behavior as observed in Fig. 1 for the model 共1兲, and is demonstrated in Fig. 8, where we fix  = 0.15 and successively increase D from 0 to 0.02 to 0.06. For D = 0 the synchronized state is stable 共bottom left panel兲. At t ⬇ 100 a chimera forms, with the center of the unsynchronized cluster at i ⬇ 200 and the center of the synchronized cluster at i ⬇ 700 共bottom middle panel兲. Once D is increased above the upper blue curve in Fig. 7 only the 共noisy兲 synchronized state is stable 共bottom right panel兲. C. Generalizations
As shown in Sec. IV B, if g共兲 =
冑2D3
冉
冊
1 , + D4 4
we could repeat the analysis of Eq. 共19兲, obtaining equations similar to, though more complicated than, Eqs. 共28兲 and 共29兲, still with the unknowns c, a, and ⍀. Indeed, for any distribution g, the double integrals 共25兲 and 共26兲 could be evaluated numerically. The form of Eq. 共22兲 is a direct result of choosing G to have one Fourier mode. If more modes were used in G 共if, for example, we were approximating a given coupling function with a finite Fourier series兲, Eq. 共22兲 would have more terms and thus more coefficients to be found.
0.06 0.05
D
0.04 0.03 0.02 0.01 0 0
−0.5
800
−0.05 0.3 0.6
0.2
0
600 1000 0
0 0
0.5
400
0.05
0.1
0.15 β
0.2
0.25
0.3
FIG. 7. 共Color online兲 Curves of pitchfork 共solid兲 and saddle-node 共dashed兲 bifurcations of solutions of Eqs. 共28兲 and 共29兲. Figure 6 corresponds to a horizontal “slice” through this figure at D = 0.01. A = 0.95.
D. Chimera states and “bumps”
Chimera states as studied in this section are very similar to “bump” states which have been studied in computational neuroscience modeling.12,11 The main difference between
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Carlo R. Laing
bump states in neural models and the chimera states studied here is that in neural models, the uncoupled unit is a model neuron which—as an input current is increased—starts to fire periodically once the current has passed a threshold,11 whereas in the chimera states studied here the uncoupled unit is a phase oscillator with uniform angular velocity. In neural models, a bump is a self-consistent state for which some neurons receive subthreshold input 共and are thus quiescent兲, while others receive superthreshold input 共and are thus firing兲. It is the coupling of phase oscillators through a sinusoidal function of the phase itself 关as in Eq. 共20兲兴, which allows some oscillators to lock and rotate at a uniform frequency 共which can be set to zero by moving to a rotating coordinate frame兲, while others drift, and thus a chimera can form. To further emphasize the similarity, compare Fig. 4 in Ref. 11 with the insets in Fig. 12 in Ref. 3 and with the middle plot in the bottom row of Fig. 8 共keeping in mind that this is a disordered system兲. The bifurcations of bumps are also very similar to those of chimerae on a ring: they both typically appear as unstable states bifurcating from a spatially uniform state; compare Fig. 10 in Ref. 12 with Fig. 12 in Ref. 3.
VII. SUMMARY
We have considered the effects of heterogeneity in the intrinsic frequencies of oscillators on chimera states in Kuramoto-type networks of coupled phase oscillators. Previous authors had only considered these states in networks of identical oscillators.1–3,10,16,21 By assuming a Lorentzian distribution of intrinsic frequencies we have generalized the results of Abrams et al.1 and Abrams and Strogatz,2,3 obtaining similar equations to them, but with an extra parameter; viz., the width of the Lorentzian distribution. All of our results show that chimerae are robust—within limits—to heterogeneity in their intrinsic frequencies, and we have shown some of the interesting bifurcations that can be induced by such heterogeneity. Importantly, in light of the recent results of Pikovsky and Rosenblum18 regarding the validity of the Ott–Antonsen ansatz 共6兲 used in this paper, our numerical results in Sec. III support the observation by Martens et al.14 that this ansatz can be used to study all attractors of a Kuramoto-type system whenever the oscillators have randomly distributed frequencies. The results presented here rely on the form of the equations studied. In particular, the results in Secs. II–V rely on the remarkable recent results of Ott and Antonsen17 showing that the infinite network can be exactly described by a finite number of ODEs, although not necessarily completely.18 Similarly, the analysis in Sec. VI depended on the form of the coupling in 共18兲, through a trigonometric function of phase differences. The challenge remains to discover similar results for oscillators not described by a single variable, and not coupled in this way.
ACKNOWLEDGMENTS
I thank the referees for their very helpful comments. 1
Abrams, D. M., Mirollo, R., Strogatz, S. H., and Wiley, D. A. “Solvable model for chimera states of coupled oscillators,” Phys. Rev. Lett. 101, 084103 共2008兲. 2 Abrams, D. M., and Strogatz, S. H., “Chimera states for coupled oscillators,” Phys. Rev. Lett. 93, 174102 共2004兲. 3 Abrams, D. M., and Strogatz, S. H., “Chimera states in rings of nonlocally coupled oscillators,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 21–37 共2006兲. 4 Acebrón, J., Bonilla, L., Pérez Vicente, C., Ritort, F., and Spigler, R., “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77, 137–185 共2005兲. 5 Baesens, C., Guckenheimer, J., Kim, S., and MacKay, R., “Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos,” Physica D 49, 387–475 共1991兲. 6 Barreto, E., Hunt, B., Ott, E., and So, P., “Synchronization in networks of networks: The onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators,” Phys. Rev. E 77, 036107 共2008兲. 7 Childs, L. M., and Strogatz, S. H., “Stability diagram for the forced Kuramoto model,” Chaos 18, 043128 共2008兲. 8 Kevrekidis, I., Gear, C., and Hummer, G., “Equation-free: The computeraided analysis of complex multiscale systems,” AIChE J. 50, 1346–1355 共2004兲. 9 Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence 共Springer, Berlin, 1984兲. 10 Kuramoto, Y., and Battogtokh, D., “Coexistence of coherence and incoherence in nonlocally coupled phase oscillators,” Nonlinear Phenom. Complex Syst. 共Dordrecht, Neth.兲 5, 380–385 共2002兲. 11 Laing, C., and Chow, C., “Stationary bumps in networks of spiking neurons,” Neural Comput. 13, 1473–1494 共2001兲. 12 Laing, C. R., “On the application of equation-free modelling to neural systems,” J. Comput. Neurosci. 20, 5–23 共2006兲. 13 Laing, C. R., and Kevrekidis, I. G., “Periodically-forced finite networks of heterogeneous coupled oscillators: a low-dimensional approach,” Physica D 237, 207–215 共2008兲. 14 Martens, E. A., Barreto, E., Strogatz, S. H., Ott, E., So, P., and Antonsen, T. M., “Exact results for the Kuramoto model with a bimodal frequency distribution,” Phys. Rev. E arXiv:0809.2129. 15 Montbrió, E., Kurths, J., and Blasius, B., “Synchronization of two interacting populations of oscillators,” Phys. Rev. E 70, 056125 共2004兲. 16 Omelchenko, O., Maistrenko, Y., and Tass, P., “Chimera states: the natural link between coherence and incoherence,” Phys. Rev. Lett. 100, 044105 共2008兲. 17 Ott, E., and Antonsen, T. M., “Low dimensional behavior of large systems of globally coupled oscillators,” Chaos 18, 037113 共2008兲. 18 Pikovsky, A., and Rosenblum, M., “Partially integrable dynamics of hierarchical populations of coupled oscillators,” Phys. Rev. Lett. 101, 264103 共2008兲. 19 Pikovsky, A., Rosenblum, M., and J. Kurths, Synchronization 共Cambridge University Press, Cambridge, 2001兲. 20 Ren, L., and Ermentrout, B., “Phase locking in chains of multiple-coupled oscillators,” Physica D 143, 56–73 共2000兲. 21 Sethia, G., Sen, A., and Atay, F., “Clustered chimera states in delaycoupled oscillator systems,” Phys. Rev. Lett. 100, 144102 共2008兲. 22 Singer, W., “Neuronal synchrony: A versatile code for the definition of relations?,” Neuron 24, 49–65 共1999兲. 23 Strogatz, S., “From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators,” Physica D 143, 1–20 共2000兲. 24 Strogatz, S., Sync: The Emerging Science of Spontaneous Order 共Hyperion, New York, 2003兲. 25 Watanabe, S., and Strogatz, S., “Integrability of a globally coupled oscillator array,” Phys. Rev. Lett. 70, 2391–2394 共1993兲. 26 Watanabe, S., and Strogatz, S., “Constants of motion for superconducting Josephson arrays,” Physica D 74, 197–253 共1994兲. 27 Wiesenfeld, K., Colet, P., and Strogatz, S., “Synchronization transitions in a disordered Josephson series array,” Phys. Rev. Lett. 76, 404–407 共1996兲.
CHAOS 19, 013114 共2009兲
Plykin-type attractor in nonautonomous coupled oscillators Sergey P. Kuznetsov Kotel’nikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Zelenaya 38, Saratov, 410019, Russian Federation
共Received 15 November 2008; accepted 29 December 2008; published online 10 February 2009兲 A system of two coupled nonautonomous oscillators is considered. Dynamics of complex amplitudes is governed by differential equations with periodic piecewise continuous dependence of the coefficients on time. The Poincaré map is derived explicitly. With the exclusion of the overall phase, on which the evolution of other variables does not depend, the Poincaré map is reduced to three–dimensional 共3D兲 mapping. It possesses an attractor of Plykin-type located on an invariant sphere. Computer verification of the cone criterion confirms the hyperbolic nature of the attractor in the 3D map. Some results of numerical studies of the dynamics for the coupled oscillators are presented, including the attractor portraits, Lyapunov exponents, and the power spectral density. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3072777兴 In mathematical theory of dynamical systems a class of uniformly hyperbolic strange attractors is known. In such an attractor all orbits are of saddle-type, they manifest strong stochastic properties and allow detailed theoretical analysis. The mathematical theory was advanced more than 40 years ago, but until now the hyperbolic strange attractors are regarded rather as a purified image of deterministic chaos than as realistic models of complex dynamics. In textbooks and reviews, examples of these attractors are traditionally represented by abstract artificial constructions like the Plykin attractor and the Smale–Williams attractor. Recently, a realistic system was suggested and implemented as electronic device, dynamics of which in stroboscopic description is associated with attractor of Smale–Williams-type. In the present article, I show how the dynamics related to an attractor of Plykin-type may be obtained in coupled nonautonomous self-oscillators. As systems of coupled oscillators occur in many fields in physics and technology, it is natural to expect that the suggested model may be realizable, for example, with electronic devices, mechanical systems, objects of laser physics and nonlinear optics. The systems with hyperbolic strange attractors may be of special interest in applications (e.g., for noise generators, chaos communication, etc.) due to the intrinsic structural stability, that means insensitivity of the chaotic motions to variations of parameters, characteristics of elements, technical fluctuations, etc. I. INTRODUCTION
Mathematical theory of dynamical systems introduces a class of uniformly hyperbolic strange attractors.1–9 In such an attractor all orbits are of saddle-type, and their stable and unstable manifolds do not touch, but can only intersect transversally. These attractors manifest strong stochastic properties and allow a detailed mathematical analysis. They are structurally stable; that means insensitivity of the structure of the attractors with respect to variation of functions and parameters in the dynamical equations. Until very recent times, the hyperbolic strange attractors were regarded rather as a 1054-1500/2009/19共1兲/013114/10/$25.00
purified image of chaos than as objects relating to complex dynamics of real-world systems. 共See discussion of the question in Ref. 9; also, a mechanical system with hyperbolic dynamics, the so-called triple linkage, has been considered there.兲 In textbooks and reviews, examples of the uniformly hyperbolic attractors are traditionally represented by mathematical constructions, the Plykin attractor, and the Smale– Williams solenoid. These examples relate to discrete-time systems, the iterated maps. The Smale–Williams attractor appears in the mapping of a toroidal domain into itself in the state space of dimension 3 or more. The Plykin attractor occurs in some special mapping on a sphere with four holes, or in a bounded domain on a plane with three holes 关Fig. 1共a兲兴.10 It is known that a variety of topologically different Plykin-type attractors may be constructed in finite twodimensional domains with holes. One of the modifications shown in Fig. 1共b兲 is of special interest for the present study and will be referred to as the Plykin–Newhouse attractor.4,11 In applied disciplines, physics and technology, people deal more often with systems operating in continuous time; they are called the flows in mathematical literature. The procedure of passage from mapping xn+1 = f共xn兲 to a flow system is called suspension.2–7 Such a passage is possible if the map is invertible. For the resulting flow system the relation xn+1 = f共xn兲 is the Poincaré map, which in the context on nonautonomous systems is called sometimes the stroboscopic map. Recently, a system was suggested and realized experimentally, in which the Poincaré map possesses an attractor of the Smale–Williams-type.12,13 It is composed of two nonautonomous van der Pol oscillators, which become active turn by turn and transfer the excitation to each other, in such manner that the transformation of the phase of oscillations on a whole cycle corresponds to expanding circle map. Computer verification of conditions guaranteeing the hyperbolic nature of the attractor was performed in Ref. 14. 共See some developments of the scheme in Refs. 15–18.兲 Until now, no explicit examples were advanced for a Plykin-type attractor to occur in a low-dimensional physi-
19, 013114-1
© 2009 American Institute of Physics
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Chaos 19, 013114 共2009兲
Sergey P. Kuznetsov
FIG. 1. Illustration of action on a plane for the map suggested in the original paper of Plykin 共a兲, and for the version of the map with the Plykin– Newhouse attractor 共b兲. Each of them may be associated with a map defined on the sphere, say, by means of the stereographic projection.
cally realizable system.32 In Refs. 20 and 21 the authors argue in favor of existence of the Plykin-type attractors in the Poincaré maps for a modified Lorenz system and for an autonomous three-dimensional system modeling dynamics of a neuron. On the other hand, an explicit example of a nonautonomous flow system with the Plykin–Newhouse attractor in the stroboscopic map has been advanced in the Ph.D. thesis of Hunt.22 The model of Hunt is defined by multiple expressions, distinct for different domains in the state space, and contains many artificially introduced smoothing factors. It is really hard to imagine that this model could be reproduced on a base of some physical system. In the present article, I show how the dynamics associated with attractor of the Plykin-type may be obtained in a system of coupled nonautonomous oscillators. As believed, it opens prospects for constructing physical and technical systems, e.g., electronic devices with the structurally stable chaotic regimes. In Sec. II a sequence of continuous transformations is defined on a two-dimensional sphere, and a system of two coupled oscillators is introduced, in which the state evolution corresponds in some sense to those transformations. The equations are written down for complex amplitudes of the oscillations. The points on the sphere represent the instantaneous states defined up to the overall phase factor. An explicit Poincaré map is derived that describes evolution of the state on one period of variation of coefficients in the nonautonomous differential equations. With exclusion of the overall phase, on which the evolution of other variables does not depend, the Poincaré map is reduced to a three-dimensional map, which possesses an attractor of the Plykin-type on an invariant sphere. In Sec. III results of computer verification of the so-called cone criterion are presented confirming the hyperbolic nature of the attractor of the three-dimensional map; it means also its structural stability. The topological type of the attractor corresponds to the construction of Plykin–Newhouse. In Sec. IV some results of numerical studies of the dynamics of the coupled oscillator system are discussed, including portraits of the attractor, Lyapunov exponents, power spectral density. In the set of equations for complex amplitudes, because of presence of a neutral direction in the state space, which is associated with the overall
FIG. 2. The unit sphere with marked points A, B, C, D, neighborhoods of which in the further construction will correspond to the holes not visited by trajectories on the attractor. The north and south poles are indicated with N and S, respectively. The angular coordinates 共 , 兲 are shown for some point M, and axes of the Cartesian coordinates x, y, z are depicted.
phase, the attractor has to be related formally to the class of partially hyperbolic ones.6,23 II. REPRESENTATION OF STATES ON A SPHERE AND EQUATIONS DESCRIBING DYNAMICS OF THE MODEL
Let us start with a system of two self-oscillators with compensation of losses from the common energy source. Let the equations for the slow amplitudes a and b read a˙ = 21 共1 − 兩a兩2 − 兩b兩2兲a, 共1兲 b˙ = 21 共1 − 兩a兩2 − 兩b兩2兲b, where is a positive parameter. Let us set b = 冑ei/2+i sin共/2兲, a = 冑e−i/2+i cos共/2兲.
共2兲
Clearly, in sustained regime of self-oscillations, the condition = 兩a兩2 + 兩b兩2 = 1 has to be valid. If we consider states satisfying = 1 and identify the states differing only in the overall phase, we can associate them with the points on a unit sphere 共Fig. 2兲. Also, on the picture the Cartesian coordinates are shown: x = cos sin , y = sin sin ,
共3兲
z = cos . Via the complex amplitudes they are expressed as x + iy = 2a*b,
z = 兩a兩2 − 兩b兩2 .
共4兲
We intend to modify the model 共1兲 in order to obtain a set of equations with coefficients periodically varying in
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Chaos 19, 013114 共2009兲
Plykin-type attractor
time, in such a way that in the stroboscopic description and in the representation on the unit sphere, the Plykin-type attractor will occur. As proved by Plykin, a uniformly hyperbolic attractor may exist on a sphere only in the presence of at least four holes, the areas being not visited by trajectories belonging to the attractor. In our construction, the holes will correspond to neighborhoods of four points A, B, C, D, with coordinates 共x , y , z兲 = 共⫾1 / 冑2 , 0 , ⫾ 1 / 冑2兲. Let us consider a sequence of the following continuous transformations on the sphere: • Flow down along circles of latitude, that is, motion of the representative points on the sphere away from the meridians NABS and NDCS towards the meridians equally distant from the arcs AB and CD. • Differential rotation around the z-axis with angular velocity depending on z linearly, in such way that the points B and C do not move, while the points A and D exchange their location. • Rotation of the sphere by 90° around the y-axis. • Flow down along circles of latitude, like at the first stage. • Inverse differential rotation around the z-axis. • Inverse rotation by 90° around the y-axis. The procedure is symmetric in the sense that the operations for stages 共I兲 and 共IV兲 are identical, while stages 共II兲 and 共III兲 differ from 共V兲 and 共VI兲 only by directions of the rotations. Intuitively, it looks reasonable that this sequence of transformations will generate a flow on the sphere accompanying with formation of filaments of fine transversal structure, the presence of which is a characteristic feature of the Plykin-type attractors. Let us construct equations for the complex amplitudes to reproduce dynamics on the above stages with corresponding motion of the points on the unit square representing the instantaneous states of the system. Duration of each of the six stages is accepted to be equal to a unit time interval. On a stage of flow down along circles of latitude we require the angular velocity of motion on the sphere to be proportional to sin 2. The simplest appropriate form of the differential equations is a˙ = − ia Im共a*2b2兲,
b˙ = ib Im共a*2b2兲.
a˙ = − 41 sb,
b˙ = 41 sa,
共7兲
where s = ⫾ 1. It corresponds to a conservative system of coupled oscillators with equal frequencies, with the coupling coefficient of such value that the energy exchange between the partial oscillators corresponds precisely to the duration of the stage. Now, we can write down equations for the complex amplitudes embracing the complete time period T = 6. For this, we compose the right-hand sides as combinations of terms 共5兲–共7兲, which are supposed to be switched in, or off, during the respective stages of the time evolution. As for the terms from Eqs. 共1兲, we account for them only on the stages of rotation. 共Their exclusion for other stages is not so significant, but we do so, because it simplifies derivation of the Poincaré map in the explicit form.兲 Finally, we arrive at the equations a˙ = − i共1 − 2 − s2兲Im共a*2b2兲a −
sb 4
1 1 + i共冑2 − 1 − 2冑2兩a兩2兲a + s2共1 − 兩a兩2 − 兩b兩2兲a, 4 2 共8兲
b˙ = i共1 − 2 − s2兲Im共a*2b2兲b + sa 4 1 1 + i共冑2 + 1 − 2冑2兩b兩2兲b + s2共1 − 兩a兩2 − 兩b兩2兲b. 4 2
共5兲
i/2+i
Indeed, substituting b=e sin共 / 2兲 and a = e−i/2+i cos共 / 2兲, after some simple transformations we get ˙ = 21 sin2 sin 2, ˙ = 0. Physically, the terms in the right-hand parts of Eq. 共5兲 give rise to a frequency shift of opposite sign for two oscillators. Magnitude of the shift is proportional to the amplitude of a low-frequency signal generated by mixing of the second harmonic components from the oscillators on a quadratic nonlinear element. On a stage of differential rotation, we set 1 a˙ = 4 i共冑2 − 1 − 2冑2兩a兩2兲a,
b˙ = 41 i共冑2 + 1 − 2冑2兩b兩2兲b,
where = ⫾ 1. In angular variables 共 , 兲, these equations 1 reduce to ˙ = 2 共冑2cos + 1兲, ˙ = 0. Note that the angular velocity ˙ depends linearly on z = cos and vanishes at z = −1 / 冑2. On this stage two subsystems must behave like uncoupled classic nonisochronous oscillators. At small amplitudes their frequencies with respect to the reference point are 1 ⌬a,b = 4 i共冑2 ⫿ 1兲. With growth of the amplitudes, the oscillation frequencies undergo a shift proportional to the squared amplitude for both subsystems. Finally, on the stages of rotation an appropriate form of the equations is
共6兲
Here the factors and s depend on time with period T = 6, and on a single period they are defined by the relations
=
冦
− 1, 1, 0,
1艋t⬍2
冧冦
4艋t⬍5 s= otherwise,
− 1,
2艋t⬍3
1,
5艋t⬍6
0,
otherwise.
冧
共9兲
Let us derive the Poincaré map, which determines transformation of the state over one period T = 6 and describes the time evolution stroboscopically. Let the initial conditions for the equations 共3兲 at tn = nT are defined as the state vector Xn = 共an , bn兲, and the state after a half of period is Xn+1/2 = F,s共Xn兲 = 共an+1/2 , bn+1/2兲. Solving Eq. 共8兲 on each successive unit interval analytically, we can represent the map F,s explicitly,
013114-4
an+1/2 =
bn+1/2 =
Chaos 19, 013114 共2009兲
Sergey P. Kuznetsov
anDei␣+/2 − sbnD*ei+/2
冑1 + 共兩an兩
2
+ 兩bn兩 兲共e − 1兲 2
xn+1/2 = Pszn ,
, 共10兲
sanDei␣+/2 + bnD*ei+/2
冋
2
2
y n+1/2 = PQ xne−共xn+yn兲/2 sin
冑1 + 共兩an兩2 + 兩bn兩2兲共e − 1兲 ,
2
册
冑 共zn 2 + 1兲 , 2
2
+ y ne共xn+yn兲/2 cos
where
冋
2
冑 共zn 2 + 1兲 2
2
zn+1/2 = PQ − sxne−共xn+yn兲/2 cos
␣ = 41 共冑2 − 1 − 2冑2兩an兩2兲,
2
冉
册
冑 共zn 2 + 1兲 , 2
P = 关共1 − e−兲冑x2n + y 2n + z2n + e−兴−1 ,
冊
Q=
1/4 2 2 2 1 兩an兩 兩bn兩 − 共an*bn兲 tanh共2兩an兩 兩bn兩 兲 D= 冑2 兩an兩2兩bn兩2 − 共anbn*兲2 tanh共2兩an兩2兩bn兩2兲 . 2
冑 共zn 2 + 1兲 2
where
 = 41 共冑2 + 1 − 2冑2兩bn兩2兲,
2
2
+ sy ne共xn+yn兲/2 sin
共14兲
The indices and s become equal ⫾1 alternately, so the mapping for the complete period is defined as follows:
Xn+1 = F共Xn兲 = F1,1共F−1,−1共Xn兲兲.
共11兲
Dynamics governed by Eqs. 共8兲 or by iterations of the map 共11兲 is invariant with respect to simultaneous constant phase shift for two oscillators, i.e., to the variable change a → aei, b → bei. Due to this, one can reduce the equations for two complex amplitudes to equations in three real variables. Performing the variable change 共4兲, we get a set of differential equations
x˙ = − 21 共z冑2 + 1兲y − 共1 − 2 − s2兲xy 2 + 21 sz
y˙ = 21 共z冑2 + 1兲x + 共1 − 2 − s2兲yx2 + s2共1 − 冑x2 + y 2 + z2兲y,
共12兲
z˙ = − 21 sx + s2共1 − 冑x2 + y 2 + z2兲z, where and s are time-dependent, as stated by formulas 共9兲. Designating at tn = nT the state vector as xn = 共xn , y n , zn兲, we can represent the three-dimensional Poincaré map as
xn+1 = f共xn兲 = f1,1共f−1,−1共xn兲兲, and the half-period map xn+1/2 = f,s共xn兲 is expressed as
共13兲
冑x2ne−共x +y 兲 + y2ne共x +y 兲 . 2 n
2 n
2 n
2 n
The maps 共11兲 and 共13兲 are invertible. The inverse maps are derived from solution of Eqs. 共8兲 and 共12兲 in the backward time; their analytic representations are omitted for brevity. III. NUMERICAL RESULTS FOR THE THREEDIMENSIONAL MAP AND HYPERBOLIC NATURE OF THE ATTRACTOR
In Figs. 3共a兲–3共c兲 portraits of the attractor are shown for the map xn+1 = f共xn兲 at = 1, = 1. They are obtained by computation of a sufficiently large number of iterations after excluding the initial transient part of the orbit. As seen from the diagram 共a兲, in the space 共x , y , z兲 the attractor is disposed on a unit sphere. In the diagram 共b兲 it is represented in the angular coordinates 共 , 兲, and in the diagram 共c兲 as an object on the plane of the complex variable W=
+ s2共1 − 冑x2 + y 2 + z2兲x,
冑x2n + y2n
x − z + iy 冑2 x + z + 冑2
.
共15兲
The last corresponds to stereographic projection from the sphere to the plane with a use of the point C as a center. This point together with a neighborhood does not belong to the attractor of the map; so, its image occupies a bounded part of the plane W. Note specific fractal-like transverse structure of the attractor. Few initial levels of this structure are easily distinguishable: The object looks like composed of strips, each of which contains narrower strips of the next level etc. As follows from the computations discussed below, it is a uniformly hyperbolic strange attractor. Its topological type corresponds to the Plykin–Newhouse attractor. The last follows from visual comparison of mutual location of filaments in diagram 共c兲 and in the Plykin–Newhouse attractor shown in diagram 共d兲. 共The last picture is taken from the paper,24 which reproduces analysis of the Hunt model,22 definitely possessing the attractor of Plykin–Newhouse.兲 To compute all Lyapunov exponents for the threedimensional map, joint iterations of Eq. 共13兲 together with a
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Plykin-type attractor
Chaos 19, 013114 共2009兲
FIG. 3. Attractor of the map 共13兲 at = 1 and = 1 in three-dimensional space on the unit square 共a兲, its representation in the angular coordinates 共 , 兲 共b兲, and a portrait on the plane in stereographic projection 共c兲. Panel 共d兲 shows for comparison the portrait of the Plykin–Newhouse attractor reproduced from Ref. 24 for the Hunt model 共Ref. 22兲. The orientation is selected especially to see better the correspondence of the structure of the filaments with that in diagram 共c兲.
collection of three equations in variations for perturbation vectors are produced. At each step, the Gram–Schmidt process is applied to obtain an orthogonal set of vectors, and normalization of the vectors to a fixed constant is performed. Lyapunov exponents are obtained as slopes of the straight lines approximating the accumulating sums of logarithms of the norm ratios for the vectors in dependence of the number of iterations.25 In particular, at = 1 and = 1 the Lyapunov exponents are, ⌳1 = 0.9575, ⌳2 = −1.2520, ⌳3 = −2, and an estimate of the attractor dimension with the Kaplan–Yorke formula yields DL = 1 + ⌳1 / 兩⌳2兩 ⬇ 1.765. To substantiate the hyperbolic nature of the attractor, let us turn to computational verification of the cone criterion known from the mathematical literature.6–8,14,22,26 Let us have a smooth map ¯x = g共x兲 that determines discrete-time dynamics on an attractor A. The criterion requires that with appropriate selection of a constant ␥ ⬎ 1, for any point x 苸 A, in the space of vectors of infinitesimal perturbations one can define the expanding and contracting cones Sx and Cx. Here Sx is a set of vectors satisfying the condition that their norms increase by factor ␥ or more under the action of the map. Cx is a set of vectors, for which the norms increase by factor ␥ or more under the action of the inverse map ˜x = g−1共x兲. The cones Sx and Cx must be invariant in the following sense. 共i兲 For any x 苸 A the image of the expanding cone from the preimage point ˜x must be a subset of the expanding cone at x. 共ii兲 For any x 苸 A the preimage of the contracting cone from the image
point ¯x must be a subset of the contracting cone at x. Let g共x兲 be a map corresponding to the k-fold iteration of the Poincaré map fk共x兲, where k is an integer selected in the course of the computations. The needed Jacobian matrices can be found in our case analytically, by differentiating Eq. 共13兲 with an application of the chain rule for the derivatives of the functional compositions. Some details of the computational procedure, which takes into account disposition of the attractor on the invariant sphere, are given in the Appendix. The calculations are organized as verification of the required conditions for a set of point on the attractor obtained from multiple iterations of the map g共x兲. We check, first, the existence of nonempty expanding and contracting cones, and secondly, the validity of inequalities corresponding to proper inclusions of these cones. If these conditions are met with ␥ = 1, the interval, is determined ␥min共x兲 艋 ␥ 艋 ␥max共x兲, in which they are true. Figure 4 shows a diagram resuming graphically results of verification of the cone criterion for the attractor of the map 共13兲 at = 1 and = 1. The computations have been performed for the map f3共x兲. The diagram represents in logarithmic scale the values ␥min共x兲 in gray and ␥max共x兲 in black versus y coordinate of the analyzed points on the attractor. Observe a gray set and a black set, one disposed strongly above, and another strongly below the horizontal line ␥ = 1. The presence of a gap separating these sets from the line ␥
013114-6
Sergey P. Kuznetsov
Chaos 19, 013114 共2009兲
FIG. 4. A graphical illustration for verification of the hyperbolic nature of the attractor for the map 共13兲, with = 1 and = 1. A positive result of the test follows from existence of the gap between the line ␥ = 1 and the black and gray sets of points, which represent, respectively, the upper and lower edges of the intervals of ␥, which met the verified conditions.
= 1 implies the positive result of the test. To express the result quantitatively, one can determine the maximum of ␥min共x兲 and minimum of ␥max共x兲 over the set of all processed points. As found, selection of the constant satisfying 0.44 ⬍ ␥2 ⬍ 2.3 ensures the required invariance of the cones. IV. NUMERICAL RESULTS FOR THE COUPLED OSCILLATORS
In accordance with the previous section, there is a correspondence between dynamics of complex amplitudes in the coupled oscillators 共8兲 and dynamics of the threedimensional mapping 共13兲 possessing the hyperbolic attractor of Plykin–Newhouse. Let us illustrate with numerical results the dynamics of the coupled oscillators concentrating on features linked with the flow nature of the system, i.e., with the continuous time evolution. Figure 5 shows plots for the amplitudes of the coupled oscillators 兩a兩 and 兩b兩 versus time obtained from numerical solution of the differential equations 共8兲 with the Runge– Kutta method at = 1 and = 1. Some small in absolute value and random in phase complex amplitudes a and b are taken as initial conditions, so the plot depicts the transient process prior to the regime of chaotic self-oscillations. In the right-hand part of the diagram the dependencies look like samples of a random process; that associates with motion on the chaotic attractor. Locally, some peculiarities can be seen because of the piecewise continuous nature of the process
FIG. 5. Plots of the real amplitudes 兩a兩 and 兩b兩 vs time in the transient process obtained from numerical solution of the differential equations 共8兲 at = 1 and = 1.
FIG. 6. 共a兲 Portrait of attractor for the system 共8兲 in three-dimensional space 共 , , t兲. In the cross section with the horizontal plane t = 0 共mod6兲 observe the object identical to that shown in Fig. 3共b兲. 共b兲 Portrait of the attractor in the plane of real amplitudes relating to one of the oscillators and separated by time interval T / 2 = 3. Technique of representation in gray scales is used: Brighter tones correspond to higher probability of visiting the pixels by the representative points. The parameter values are = 1, = 1.
composed of successive stages. In particular, the horizontal plateaus relate to the stages of evolution, on which the amplitudes 兩a兩 and 兩b兩 remain constant according to Eqs. 共6兲. Note that the realizations for 兩a兩 and 兩b兩 are interconnected; in the sustained regime they obey the relation 兩a兩2 + 兩b兩2 = 1. Figure 6 presents two versions of portraits of the attractor for the system 共8兲 at = 1 and = 1. As the dimension of the state space is sufficiently high 关vector X = 共a , b兲 is fourdimensional, and the extended state space of the nonautonomous system is five-dimensional兴, depicting the image to resolve subtle fractal transverse structure intrinsic to the attractor is not a trivial task. For this, we apply presentation of the object in gray scales. Brighter tones correspond to pixels visited by the representative point with higher probability.27 In panel 共a兲 this technique is used to draw the three-dimensional portrait. Angular coordinates 共 , 兲 are plotted in a horizontal plane. The third variable plotted along the vertical axis is time, within one full period of variation of coefficients in Eqs. 共8兲. The picture reminds us of rising and swirling smoke. In the cross section with a horizontal plane t = 0 共mod6兲 observe a fractal-like formation identical to the
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Plykin-type attractor
FIG. 7. Four Lyapunov exponents vs parameter at = 1 共a兲 and vs parameter at = 1 共b兲 for the Poincaré map represented in terms of complex amplitudes. A zero exponent occurs due to invariance of the equations with respect to the overall phase shift.
attractor of the three-dimensional map depicted in angular coordinates in Fig. 3共b兲. One more portrait is shown in the panel 共b兲 on the plane of two values of real amplitude 兩a共t兲兩 and 兩a共t − 3兲兩, which relate to instants separated by a half of time period of variation of the coefficients in Eqs. 共8兲. Here, again one can distinguish the fractal-like transverse structure linked with the dynamics of the Plykin-type. This method of visualization may be appropriate in experiments with systems of the class under consideration. The Lyapunov exponents i of Eqs. 共8兲 are linked with the exponents for the Poincaré map 关see Eqs. 共10兲 and 共11兲兴 by an evident relation i = ⌳i / T, where T = 6 is the period of variation of the coefficients in the equations. The procedure of computation of the Lyapunov exponents ⌳i is analogous to that used for the three-dimensional map. Joint iterations of the map 共11兲 together with a collection of four equations in variations are produced. At each step, the Gram–Schmidt process is applied to the set of vectors, and normalization of them to a fixed constant is performed. Figure 7 presents the results graphically. The first plot 共a兲 shows four Lyapunov exponents ⌳i dependent on parameter at fixed = 1. In the range ⬍ c ⬇ 2.03 one of the exponents is positive that means chaos. Among other exponents one is zero 共up to numerical errors兲, and two are negative. Note a smooth dependence of the largest exponent on the parameter. For larger 共strong dissipation bringing in during the flow down stages兲 chaos disappears. The second plot 共b兲 shows the Lyapunov exponents versus at fixed = 1. Observe that variation of notably effects only one exponent, which corresponds, obviously, to an approach of orbits to the invariant sphere. The presence of a zero exponent reflects invariance of the equations with respect to the overall phase shift. Of course, the results of the computations agree with the data from the previous section: At identical and three nonzero exponents are equal, up to numerical errors, to those obtained for the three-dimensional map. Figure 8 shows a plot of spectral density in logarithmic scale versus a frequency for a signal generated by one of the oscillators. It relates to a regime of dynamics on the Plykin– Newhouse attractor interpreted in terms of the threedimensional map. This spectrum is one more characteristic that is interesting in the context of possible experiments. In computations we used the standard method recommended for nonparametric statistical estimates of the power spectral density. It is based on subdividing the whole realiza-
Chaos 19, 013114 共2009兲
FIG. 8. The power spectral density for one of the coupled oscillators vs the frequency computed by processing a realization obtained from numerical integration of Eqs. 共8兲 at = 1 and = 1.
tion on a number of parts of equal duration. For each part, the signal is multiplied by a smooth function vanishing at the ends of the interval 共“window”兲, then Fourier transform is applied, and finally, the squared amplitudes of the frequency components are averaged over all the parts. A sample of the time series for the complex variable a共t兲 was obtained from numerical solution of Eqs. 共8兲 by the finite-difference method. It corresponds to motion on the attractor and contains 6 · 105 data points with time step ⌬t = 0.01. As seen from the picture, the spectrum looks continuous that corresponds to chaotic nature of the generated signal. The spectrum is almost perfectly symmetric about the reference frequency, where the spectral density is maximal. The two main side maximums have a level below the central one by about 7 dB, and their frequencies approximately correspond to the inverse value of the period of variation of the coefficients in Eqs. 共8兲: f ⬇ ⫾ 1 / 6. Apparently, this periodicity is a reason for the rugged form of the spectrum. A plot for the power spectral density for the second oscillator looks exactly the same because of the symmetry of the system.
V. CONCLUSION AND DISCUSSION
In the present article a system of two coupled nonautonomous nonlinear oscillators is introduced manifesting chaotic dynamics, which is in a direct relation with the concepts of the hyperbolic theory. With the exclusion of the overall phase, the Poincaré map reduces to a three-dimensional map possessing attractor of Plykin-type disposed on an invariant sphere. As the systems of coupled oscillators occur in many fields in physics and technology, it is natural to expect that the suggested model may be realizable. Particularly, it may relate to electronic devices, mechanical systems, objects of laser physics, and nonlinear optics. The systems with hyperbolic chaos may be of special interest for applications due to their robustness, or structural stability, that means insensitivity of the devices to variations of parameters, characteristics of elements, technical fluctuations, noise, etc. Appearance of concrete examples of systems with hyperbolic strange attractors makes it reasonable to apply for them the whole arsenal of computational methods accumulated in nonlinear dynamics and its applications. This is of evident interest both from the point of view of complementation of the mathematical concepts with concrete and visible context 共see, e.g., Ref. 29兲, and for exploiting these concepts in applications. In the present work, such results of computer
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Chaos 19, 013114 共2009兲
Sergey P. Kuznetsov
studies are presented as realizations, attractor portraits, Lyapunov exponents, estimate of dimension, power spectral density. It is worth mentioning some possible modifications of the model. • It is easy to suggest a version of the equations, in which the temporal dependence of the coefficients will be piecewise smooth. For this, one can introduce in the equations x˙ = f共x , t兲 a smooth common time-dependent factor vanishing at the junctions of the stages, an integral of which over a stage duration equals 1. An appropriate variant is x˙ = 共2 sin2 t兲f共x , t兲. The Poincaré map remains the same, and the nature of the attractor is not changed too. 共Hunt used a similar trick in his thesis.22兲 • As mentioned, the “self-oscillatory” term in the equations proportional to may be retained on all stages of the dynamics. • Working with a version of the model described by the three-dimensional set of equations, it is possible to simplify the form of the nonlinearity excluding the operation of extracting the square root, and setting the respective term to be 共1 − x2 − y 2 − z2兲. This modification does not influence dynamics on the attractor belonging to the invariant set x2 + y 2 + z2 = 1. 共In amplitude equations for a and b, such modification leads rather to complication because of an increase of the degree of the nonlinear terms.兲 • Taking into account structural stability of the hyperbolic attractor, many other modifications can be done, which do not change the nature of the attractor, while the variations are not too large. In particular, it is possible to introduce a model with smooth analytical variation of the coefficients in time in the nonautonomous equations.28 Formally, in our complex amplitude equations the attractor should not be interpreted as uniformly hyperbolic, because of the presence of a neutral direction in the state space associated with the overall phase. Instead, it has to be related to the class of partially hyperbolic attractors.6,23 Nevertheless, in the form used here invariance of the equations with respect to the overall phase is exact; it means that one can accept a rightful agreement not to distinguish states distinct only in the phase, and, in this sense, treat the dynamics as true hyperbolic. However, in systems, for which description in terms of slow complex amplitudes is an approximation, and the breakdown of the phase-shift symmetry occurs, one can expect the appearance of peculiarities associated with features of the partially hyperbolic attractor. If the deflections from the slow-amplitude approximation are small, I should suppose, nevertheless, that the dynamics of the basic variables will retain its character because of intrinsic robustness, while the overall phase will manifest slow diffusive-like random walk. Rigorously speaking, this question concerning longtime dynamics of time-dependent perturbations, for sure, requires special research. Dependent on the parameters, the suggested model can manifest chaos and regular 共periodic兲 dynamics. So, it may serve as an object for principal and interesting studies of scenarios of the onset of hyperbolic strange attractors in the
course of parameter variations 共see, e.g., Refs. 30 and 31兲. Insufficient progress in this research direction may be explained particularly by the fact that no realistic examples of concrete systems undergoing such transitions were known. ACKNOWLEDGMENTS
A part of this work was performed during a visit of the author to the Group of Statistical Physics and Theory of Chaos in Potsdam University. The research is supported by RFBR-DFG Grant No. 08-02-91963. APPENDIX: DETAILS OF COMPUTATIONAL PROCEDURE FOR VERIFICATION OF THE HYPERBOLICITY CRITERION
For a three-dimensional dissipative map ¯x = g共x兲, x, ¯x 苸 R let us consider the procedure for verification of the cone criterion, bearing in mind the situation that one expanding and two contracting directions present in the state space, and the attractor is located on the invariant sphere. The map is supposed to be invertible: any state vector x has a unique image ¯x = g共x兲 and a unique preimage ˜x = g−1共x兲. Let the derivative matrix of the map g at x be v = dx共g兲, which acts in the tangent space of vectors u = 兵u1 , u2 , u3其. Via the auxiliary symmetric matrix bˆ = vTv, where the superscript T means the transpose, the norm of the vector ¯u = vu is expressed as 3
¯ 储2 = uTbˆ u. 储u
共A1兲
The expanding cone at the point x is a set of vectors Sx = 兵u兩uTbˆ u 艌 ␥2uTu其.
共A2兲
With the same matrix bˆ one can define a preimage of the contracting cone relating to the point ¯x = g共x兲, namely, C¯x⬘ = 兵u兩␥2uTbˆ u 艋 uTu其.
共A3兲
Now, let us consider an inverse map ˜x = g−1共x兲 and its matrix derivative w = dx共g−1兲. Via the auxiliary symmetric matrix aˆ = wTw we represent the norm of the vector ˜u = wu as ˜ 储2 = uTaˆ u. 储u
共A4兲
With the matrix aˆ we define the contracting cone at x as a set Cx = 兵u兩uTaˆ u 艌 ␥2uTu其,
共A5兲
and a cone that is an image of the expanding cone relating to the point ˜x = g−1共x兲, namely, S˜x⬘ = 兵u兩␥2uTaˆ u 艋 uTu其.
共A6兲
In computations it is necessary to check, first, the existence of nonempty cones satisfying the definitions, and, second, validity of the inclusions S˜x⬘ 傺 Sx and C¯x⬘ 傺 Cx. As the attractor is placed on an invariant sphere S2, let us assume that x 苸 S2. To define a convenient orthogonal basis 兵i1 , i2 , i3其 we take as i3 a unit vector directed along the radius at x, and the unit vectors i1 , i2 are taken in the tangent plane. To be concrete, we require the matrix element bˆ12 to vanish, and the inequality bˆ11 ⬎ bˆ22 to hold.
013114-9
Chaos 19, 013114 共2009兲
Plykin-type attractor
The conditions of required inclusion of the cones are formulated in terms of quadratic forms associated with the matrices b = bˆ − ⌫2eˆ and a = aˆ − ⌫2eˆ , where eˆ is the unit matrix. A constant factor ⌫ is assumed to be equal ␥, or 1 / ␥, considering the expanding, or contracting cones, respectively. Note that the matrices a and b are symmetric: bij = b ji, aij = a ji. Equations b11u21 + b22u22 + b33u23 + 2b13u1u3 + 2b23u2u3 = 0
共A7兲
= 冑b11u⬘1,
2 + 2 = 1.
−
+
共A9兲
the quadratic form in Eq. 共A7兲 is reduced to a standard form,
⬘ u⬘32 = 0, b11u1⬘2 + b22u2⬘2 + b33
共A10兲
while Eq. 共A8兲 becomes
⬘ u3⬘2 + 2a12u1⬘u2⬘ + 2a13 ⬘ u1⬘u3⬘ a11u1⬘2 + a22u2⬘2 + a33 ⬘ u2⬘u⬘3 = 0. + 2a23
0 =
0 =
冑− b11b⬘33
+
a11 = 0, b11
共A17兲
⬘ + a13 ⬘ a23 ⬘ a12a33 ⬘ − a22a33
⬘2 a23
⬘ a22 + a12a23 ⬘ a13 ⬘ − a23 ⬘2 a22a33
冑
−
b22 , b11
冑
b⬘ − 33 . b11
共A18兲
In variables ˜ = − 0, ˜ = − 0, the equation becomes 共A11兲
Here
−
2 −1 2 ⬘ = b33 − b−1 b33 11 b13 − b22 b23
⬘ 2a13
and its center is located at
determine the borders of the cones. By variable change u3⬘ = u3
共A16兲
⬘ 2a23 a⬘ 2a12 a22 2 + − 33 2 + 冑 冑b22b33 b22 ⬘ b − b11b22 ⬘ 33
共A8兲 u2⬘ = u2 + b−1 22 b23u3,
共A15兲
Then, the equation for the second ellipse is
a11u21 + a22u22 + a33u23 + 2a12u1u2 + 2a13u1u3 + 2a23u2u3
u1⬘ = u1 + b−1 11 b13u3,
= 冑− b33u⬘3
and setting u1⬘ = 1 / 冑b11 transforms the first ellipse to a unit circle,
and
=0
= 冑− b22u2⬘,
共A12兲
⬘ 2a23 a⬘ a22 2 ˜ + ˜˜ − 33˜2 = R2 , 冑 b22 ⬘ b33 ⬘ b22b33
共A19兲
where
and −1 ⬘ = a13 − a11b−1 a13 11 b13 − a12b22 b23 ,
R2 = −
−1 ⬘ = a23 − a12b−1 a23 11 b13 − a22b22 b23 ,
共A13兲 −
2 −2 2 −1 ⬘ = a33 + a11b−2 a33 11 b13 + a22b22 b23 − 2a13b11 b13
In the cross section by a plane u1⬘ = const, Eqs. 共A10兲 and 共A11兲 determine some curves of the second order; their types and mutual location have to be revealed in the course of computations. To have a situation of inclusion required by the criterion, these curves must be ellipses. First, in computations we check the inequalities b11 ⬎ 0, b22 ⬍ 0, b33 ⬘ ⬍ 0. If they are true, Eq. 共A10兲 defines an ellipse. To determine the type of the curve given by Eq. 共A11兲, we compute the invariants
D=
冏
⬘ a22 a23 ⬘ a33 ⬘ a23
冏
,
2a12
⬘ 2a13
0 . 冑− b11b22 0 − 冑− b11b33 ⬘
共A20兲
Computing the lesser root of the square equation
−1 −1 − 2a23b−1 22 b23 + 2a12b11 b13b22 b23 .
⬘, I = a22 + a33
⬘ 2a23 a⬘ a11 a22 2 + 0 − 00 + 33 20 冑b22b⬘33 b11 b22 ⬘ b33
冨
⬘ a11 a12 a13
⬘ A = a12 a22 a23 ⬘ a23 ⬘ a33 ⬘ a13
冨
共A14兲
and check that D ⬎ 0, and A / I ⬎ 0. Then, in accordance with the theory of conic sections, Eq. 共A11兲 also defines an ellipse. Let us formulate a convenient and simple sufficient condition of location of the second ellipse inside the first one. Renormalizing variables
⬘ /b33 ⬘ 兲l + 共a22a33 ⬘ − 4a23 ⬘2兲/共b22b33 ⬘ 兲 = 0, l2 + 共a22/b22 + a33 共A21兲 one finds out the semimajor axis R / 冑lmin. A sufficient condition for the ellipse to be located inside the unit disk is inequality
冑20 + 20 + R/冑lmin ⬍ 1.
共A22兲
If all the named conditions are true at ⌫ = ␥ and at ⌫ = 1 / ␥, one can deduce about the correct inclusion for the expanding and contracting cones at the analyzed point x. Indeed, in variables , the cross section of the expanding cone Sx is the closed unit disk, and the cross section of the cone S˜x⬘ is represented by the closure of interior of the small ellipse obtained at ⌫ = ␥, so, the required inclusion S˜x⬘ 傺 Sx is valid 关Fig. 9共a兲兴. On the other hand, cross section of the contracting cone Cx is a closure of exterior of the small ellipse obtained at ⌫ = 1 / ␥. Cross section of the cone C¯x⬘ corresponds to a closure of exterior of the unit circle. Hence, the inclusion C¯x⬘ 傺 Cx is valid 关Fig. 9共b兲兴.
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Chaos 19, 013114 共2009兲
Sergey P. Kuznetsov 4
FIG. 9. Cross sections of the cones for the case of the three-dimensional map with one expanding and two contracting directions. A picture of proper inclusion is shown 共a兲 for expanding cones S˜x⬘ 傺 Sx and 共b兲 for contracting cones C¯x⬘ 傺 Cx. A circle circumscribed around the ellipse is located inside the unit disk if condition 共A4兲 is valid.
Computations in the present work were organized as verification of the cone criterion for a set of points on the attractor obtained from multiple iterations of the map g共x兲. The inclusions were checked with a help of the inequality 共A22兲. Because of the smoothness of the map under study, the objects considered in the context of the cone criterion 共matrices, quadratic forms, and their invariants兲 depend on the state variables in smooth manner, as they are determined by the dynamics on finite time intervals. As follows, validity of the conditions at some point x with distant from 1 constant ␥ implies that the cone criterion holds as well in a neighborhood of x 共as wider, as larger the value 兩␥ − 1兩 is兲. A positive result of the test for a representative set of points implies validity of the cone criterion on the whole attractor, if it is completely covered by the union of the mentioned neighborhoods. Practically, such situation is achieved by an increase of the number of iterations and, respectively, a number of tested points on the attractor. It is convenient not to fix in advance the constant ␥, but to arrange the computations as follows. First, at each point x we compute the matrices aˆ and bˆ and check all the formulated conditions for ␥ = 1. If they hold, the program determines an allowable interval of ␥. For this, the program simply enumerates the ␥ values with a small step in a sufficiently wide range. The attractor is recognized as hyperbolic if a gap of a finite width separates the obtained sets of top and bottom edges of the intervals from the axis ␥ = 1 on the plot of ␥ versus some dynamical variable characterizing location of the analyzed point x. J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 共1985兲. R. L. Devaney, An Introduction to Chaotic Dynamical Systems 共AddisonWesley, New York, 1989兲. 3 L. Shilnikov, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 1353 共1997兲. 1 2
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 共Springer, Berlin, 2002兲. 5 L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics 共World Scientific, Singapore, 1998兲. 6 A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems 共Cambridge University Press, Cambridge, 1995兲. 7 V. Afraimovich and S.-B. Hsu, Lectures on Chaotic Dynamical Systems, AMS/IP Studies in Advanced Mathematics 共American Mathematical Society, Providence; International, Somerville, 2003兲, Vol. 28. 8 B. Hasselblatt and Y. Pesin, “Hyperbolic dynamics,” Scholarpedia, http:// www.scholarpedia.org 共2008兲. 9 T. J. Hunt and R. S. MacKay, Nonlinearity 16, 1499 共2003兲. 10 R. V. Plykin, Math. USSR. Sb. 23, 233 共1974兲. 11 S. E. Newhouse, Lectures on Dynamical Systems, Dynamical Systems, C.I.M.E. Lectures Bressanone. Progress in Mathematics, No 8 共Birkhäuser-Boston, Boston, 1980兲, p. 1. 12 S. P. Kuznetsov, Phys. Rev. Lett. 95, 144101 共2005兲. 13 S. P. Kuznetsov and E. P. Seleznev, JETP 102, 355 共2006兲. 14 S. P. Kuznetsov and I. R. Sataev, Phys. Lett. A 365, 97 共2007兲. 15 O. B. Isaeva, A. Yu. Jalnine, and S. P. Kuznetsov, Phys. Rev. E 74, 046207 共2006兲. 16 S. P. Kuznetsov and A. Pikovsky, Physica D 232, 87 共2007兲. 17 S. P. Kuznetsov and A. Pikovsky, Europhys. Lett. 84, 10013 共2008兲. 18 S. P. Kuznetsov and V. I. Ponomarenko, Tech. Phys. Lett. 34, 771 共2008兲. 19 J. T. Halbert and J. A. Yorke, “Modeling a chaotic machine’s dynamics as a linear map on a ⬘square sphere’,” http://www.math.umd.edu/~halbert/ taffy-paper-1.pdf. 20 C. A. Morales, Ann. Inst. Henri Poincare, Sect. A 13, 589 共1996兲. 21 V. Belykh, I. Belykh, and E. Mosekilde, Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, 356 共2005兲. 22 T. J. Hunt, “Low dimensional dynamics: Bifurcations of Cantori and realizations of uniform hyperbolicity,” Ph.D. thesis, University of Cambridge, 2000. 23 Y. Pesin and B. Hasselblatt, “Partial hyperbolicity,” Scholarpedia, http:// www.scholarpedia.org 共2008兲. 24 J. S. Aidarova and S. P. Kuznetsov, “Chaotic dynamics of the Hunt model, an artificially constructed system with hyperbolic attractor,’’ Izv. VUZov—Prikladnaja Nelineinaja Dinamika 16, 176 共2008兲 共in Russian兲, Saratov State University, ISSN 0869–6632, English translation. Preprint is available http://xxx.lanl.gov/abs/0901.2727. 25 G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, Meccanica 15, 9 共1980兲. 26 J. G. Sinai and E. P. Vul, Physica D 2, 3 共1981兲. 27 S. P. Kuznetsov, Dinamicheskii Khaos 共Moscow, Fizmatlit, 2006兲 共in Russian兲. 28 S. P. Kuznetsov A non-autonomous flow system with Plykin type attractor, ‘‘Communications in Nonlinear Science and Numerical Simulations,’’ Elsevier 共to be published兲. Preprint is available http://xxx.lanl.gov/abs/ 0901.3533. 29 Y. Coudene, Not. Am. Math. Soc. 53, 8 共2006兲. 30 S. Newhouse, D. Ruelle, and F. Takens, Commun. Math. Phys. 64, 35 共1978兲. 31 A. L. Shilnikov, L. P. Shilnikov, and D. V. Turaev, Mosc. Math. J. 5, 269 共2005兲. 32 A special comment is needed to the work of Halbert and Yorke 共Ref. 19兲 announcing a physical realization of the Plykin attractor. As a physical object, the taffy-pulling machine they discuss is not a low-dimensional system, but contains a piece of continuous medium undergoing deformations in such way that the motion of local elements of the medium obeys a map with the Plykin-type attractor. In other words, it is an ensemble of elements, each of which carries out motion on the Plykin-type attractor. Thus, referring to the physical realization of the attractor, the authors stand for another meaning than that we have in mind here 共as well as other authors, Refs. 9 and 20–22兲.
CHAOS 19, 013115 共2009兲
Coherence resonance induced by rewiring in complex networks Mi Jiang and Ping Ma State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100871, China
共Received 1 October 2008; accepted 8 January 2009; published online 11 February 2009兲 We report a novel coherent excitation phenomenon in a heterogeneous network of coupled FitzHugh–Nagumo elements. It is demonstrated that dynamical rewiring in the network can play a constructive role to bring on coherent excitations. The coherence factor as the function of rewiring time interval represents a nontrivial phenomenon which is a fingerprint of coherence resonance. We call this resonant behavior caused by dynamical wiring changes the network-rewiring-induced coherence resonance. The mechanism can be understood by the effective noise played by the rewiring process. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3076398兴 Coherence resonance 共CR兲 has been well known in excitable systems driven by external noises. Typically, optimized moderate noise strength generates best regular oscillations. In this report, we demonstrate that in a network composed of excitable elements, the rewiring of connections in the network can effectively play the role of noise and induce coherence resonance at an optimized rewiring rate. The networkrewiring-induced resonant behavior could be significant in the understanding of the abundant effects of evolving complex networks. Recent advances in the study of complex networks have witnessed a shift of research interest from static networks to evolving ones.1–7 In the real-world systems, such as, biological, epidemiological, and social networks, both the coupling between the nodes and the topological structure of network can be dynamically evolvable. Due to the relevance to a wide variety of phenomena, synchronization behaviors in these evolving complex networks have attracted extensive attention.1–4 Recently, the synchronization over a network of moving chaotic agents was reported4 and provides a promising model for clock synchronization in mobile robots or task coordination of swarming animals. Dynamical behaviors in neuronal networks, especially the strength of connections between neurons is activity-dependent and changeable. Ongoing research of structural plasticity in the adult brain, including synapse formation/elimination and remodelling of axons and dendrites, revealed that the cortical wiring diagram can also be changed by learning activities.8–10 Further understanding of these advances in diverse disciplines requires extensive investigations on evolving complex networks. In this paper, we demonstrate that dynamical wiring changes in complex networks can induce constructively coherent synchronized excitations in the network. The behavior is an exceptional example of coherence resonance 共CR兲.11–14 CR is also called autonomous stochastic resonance because no external periodic signal is needed in contrary to the conventional stochastic resonance. This nontrivial phenomenon was recently demonstrated in small-world networks driven by spatially correlated noise.15 Experimental evidence of CR has also been observed in the coherence between spinal and cortical neurons in the somatosensory system of the anesthetized cat.16 With a small-world network composed of non1054-1500/2009/19共1兲/013115/4/$25.00
identical elements, we report that when the connections in the network are rewired in a time interval , the best coherence of excitations can be achieved at an intermediate optimized value of the rewiring interval . The phenomenon is thus a coherence resonance induced by rewiring in complex networks. The system we consider is a network composed of N FitzHugh–Nagumo 共FHN兲 elements. FHN is a typical model for excitable media, especially for neuronal systems.17,18 The dynamical evolution of an element in the network is determined by the following equations: N
x˙i = xi −
x3i /3
− y i + g 兺 Lij共t兲x j ,
共1兲
j=1
y˙ i = xi + ai ,
共2兲
where the Laplacian matrix of the network Lij共t兲 = Aij共t兲 − ki共t兲␦ij, Aij共t兲 is the adjacent matrix of the network, and g measures the coupling strength. ki共t兲 = 兺Nj=1Aij共t兲, is the degree of node i. ai is the control parameter. Each isolated FHN element in the network has only one stable fixed point when 兩a兩 ⬎ 1.0 while a limit cycle appears when 兩a兩 ⬍ 1.0. When 兩a兩 is slightly larger than 1.0, the system is excitable. In our model, we assume that ai is inhomogeneous and is distributed uniformly in the range ai 苸 共1 , 1.1兲. The system is therefore composed of N elements with nonidentical excitability. They are initially organized into a network of Watts–Strogatz small-world topology,19 which is constructed by rewiring a ring of 2K-nearest neighbor nodes with probability p 共0 ⬍ p ⬍ 1兲. We further assume a dynamical rewiring scenario in the network, that is, as the system evolves according to Eqs. 共1兲 and 共2兲, where each connection in the network is rewired with probability pre every time interval . When a link is rewired, both ends of this link are shifted to two new nodes which are selected uniformly at random and have not connected in the last time interval . Equations 共1兲 and 共2兲 are integrated numerically by using Euler’s algorithm with = 0.01 and with different coupling strength g and rewiring probability pre. The initial conditions for the elements are always random.
19, 013115-1
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S
4 3 2
pre=0.1
1
pre=0.2 pre=0.3
MFI
15
pre=0.5
10
pre=0.8
5 0
FIG. 1. Spatiotemporal plot 共vertical for the nodes and horizontal for time兲 of persistent excitations of the elements when the connections are rewired at different rewiring intervals. The rewiring process is switched on at t = 30 and off at t = 170. The white zones represent excitations. From top to bottom = 0.04, = 0.11, and = 0.3, respectively. Other parameters: = 0.01, N = 100, pre = 0.3, g = 0.2.
When the network is static without any rewiring, i.e., pre = 0, no excitation persists and the network always resorts to the stable state with each element setting down to its stable fixed point. With a nonzero pre, we observe that a too fast 共small 兲 or too slow 共large 兲 rate of rewiring of the network cannot induce sustained pulse trains. It is with a moderate rewiring interval that persistent excitations appear. Figure 1 demonstrates the spatiotemporal excitations of elements with three different rewiring intervals. Both before switching on the rewiring and after switching off the rewiring no emitted spike trains are observed. Obviously it is the rewiring process that results in the excitations. At = 0.04 or = 0.3, the excitations are continuously sustained but are rather irregular 共top and bottom panels兲. Quite regular pulses come out at an optimal interval with = 0.11 共middle panel兲. The phenomenon is in close resemblance to CR induced by either external or intrinsic noise,11–14 or by effective noise of fast deterministic dynamics.20 Besides, the spatiotemporal excitations in the network are well synchronized, with the pulses fired almost simultaneously. The phenomenon of synchronized excitation is closely related to the high degree of integrity of the network. Due to constant rewiring of the links, nodes in the network can communicate effectively with each other so that local activities can be integrated in a global and coherent dynamics. To quantitatively measure the coherence of the emitted spike trains in the network, we adopt the parameter of sharpness Si for each element i in the network,13 which is calculated from the mean and variance of the pulse intervals of adjacent firing events Tik = tik+1 − tik according to Si = 具Tik典t/冑Var共Tik兲
共i = 1,2, . . . ,N兲.
共3兲
The coherence factor of the whole network S is similarly defined, but is averaged over both time and space. Besides
0.1
0.2
τ
0.3
0.4
FIG. 2. 共Color online兲 Coherence factor S and mean firing interval 共MFI兲 as a function of rewiring time interval when the connections are rewired with different probabilities pre. The network is composed of N = 100 excitable elements with coupling strength g = 0.2. Results are averaged over 50 realizations.
we have determined the time between two successive firings and calculated the mean firing interval 共MFI兲, which can be regarded as the measurement of the time scale of interspike period. Figure 2 共top panel兲 illustrates the coherence factor S as a function of the rewiring time interval . There is always a maximal coherence which is achieved at a moderate value of . The tent-shaped S- curve is a clear fingerprint of CR.11–14 The behavior is exceptional in that the noise intensity is replaced here by the rewiring rate of the network. We call this collective phenomenon caused by rewiring the networkrewiring-induced coherence resonance 共NRCR兲. On the other hand, the MFI- curve 共bottom panel兲 also represents a nontrivial behavior but shows an opposite trend to S in that the same opt corresponds to the smallest interspike period, which is exactly the point to understand the best coherence of S. Note that the rewiring time interval to support sustained excitations is much smaller than the time scale of the interspike period. Actually, approximately 50 times of rewiring are needed to support one excitation. We checked the effect of the rewiring probability Pre. As illustrated in Fig. 2, when the network is rewired with a larger probability pre, a larger value of is required in order to achieve the maximal coherence level and minimal interspike period. There is also an optimal rewiring probability, approximately at pre ⬇ 0.3, when the coherence factor S reaches the highest peak 共black triangles in Fig. 2兲. This implies that the network which is rewired to an intermediate extent would best benefit the coherence of spike trains. Besides, the minimal value of which is needed to support excitations in the network grows when more percentage of connections are rewired 共larger probability pre兲. The coherent excitations in the network also depend nontrivially on the coupling strength g between the elements. Too small or too large values of g would not sustain the spike trains for the elements. Persistent excitations are supported only in the range 0.1艋 g 艋 0.3. The NRCR behavior would thus not take place when the coupling strength is too weak or
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−0.8 −1
3.5
0.25
0.2
2.5 2
0.15
2 0
S
g
3
−1.2
−2 2 0
1.5 0.1
0.1
0.2
τ
0.3
0.4
FIG. 3. Coherence factor S vs the rewiring interval and coupling strength g, with the white for high value of S. Other parameters: rewiring probability pre = 0.3 and network size N = 100. The results have been averaged over 50 realizations.
too strong. Figure 3 depicts the distribution of coherence factor S in the g- plane. S reaches its peak roughly at coupling strength g ⬇ 0.2 and ⬇ 0.1. The randomness of the initial small-world network can be tuned by adjusting the probability p from regular 共p = 0兲 to small-world 共0 ⬍ p ⬍ 1兲, and to completely random 共p = 1兲 networks. Our simulations reveal that the parameter p has no significant influence on the behavior of coherent excitations described above, i.e., the NRCR phenomenon is not dependent on the initial structure of the network. In fact, the network with the scenario of rewiring will tend to the Erdős– Rényi random network so that the initial topology has neglected effects. Nevertheless, the heterogeneity in the FHN elements of the network is crucial for the appearance of NRCR. The parameter ai in Eq. 共2兲 determines the excitability and the stationary state of element i. To ensure persistent excitations and NRCR, ai should be inhomogeneous over the network. In the above results, the size of the network and the number of the connections are both constant. We find that the rewiring-induced excitations in the network depend on the density of connections in the network as well. The connection density is characterized by the ratio of the number of currently existing connections to the number of all possible links, which reads = NK / C2N = 2K / 共N − 1兲. We observe that a sparsely linked network is more efficient to induce sustain persistent excitations, i.e., NRCR appears only when the link density is relatively small. This can be represented by the effects of N and K, respectively, on the persistent excitations. As illustrated in Fig. 4, for N 艋 20 or K 艌 4, the persistent excitations are highly suppressed; especially when N = 10 and K = 5 共the top and bottom panel兲 no excitations can be induced. The link density determines the degree of integrity of the network. A too densely linked network is highly united and any differences in the states of the nodes would diffuse fast and fade away quickly. The network would resort finally to the rest state. In sparse networks, rewiring can dramatically increase the number of new functional connections to ensure the appropriate intensity of effective noise. CR behaviors have been reported from excitable systems driven by external noises or from systems influenced by in-
−2 0 −1 −2 0
40
80
t
120
160
200
FIG. 4. The temporal behavior of a characteristic node in the network showing the influence of N and K on the persistent excitations. The rewiring process is switched on at t = 30 and off at t = 170. From the top frame to bottom frame, 共N , K兲 is 共10, 3兲, 共20, 3兲, 共100, 4兲, 共100, 5兲, respectively. Other parameters: g = 0.2, pre = 0.2, = 0.12.
trinsic noises that lie close to the onset of Hopf bifurcation. The NRCR phenomenon is different from the conventional CRs in that it emerges without the influence of external or internal noises. Instead, the rewiring of connections is effectively equivalent to the role of noise. It is also different from the deterministic coherence resonance20 in which the fast deterministic dynamics plays the role of effective noise. However, we can still understand the mechanism of NRCR based on the effective noise. As the excitability and stationary state of the nodes are inhomogeneous in the network, the random rewiring of links brings constantly random and temporary contacts between elements at different dynamical states. A node in the network is effectively affected by random perturbations. To characterize the effective noise, we directly define Inti共t兲 = g兺Nj=1Lij共t兲x j as the intensity of the noise acting on node i and accordingly the intensity of the effective noise in the whole network can be defined as Int = 1 / N limT→⬁ 兺Ni=11 / T兰T0 Int2i 共t兲dt. Compared to Eqs. 共1兲 and 共2兲, this definition views the integrated inputs, including the interconnection architecture and its temporal change, of the whole system as the effective noise. Although certainly there are other modified definitions, this method of measuring the effective noise is easy and reliable to work out the mechanism of NRCR phenomenon. Figure 5 demonstrates that the dependence of the “noise strength” Int on is also tent-shaped for different pre. Particularly compared to Fig. 2, it is noted that the optimal rewiring interval opt corresponding to the best coherence of excitations also ensures the largest effective noise intensity. On the other hand, we know that the FHN system has two characteristic times: the activation time ta and the excursion time te and the firing interval is the sum of them.11 In our simulations there is always te ⬇ 4.5, namely the fluctuations of te is negligible. For relatively strong effective noise the contribution of te is dominant, which can be proved from the MFI- curve 共bottom panel in Fig. 2兲. For different pre, opt all correspond to the minimal value of MFI⬇ 4.5, which en-
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6.2
x 10
Int
5.8
5.4 pre=0.1 pre=0.3
5
pre=0.5
4.6
0
0.1
0.2
τ
0.3
0.4
FIG. 5. 共Color online兲 The intensity of effective noise, Int, as a function of rewiring time interval when the network topology are rewired with different probabilities pre. The network is composed of N = 100 excitable elements with coupling strength g = 0.2.
sures MFI⬇¯te so that the coherence factor S is largely determined by te. Because of the same reason of the almost constancy of te, S reaches the top value for the opt. However, according to the MFI- curve in Fig. 2 and Int- curve in Fig. 5, the time scales of ta and te are neck and neck for weak noises while the fluctuation of ta is relatively large. Therefore the coherence factor decreases. Regarding different rewiring probability Pre, as shown in Fig. 5, opt always corresponds to the largest noise intensity and there exists an optimal Pre to ensure the strongest noise as well. For the same reason that relatively larger noise intensity leads to better coherence of excitations, it is easy to understand the existence of the optimal rewiring probability pre ⬇ 0.3. We have also computed the noise intensity for different coupling strength g. It is shown that the trend of noise amplitude versus g and is similar to Fig. 3. Coupling strength g that is too large 共small兲, would make persistent excitations impossible since the effective noise amplitude is too large 共small兲 to maintain the stability of spike trains 共induce to exceed the excitation threshold兲. In our rewiring scheme, both ends of a connection are shifted to newly selected nodes, providing each FHN element the same opportunity to receive and respond to the effective noise induced by the random rewiring of the network. Simulations for the case that connections are rewired by shifting only one end to new nodes demonstrate that the amplitude of effective noise is much smaller, approximately equal to the value out of the
domain of supporting persistent excitations corresponding to two nodes rewiring. As a result the phenomenon of persistent excitations is largely attenuated. In conclusion, we have demonstrated a novel coherence resonance phenomenon with a network composed of excitable FitzHugh–Nagumo elements. It is shown that the dynamical rewiring process in networks can play a constructive role to induce collective coherent excitations. The rewiring of connections in the network can be viewed as effective noise, and resonance comes out at optimal and intermediate rewiring rates. The phenomenon reported represents an exceptional variation of CR behavior that has been investigated in a wide variety of systems. In the literature of the research on the evolving complex networks, our findings suggest a prominent influence of the topological structural changes that networks could bring on. The authors thank H. Wang for useful discussions. This work is supported by the National High Technology Research and Development Program of China 共Grant No. 2007AA03Z238兲, the National Natural Science Foundation of China 共Grant No. 10674006兲, and the State Key Development Program for Basic Research of China 共Grant No. 2006CB601007兲. J. D. Skufca and E. M. Bollt, Math. Biosci. Eng. 1, 347 共2004兲. M. Porfiri, D. J. Stilwell, E. M. Bollt, and J. D. Skufca, Physica D 224, 102 共2006兲. 3 F. Sorrentino and E. Ott, Phys. Rev. Lett. 100, 114101 共2008兲. 4 M. Frasca, A. Buscarino, A. Rizzo, L. Fortuna, and S. Boccaletti, Phys. Rev. Lett. 100, 044102 共2008兲. 5 I. V. Belykh, V. N. Belykh, and M. Hasler, Physica D 195, 188 共2004兲. 6 C. Moore, G. Ghoshal, and M. E. J. Newman, Phys. Rev. E 74, 036121 共2006兲. 7 L. B. Shaw and I. B. Schwartz, Phys. Rev. E 77, 066101 共2008兲. 8 A. M. Turner and W. T. Greenough, Brain Res. 329, 195 共1985兲. 9 G. W. Knott, C. Quairiaux, C. Genoud, and E. Welker, Neuron 34, 265 共2002兲. 10 D. B. Chklovskii, B. W. Mel, and K. Svoboda, Nature 共London兲 431, 782 共2004兲. 11 A. S. Pikovsky and J. Kurths, Phys. Rev. Lett. 78, 775 共1997兲. 12 Z. Liu and Y-C. Lai, Phys. Rev. Lett. 86, 4737 共2001兲. 13 B. Hu and C. Zhou, Phys. Rev. E 61, R1001 共2000兲. 14 M. A. Arteaga, M. Valencia, M. Sciamanna, H. Thienpont, M. LópezAmo, and K. Panajotov, Phys. Rev. Lett. 99, 023903 共2007兲. 15 O. Kwon, H-H. Jo, and H-T. Moon, Phys. Rev. E 72, 066121 共2005兲. 16 E. Manjarrez, J. G. Rojas-Piloni, I. Méndez, L. Martínez, D. Vélez, D. Vázquez, and A. Flores, Neurosci. Lett. 326, 93 共2002兲. 17 A. C. Scott, Rev. Mod. Phys. 47, 487 共1975兲. 18 M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 共1993兲. 19 D. J. Watts and S. H. Strogatz, Nature 共London兲 393, 440 共1998兲. 20 J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, Phys. Rev. Lett. 93, 144101 共2004兲. 1 2
CHAOS 19, 013116 共2009兲
Experimental verification of rank 1 chaos in switch-controlled Chua circuit Ali Oksasoglu,1,a兲 Serdar Ozoguz,2,b兲 Ahmet S. Demirkol,2,c兲 Tayfun Akgul,2,d兲 and Qiudong Wang1,e兲 1
Department of Mathematics, The University of Arizona, Tucson, Arizona 85721, USA Faculty of Electrical-Electronics Engineering, Istanbul Technical University, 34469 Istanbul, Turkey
2
共Received 23 September 2008; accepted 31 December 2008; published online 12 February 2009兲 In this paper, we provide the first experimental proof for the existence of rank 1 chaos in the switch-controlled Chua circuit by following a step-by-step procedure given by the theory of rank 1 maps. At the center of this procedure is a periodically kicked limit cycle obtained from the unforced system. Then, this limit cycle is subjected to periodic kicks by adding externally controlled switches to the original circuit. Both the smooth nonlinearity and the piecewise linear cases are considered in this experimental investigation. Experimental results are found to be in concordance with the conclusions of the theory. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3073967兴 This paper is about experimentally demonstrating the use of a new chaos theory, namely, the theory of rank 1 maps, in practical systems. One of the most important practical implications of the theory is that it provides a recipelike procedure to obtain chaos from a weakly stable limit cycle. The setting of the theory is easily satisfied by use of externally controlled switches in a given autonomous system. The practical system used in this investigation is the well-known Chua circuit. Our investigation in this paper has shown that the predictions of the theory are in great agreement with our experimental findings. Thus, it can reasonably be claimed that this theory can provide a practical means of generating chaos from many practical electronic oscillators. I. INTRODUCTION
In a sequence of papers published recently, Oksasoglu and co-workers proposed a generic scheme of creating rank 1 chaos in practical circuits by using periodically controlled switches.1–4 In these studies, the theory of rank 1 maps was applied to rigorously verify the existence of rank 1 attractors. In addition, extensive numerical simulations were conducted in search of the strange attractors implicated under the guidance of the theory. The results of these numerical simulations were found to be in perfect match with the conclusions of the theory. The theory of rank 1 maps used in these studies is a new chaos theory developed in recent years by Wang and Young.5–7 This new chaos theory is based on the Jakobson theory on quadratic maps8 and the studies of Benedicks and Carleson on strongly dissipative Hénon maps.9 In this paper, we provide the first experimental evidence of rank 1 chaos in a switch-controlled circuit, namely, the switch-controlled Chua circuit10 as proposed by Oksasoglu a兲
Electronic mail:
[email protected]. Also at Honeywell Inc., Tucson, AZ 85737. b兲 Electronic mail:
[email protected]. c兲 Electronic mail:
[email protected]. d兲 Electronic mail:
[email protected]. e兲 Electronic mail:
[email protected]. 1054-1500/2009/19共1兲/013116/9/$25.00
and Wang in Ref. 4. To demonstrate the applicability of the theory of the rank 1 maps, a Hopf limit cycle in the smooth case and an arbitrary limit cycle in the piecewise linear 共PWL兲 case are used in this investigation. Since a local nonlinearity is needed to create Hopf bifurcations, a Chua circuit with a nonlinear resistor of cubic nonlinearity is constructed following the procedure outlined in Ref. 11. Then, the parameters of the circuit are so chosen that it has a weakly stable oscillation coming out of a supercritical Hopf bifurcation. For the case of arbitrary 共non-Hopf兲 limit cycles, a local nonlinearity is not necessary; hence, the use of the original Chua circuit with a three-segment PWL resistor is sufficient for this purpose. In both cases, once a limit cycle is created, switches controlled by externally applied periodic pulses are added to the circuit in such a way as to modulate the state variables, namely, the capacitor voltages and the inductor current. The addition of the periodically controlled switches to an existing nonlinear system provides a natural setting for the application of the theory of rank 1 maps. In other words, the use of periodically controlled switches generates the kicking effect proposed by Wang and Young7,12 to create rank 1 chaos. Consequently, in the range of parameters where the theory of rank 1 maps applies, strange attractors do appear in such a way as predicted by the theory. A great majority of the existing studies on chaotic attractors are based on breaking the homoclinic loop by small perturbations to yield a transverse homoclinic orbit 共transversal intersections of the stable and unstable manifolds兲 in the phase space 共see, e.g., Ref. 13兲. The rank 1 attractors presented in this paper are, however, of a different kind. They are generated by small disturbances that are periodically applied to a weakly stable limit cycle. When small disturbances in the form of periodic kicks are introduced 共by use of externally controlled switches in this case兲, the shape of the weakly stable limit cycle is slightly deformed. Then, the natural force of shearing created by the nonlinearity of the original system goes to work and exaggerates the initial deformation to create chaos. The dynamical properties of the rank 1 attractors created that way are dominated by the so-
19, 013116-1
© 2009 American Institute of Physics
013116-2
Chaos 19, 013116 共2009兲
Oksasoglu et al.
i
I (t) v
v (t)
FIG. 1. A switch-controlled state variable modulation scheme.
FIG. 2. Switched-controlled Chua circuit.
called Sinai–Ruelle–Bowen measures14 representing the statistical law of the system. It is worth noting that the use of various kicking schemes in the study of chaotic dynamics is not uncommon.15–17 However, the study of the strange attractors in this paper differs from others in that it is supported by a comprehensive theory of dynamics that has a long history. The theory itself is little known outside the pure mathematical side of the dynamical systems community and has only been recently developed into a form that is applicable to concrete systems of differential equations. We refer the reader to a recent tutorial paper for more background information on the theory and its potential applications to circuits and systems.18 II. THEORETICAL SETTING AND IMPLEMENTATION APPROACH
共1兲
where u 苸 Rn, n ⱖ 2, represents the system state variables and 苸 Rm, m ⱖ 1, the system parameters. It is assumed that there is a = 0 at which the system of Eq. 共1兲 goes through a supercritical Hopf bifurcation. This system is then modified to obtain the nonautonomous system of the form du = f 共u兲 + ⌽共u兲PT,p共t兲, dt
C
dvc = is共t兲 − vcG1 PT,p共t兲, dt 共3兲
diL = vs共t兲 − iLR2 PT,p共t兲. L dt In the scheme of Fig. 1, the resulting ⌽共u兲, the shape of the forcing, becomes ⌽共u兲 = −u. III. SWITCH-CONTROLLED CHUA CIRCUIT
In this section, we briefly discuss the setting of the theory and a practical approach originally introduced in Ref. 4 to generically satisfy the requirements of the theory. We first start with an autonomous system given by du = f 共u兲, dt
each switch is controlled by the periodic pulse train PT,p共t兲. In this case, the governing equations for the capacitor voltage and the inductor current are given by
共2兲
where PT,p共t兲 is a periodic pulse train with a pulse width of p and a period of T, ⌽共u兲 is a function that determines the shape of the forcing, and is used to control the magnitude of the forcing. Let T Ⰷ p so that a pulse of pulse width p is followed by a long relaxation period T − p. We regard the system of Eq. 共2兲 as the kicked version of the system of Eq. 共1兲. When the system of Eq. 共1兲 is an electrical circuit whose state variables are the capacitor voltages and inductor currents, the system of Eq. 共2兲 can be implemented for certain ⌽共u兲 by modulating the state variables through switches externally controlled by PT,p共t兲. This scheme, depicted in Fig. 1, was proposed by Oksasoglu and Wang in Ref. 4. In Fig. 1,
For the experimental investigations of this paper, we apply the above-outlined scheme to the well-known Chua circuit.10 The modified circuit, which is referred to as the switch-controlled Chua circuit, is depicted in Fig. 2. The switches Si are controlled by a periodic pulse train with p0 and T0 being the physical pulse width and the period, respectively. A. Smooth case
Due to the need for a local nonlinearity for Hopf bifurcations to occur, the PWL resistor 共Chua’s diode兲 in the circuit of Fig. 2 is replaced with a nonlinear resistor whose v − i characteristic is given by in共v1兲 = g共v1兲 = a1v1 + a3v31 .
共4兲
The physical implementation of this cubic nonlinearity is achieved by using the design approach given in Ref. 11. The governing equations for the switch-controlled circuit can be given by C1
dv1 = G共v2 − v1兲 − g共v1兲 − G1v1 , dt 共5兲
dv2 C2 = i + G共v1 − v2兲 − G2v2, dt for nT0 ⱕ t ⬍ nT0 + p0 and by
di L = − v 2 − R 3i dt
013116-3
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Chaos 19, 013116 共2009兲
Experimental verification of rank 1 chaos
dv1 = G共v2 − v1兲 − g共v1兲, dt 共6兲
dv2 = i + G共v1 − v2兲, C2 dt
di L = − v2 dt
for nT0 + p0 ⱕ t ⬍ 共n + 1兲T0, n = 0 , 1 , 2 , . . .. Putting Eqs. 共5兲 and 共6兲 together, we obtain ⬁
dv1 = G共v2 − v1兲 − g共v1兲 − G1v1 兺 Fn,p0,T0共t兲, C1 dt n=0 dv2 = i + G共v1 − v2兲 − G2v2 兺 Fn,p0,T0共t兲, dt n=0
共7兲
Fn,T0,p0共t兲 =
再
1, nT0 ⱕ t ⬍ nT0 + p0 0, elsewhere.
v2 y= , V0
i z= , I0
2 = − ␣2b1共b1 − 1兲 ⬎ 0.
共13兲 共14兲
According to the standard theory of Hopf bifurcations, Eq. 共10兲 has a center manifold, on which the equation for the flow can be transformed into the following normal form:
冎
共8兲
By setting v1 x= , V0
共12兲
b1 苸 共0,1兲.
di = − v2 − R3i 兺 Fn,p0,T0共t兲, dt n=0
where
␣共b1 − 1兲共␣b1 + 1兲 ⬎ 0, 
Thus, a necessary condition for a Hopf bifurcation to occur is
⬁
L
0 = −
the eigenvalues of the linear part of Eq. 共10兲 are ⫾i and −共␣b1 + 1兲, where
⬁
C2
Ref. 4 and give the values of parameters for the system of Eq. 共10兲, for which rank 1 chaos is likely to occur. For this purpose, first, we consider the autonomous part of the system of Eq. 共10兲 and look for the values of parameters for a supercritical Hopf bifurcation at 共x , y , z兲 = 共0 , 0 , 0兲. For the computations to follow, the parameter is regarded as the bifurcation parameter. Observe that at
t t→ n
共9兲
dz = 共a共兲 + 共兲冑− 1兲z + k1共兲z2¯z + k2共兲z3¯z2 + ¯ , 共15兲 dt where k1共兲 , k2共兲 are complex numbers. The fact that there is a well-defined computational process to reach the indicated normal form is important to us. Let us write k1共兲 = − E共兲 + F共兲冑− 1.
we obtain the following dimensionless set of equations: dx = ␣关y − h共x兲兴 − 1xPT,p共t兲, dt
共16兲
From the computations given in Ref. 4, we have E共0兲 = − c1共1 + 2␣b1 − ␣兲,
dy = ␥关x − y + z兴 − 2yPT,p共t兲, dt
F共0兲 = − c1
共10兲
共1 + 2␣b1兲, ␣b1 共17兲
where dz = − y − 3zPT,p共t兲, dt
c1 =
where
p = p 0 n,
T = T 0 n,
h共x兲 = b1x + b3x3 , a1 b1 = 1 + , G
− 3␣b3 关1 + 2␣b1 − ␣兴 ⬍ 0. 8b1共␣ b1 + 2␣b1 + 1兲
a3V20 b3 = , G
2
共11兲
␣= =
G , C 1 n Rn , Ln
␥= 1 =
G = 1.0, C 2 n
␣Rp , R1
2 =
=
␥Rp , R2
R , Rn
Rn =
3 =
共18兲
Furthermore, in order to have a weakly stable periodic solution coming out of the origin, it is also necessary to have E共0兲 ⬎ 0 yielding
⬁
1 PT,p共t兲 = 兺 Fn,T,p共t兲, p n=−⬁
− 3␣b3 . 8b1共1 + 2b1␣ + b1␣2兲
V0 , I0
R3 p . R
B. Conditions for Hopf bifurcation and rank 1 chaos
In this subsection, the conditions for the Hopf limit cycle and rank 1 chaos are derived. These conditions are obtained by following an explicit, recipe-like procedure given in
共19兲
Consequently, for a supercritical Hopf limit cycle to occur, we must have b1 ⬎
␣−1 if b3 ⬎ 0, 2␣
b1 ⬍
␣−1 if b3 ⬍ 0. 2␣
共20兲
As was shown in Ref. 12, in order for rank 1 attractors to exist, a relatively large twist number, as defined below, is needed,
ª
冏 冏
F共0兲 . E共0兲
共21兲
Therefore, to find rank 1 attractors, the values of parameters are adjusted in such a way to make the following large:
013116-4
Chaos 19, 013116 共2009兲
Oksasoglu et al. 0.6
0.1 0.08
0.4
0.06 0.04
0.2
y(t)
y(t)
0.02 0
0
−0.02
−0.2
−0.04 −0.06
−0.4
−0.08 −0.1
−0.6
−0.25 −0.2 −0.15 −0.1 −0.05
0 x(t)
0.05
0.1
0.15
0.2
−1
FIG. 3. A Hopf limit cycle from numerical simulations 共i = 0兲.
冏 冏冏 冏冏
冏
F共0兲 Im共k1兲 共1 + 2␣b1兲 = = . E共0兲 Re共k1兲 ␣b1共1 + 2␣b1 − ␣兲
共ii兲
共iii兲
共22兲
Parameter values for Hopf bifurcation: b3 ⫽ 0,  ⬎ 0, ␣ ⬎ 1 are arbitrarily fixed, and is around 0 = −␣共b1 − 1兲共␣b1 + 1兲 / . Strong shearing: choose b1 苸 共0 , 1兲 sufficiently close to b1 = 共␣ − 1兲 / 2␣ either from above or below depending on the sign of b3 关see Eq. 共20兲 for stability criterion兴. Parameters of forcing: choose relatively small, e.g., ⬍ 1.
With the guidance of the steps above, the following parameter values are chosen and fixed:
␣ = 2.0,
 = 2.0,
␥ = 1.0,
0 x(t)
0.5
1
FIG. 4. A non-Hopf limit cycle from numerical simulations 共i = 0兲.
In summary, the values of parameters are determined using the following guidelines as given in Ref. 18: Let ␣, , ␥, , b1, b3 be the parameters of the autonomous part of Eq. 共10兲 and p, = 1, T be the parameters of the periodic forcing. The values of all parameters except T are fixed as follows: 共i兲
−0.5
in共v1兲 = g共v1兲 = Gbv1 + 0.5共Ga − Gb兲共兩v1 + Vb兩 − 兩v1 − Vb兩兲. 共25兲 Using the change of variables given in Eq. 共9兲 results in the same dimensionless system as that of Eq. 共10兲: dx = ␣关y − h共x兲兴 − 1xPT,p共t兲, dt dy = ␥关x − y + z兴 − 2yPT,p共t兲, dt
共26兲
dz = − y − 3zPT,p共t兲. dt However, in this case, the nonlinear function h共x兲 is given by h共x兲 = m1 + 0.5共m0 − m1兲共兩x + B p兩 − 兩x − B p兩兲,
共27兲
where m0 = 1 +
Ga , G
m1 = 1 +
Gb , G
Bp =
Vb . V0
共28兲
b1 = 0.242, 共23兲
The rest of the system parameters are as given by Eq. 共11兲. The autonomous part of Eq. 共26兲, obtained by setting 1 = 2 = 3 = 0, has a limit cycle, as shown in Fig. 4, for
A Hopf limit cycle numerically obtained for the values of Eq. 共23兲 is shown in Fig. 3. This is the limit cycle that is going to be kicked to obtain rank 1 chaos. In this case, the twist constant is roughly
共␣, ␥, , ,m0,m1,B p兲 = 共2.0,1.0,2.0,1.12,− 0.75,− 0.225,1.0兲.
0 = 1.124 872,
ª
b3 = − 1.0,
= 0 − 0.005.
冏 冏
F共0兲 = 108. E共0兲
共24兲
C. Piecewise linear case
In this case, the v − i characteristics of the nonlinear resistor of Fig. 2 is given by
共29兲 This limit cycle is the one that is kicked to obtain rank 1 chaos. IV. NUMERICAL SIMULATIONS FOR RANK 1 CHAOS
In this section, the results of our numerical simulations for both smooth and PWL cases are presented. Although various single- or multiswitch control schemes can be formulated by setting selected i to zero, in our investigations, only S1 and S2 are employed by setting 3 = 0. The numerical results of this section are obtained by directly solving Eqs.
Chaos 19, 013116 共2009兲
Experimental verification of rank 1 chaos (a)
(b)
0
0.05
−0.045
y(kT)
y(kT)
y(kT)
0.1
−0.04
0.1 0.05
−0.05
0 −0.05
−0.05 −0.055
−0.1
−0.1
−0.06 −0.2 −0.1
0 0.1 x(kT)
0.2
−0.23
(c)
−0.22 x(kT)
−0.25
−0.21
0 −0.1
60
0.1
40
8000
−0.05
0 x(kT)
0
0.05
2000 k
3000
2000
4000
FIG. 5. A single-switch case rank 1 attractor from numerical simulations: smooth case 共1 = 0.5, 2 = 3 = 0, T = 501.5兲. 共a兲 Time-T map on x-y plane. 共b兲 Magnification of the indicated area in 共a兲. 共c兲 Time evolution of xk. 共d兲 Frequency spectrum of xk.
0.2
0.25
(c)
100 50
0
1000
0.15
150
−0.2
0
0.1
200
0
−0.2 6000
−0.1
−0.1
20
4000 k
−0.15 (b)
0.2
x(kT)
X(ω)
x(kT)
0.1
2000
−0.2
(d) 80
0.2
0
(a)
0.15
0.15
X(ω)
013116-5
4000 k
6000
0
8000
0
1000
2000 k
3000
4000
FIG. 6. A single-switch case rank 1 attractor from numerical simulations: smooth case 共1 = 0.5, 2 = 3 = 0, T = 97.0兲. 共a兲 Time-T map on x-y plane. 共b兲 Time evolution of xk. 共c兲 Frequency spectrum of xk.
B. Arbitrary limit cycle
共10兲 and 共26兲. Computations are performed using the fourthorder Runge–Kutta routine starting at t0 = 0. For both cases, only the kicking parameters 共i and T兲 are varied to obtain rank 1 chaos. The remaining system parameters are fixed as given in Eq. 共23兲 for the smooth case 共Hopf limit cycle兲 and Eq. 共29兲 for the PWL case. For each chaotic picture presented, the time-T map obtained for one discrete orbit starting near the attractor is given along with the time evolution and the frequency spectrum of the attractor.
In this subsection, the rank 1 attractors shown in Figs. 8–10 are numerically obtained by kicking the non-Hopf limit cycle given in Fig. 4. The approach taken in presenting the figures of this subsection is the same as that of Sec. IV A. In other words, the attractor depicted in Fig. 8 is for the singleswitch 共S1兲 scheme with a large value of T, and those of Figs. 9 and 10 are for lower T values with the single-switch 共S1兲 and double-switch 共S1 and S2兲 kicking schemes, respectively. The parameter values used in these cases are 1 = 0.5 and T = 253 for Fig. 8, 1 = 0.5 and T = 71.5 for Fig. 9, and 1 = 0.36, 2 = 0.17, and T = 72 for Fig. 10. As before, for the cases of lower T 共Figs. 9 and 10兲, the arms of the attractors are readily visible, whereas for the case of large T 共Fig. 8兲,
A. Hopf limit cycle (a)
0.15 0.1 y(kT)
0.05 0 −0.05 −0.1 −0.25
−0.2
−0.15
−0.1
−0.05
0 x(kT)
(b)
0.05
0.2
100
0.1
80
0
40
−0.2
20 2000
4000 k
6000
8000
0.15
0.2
0.25
3000
4000
(c)
60
−0.1
0
0.1
120
X(ω)
x(kT)
The rank 1 attractors presented in this subsection are numerically obtained for different parameter values and kicking schemes. The limit cycle kicked in this case is the Hopf limit cycle shown in Fig. 3. The attractor of Fig. 5共a兲 is obtained by employing only S1 with 1 = 0.5 and T = 501.5. In this case, a high value of T is used to capture the true rank 1 nature of the attractor. Even though the attractor in Fig. 5共a兲 appears to be a simple closed curve, it is, in fact, a chaotic attractor of a very complicated structure. This complicated structure is evident from Fig. 5共b兲, which is the magnification of the indicated area in Fig. 5共a兲 and the continuous characteristic of the frequency spectrum shown in Fig. 5共d兲. For more on this, we refer the reader to Ref. 18. Using lower values of T reveals more of the chaotic structure of the attractors. The attractors in the next two figures, Figs. 6 and 7, are such examples for the single-switch 共S1兲 and the double-switch 共S1 and S2兲 cases, respectively. The parameter values used in these cases are 1 = 0.5 and T = 97 for Fig. 6 and 1 = 0.5, 2 = 0.34, and T = 87.5 for Fig. 7. As stated before, with lower values of T, the chaotic structure of these attractors is readily visible.
0
0
1000
2000 k
FIG. 7. A two-switch case rank 1 attractor from numerical simulations: smooth case 共1 = 0.5, 2 = 0.34, 3 = 0, T = 87.5兲. 共a兲 Time-T map on x-y plane. 共b兲 Time evolution of xk. 共c兲 Frequency spectrum of xk.
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Oksasoglu et al. (a)
(a)
(b) −0.34
0.6
0.6
0
−0.36
0.4
−0.38
0.2
y(kT)
0.2
y(kT)
−0.4
−0.2
−0.4
−0.42
−0.4 −0.6
−0.6
−0.44 −1
−0.5
0 x(kT)
0.5
1
−1.15
(c)
−1.1 x(kT)
−1
−1.05
1
150
−0.5 −1 0
2000
4000 k
6000
−0.5
50
−1
0
8000
0
1000
2000 k
3000
4000
FIG. 8. A single-switch case rank 1 attractor from numerical simulations: PWL case 共1 = 0.5, 2 = 3 = 0, T = 253兲. 共a兲 Time-T map on x-y plane. 共b兲 Magnification of the indicated area in 共a兲. 共c兲 Time evolution of xk. 共d兲 Frequency spectrum of xk.
they are radially pushed down to give the attractor the appearance of a simple closed curve, whose complicated structure is also evident from Figs. 8共b兲 and 8共d兲. V. CIRCUIT IMPLEMENTATION AND EXPERIMENTAL RESULTS
In this section, the circuit implementations and the experimental results for both the smooth and PWL cases are presented. For both cases, the circuit of Fig. 2 is realized with different implementations for the nonlinear resistor. In the smooth case, the nonlinear resistor is implemented to realize a v − i characteristic of the form i = a1v1 + a3v31. In the
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.5
0 x(kT)
(b)
0.5 (c)
200
1
1
150 X(ω)
0.5 0 −0.5
100 50
−1 0
2000
4000 k
6000
8000
0
0
1000
2000 k
3000
4000
FIG. 9. A single-switch case rank 1 attractor from numerical simulations: PWL case 共1 = 0.5, 2 = 3 = 0, T = 71.5兲. 共a兲 Time-T map on x-y plane. 共b兲 Time evolution of xk. 共c兲 Frequency spectrum of xk.
(c)
150 100 50
0
2000
4000 k
6000
0
8000
0
1000
2000 k
3000
4000
FIG. 10. A two-switch case rank 1 attractor from numerical simulations: PWL case 共1 = 0.36, 2 = 0.17, 3 = 0, T = 72兲. 共a兲 Time-T map on x-y plane. 共b兲 Time evolution of xk. 共c兲 Frequency spectrum of xk.
PWL case, its implementation realizes a three-segment PWL v − i characteristic. The passive element values in both cases are chosen to be C1 = 2.2 nF,
C2 = 4.4 nF,
L = 5 mH,
R = 1.5 k⍀. 共30兲
With the values above, the frequency scaling constant is found to be
n =
1 = 151 515 rad/s, RC2
共31兲
resulting in the following nondimensional parameter values:
␣ = 2.0,
(a)
−1
0
100
1
200
0.5 x(kT)
0
0.5
250
1
200 X(ω)
x(kT)
0 x(kT)
250
0.5
y(kT)
−0.5 (b)
(d) 300
x(kT)
0 −0.2
X(ω)
y(kT)
0.4
␥ = 1.0,
 = 1.98.
共32兲
In addition, the pulse width of the kicking pulse train is chosen to be p0 = 3.5 s, giving p = p0n = 0.523. As before, for both cases, three experimentally obtained rank 1 attractors are presented. As in the case of numerical simulations, the first two of these figures are for the singleswitch case, one with a high and another with a low value of T. The third figure is for the double-switch case with a low value of T. Note that T = nT0 where T0 is the period of the kicking pulse train. In the single-switch scheme, only the switch S1 is activated. In the case of double-switch scheme, the switches S1 and S2 both are activated. In these experimental simulations, first, the values of 1 and 2 共for the double-switch scheme only兲 are fixed. This is accomplished by fixing the values of R1 and R2 in Fig. 2. Then, to generate chaotic attractors, only the period T0 共i.e., T兲 of the kicking pulse train is varied. As in the case of numerical simulations, the time-T maps of the experimentally obtained chaotic attractors are given in the figures to follow. This is achieved by sampling the experimentally obtained flow of the system 共circuit of Fig. 2兲 every T0 seconds.
013116-7
Chaos 19, 013116 共2009兲
Experimental verification of rank 1 chaos
FIG. 11. Cubic resistor implementation.
A. Smooth case
The circuit implementation of the cubic v − i characteristic, in = a1v1 + a3v31, of the nonlinear resistor in Fig. 2 can be accomplished by use of analog multipliers. Here, the same design approach given in Ref. 11 is followed. The specific analog multipliers used for this purpose are AD633 of Analog Devices. For operational amplifiers, AD711 are used. The biasing used for all of the active elements is ⫾5 V. The resulting implementation of this cubic v − i characteristic is given in Fig. 11. For the controlled switches of Fig. 2, Texas Instruments’ CD4016 is used. In order to stay in the vicinity of the normalized parameter values given in Eq. 共23兲, the physical element values for Fig. 11 are chosen to be Ra = Rb = Rd = Re = 2.2 k⍀,
Rc = 2.07 k⍀.
共33兲
These values result in the following nondimensional parameter values: b1 = 0.242,
b3 = − 1.0,
= 1.12.
共34兲
With this choice of parameter values, an experimentally obtained Hopf limit cycle is shown in Fig. 12. In Fig. 12, the horizontal axis corresponds to v1共t兲 of Fig. 2 with 0.5 V / div and the vertical axis to v2共t兲 of Fig. 2 with 0.25 V / div. This
FIG. 12. A Hopf limit cycle from experimental simulations 共i = 0兲.
FIG. 13. Time-T map for a single-switch case rank 1 attractor from experimental simulations: smooth case 共1 = 0.787, 2 = 3 = 0, T = 212兲.
Hopf limit cycle is the one that is kicked to experimentally obtain the rank 1 attractors given in this subsection. The attractor shown in Fig. 13 is obtained for the singleswitch scheme with a high value of T = 212. In this case, the magnitude of kicking, 1, is set to 1 = 0.787 by choosing R1 = 3020 ⍀ in Fig. 2. Note that, as was seen in the corresponding numerical simulation of Fig. 8, the arms of the attractor in Fig. 13 are pressed down. By lowering the value of T, attractors with more visible structures are obtained as shown in Figs. 14 and 15. The one in Fig. 14 is obtained for the single-switch scheme with T = 60 and 1 = 0.787 共R1 = 3020 ⍀兲. The attractor shown in Fig. 15 is obtained for the double-switch scheme with T = 112, 1 = 0.526 共R1 = 2020 ⍀兲, and 2 = 0.398 共R2 = 2000 ⍀兲.
FIG. 14. Time-T map for a single-switch case rank 1 attractor from experimental simulations: smooth case 共1 = 0.787, 2 = 3 = 0, T = 60兲.
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FIG. 17. A non-Hopf limit cycle from experimental simulations 共i = 0兲.
FIG. 15. Time-T map for a two-switch case rank 1 attractor from experimental simulations: smooth case 共1 = 0.526, 2 = 0.398, 3 = 0, T = 112兲.
B. Piecewise linear case
The physical circuit implementation of the PWL resistor can, in general, be considered more feasible due to the fact that, unlike the case of smooth nonlinearity, it requires no analog multipliers but passive resistors and operational amplifiers only. Such a circuit realization of a three-segment PWL resistor is shown in Fig. 16. In this realization, high
FIG. 16. A three-segment PWL resistor implementation.
performance current feedback operational amplifiers are used. The passive element values for the circuit of Fig. 16 are chosen to be Ra = 1.5 k⍀,
Rb = 1.7 k⍀,
Rm = 5.3 k⍀,
共35兲
corresponding to m0 = − 0.88,
m1 = − 0.4.
共36兲
With this choice of parameters, the experimentally obtained non-Hopf limit cycle that is to be kicked is shown in Fig. 17. The attractor shown in Fig. 18 is experimentally obtained for the single-switch and high-T case where 1 = 0.787 共R1 = 3020 ⍀兲 and T = 155. The attractor shown in Fig. 19 is for the single-switch and low-T case where 1 = 0.66 共R1 = 2410 ⍀兲 and T = 82. For the double-switch case, R1 and R2 are set to 2410, and 2200 ⍀, corresponding to 1 = 0.66 and 2 = 0.36, respectively. The chaotic attractor obtained in this case is for T = 75 and shown in Fig. 20. Once again, in the high-T case, the arms of the attractors are radially pressed down to give it the look of a simple closed curve.
FIG. 18. Time-T map for a single-switch case rank 1 attractor from experimental simulations: PWL case 共1 = 0.787, 2 = 3 = 0, T = 155兲.
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Experimental verification of rank 1 chaos
FIG. 19. Time-T map for a single-switch case rank 1 attractor from experimental simulations: PWL case 共1 = 0.66, 2 = 3 = 0, T = 82兲.
VI. CONCLUSION
In this paper, we have provided the first experimental proof of rank 1 chaos by using a switch-controlled Chua circuit. Both the smooth and the PWL cases have been investigated under different kicking schemes. In the smooth case, first, a weakly stable Hopf limit cycle coming out of a fixed point is generated. Then, under the guidance of the theory, by following a recipe-like procedure, this limit cycle is subjected to periodic kicks to obtain rank 1 attractors. The theory is also applicable in the case of arbitrary limit cycles. This makes it possible to obtain chaotic attractors in systems where the employed nonlinearity is not smooth. The generic setting of the theory is satisfied by adding externally controlled switches to the original circuit. Even though the theory is equally applicable to smooth nonlinearity and PWL cases, each case has its own advan-
FIG. 20. Time-T map for a two-switch case rank 1 attractor from experimental simulations: PWL case 共1 = 0.66, 2 = 0.36, 3 = 0, T = 75兲.
tages and disadvantages. The advantage in the case of smooth nonlinearity is that, for this case, the theory provides a recipe-like procedure where the precise analytical computations for the emergence of chaos can easily be carried out 共see Sec. III B兲. Therefore, in the smooth case, it is easier to follow the guidance of the theory, and hence to explicitly know in what manner to control the system parameters to generate chaos. However, the disadvantage in this case is the fact that the physical realization of the circuit can be relatively more difficult. As for the PWL case, one advantage is the ease of physical circuit realization. Nonetheless, the main difficulty in this case is that, currently, there is no recipe-like procedure available from which explicit analytical conditions for chaos can be derived. Despite this relative difficulty, the plethora of PWL systems in practice makes them a natural target for the application of the theory of rank 1 chaos. It is also worth noting that the results of the experimental simulations are in perfect agreement with the predictions of the theory. It also seems that the geometric complexity of the resulting attractors increases with the number of switches employed. Another point worth mentioning here is that the width of the applied pulses p0 is not crucial 共other than practical concerns for the physical switches used兲 as long as it is followed by a much longer relaxation interval, i.e., T0 Ⰷ p0. For more background information on the theory of rank 1 chaos and its potential applications to circuits and systems, we refer the reader to a recent tutorial paper.18 ACKNOWLEDGMENTS
This project was partially supported by The Scientific and Technological Research Council of Turkey 共TUBITAK兲 under Grant No. 106E093. 1
Q. Wang and A. Oksasoglu, Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, 83 共2005兲. 2 A. Oksasoglu and Q. Wang, Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 2659 共2006兲. 3 A. Oksasoglu, D. Ma, and Q. Wang, Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 3207 共2006兲. 4 A. Oksasoglu and Q. Wang, “Chaos in switch-controlled Chua’s circuit,” J. Franklin Inst. 共unpublished兲. 5 Q. Wang and L.-S. Young, Commun. Math. Phys. 218, 1 共2001兲. 6 D. Wang and L.-S. Young, J. Anal. Math. 167, 349 共2008兲. 7 D. Wang and L.-S. Young, Commun. Math. Phys. 225, 275 共2002兲. 8 M. Jakobson, Commun. Math. Phys. 81, 39 共1981兲. 9 M. Benedicks and L. Carleson, J. Anal. Math. 133, 73 共1991兲. 10 L. O. Chua, J. Circuits Syst. Comput. 4, 117 共1994兲. 11 G. Q. Zhong, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 41, 934 共1994兲. 12 D. Wang and L.-S. Young, Commun. Math. Phys. 240, 509 共2002兲. 13 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, 5th ed. 共Springer-Verlag, New York, 1997兲. 14 L.-S. Young, J. Stat. Phys. 108, 733 共2002兲. 15 H. G. Schuster and W. Just, Deterministic Chaos: An Introduction, 4th ed. 共Wiley-VCH, Weinheim, 2005兲, Chap. 2.2. 16 A. Venkatesan, S. Parthasarathy, and M. Lakshmanan, Chaos, Solitons Fractals 18, 891 共2003兲. 17 S. Parthasarathy and K. Manikandakumar, Chaos 17, 043120 共2007兲. 18 Q. Wang and A. Oksasoglu, Int. J. Bifurcation Chaos Appl. Sci. Eng. 18, 1261 共2008兲.
CHAOS 19, 013117 共2009兲
Effect of chemical synapse on vibrational resonance in coupled neurons Bin Deng,a兲 Jiang Wang,b兲 and Xile Weic兲 School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People’s Republic of China
共Received 29 October 2008; accepted 8 January 2009; published online 12 February 2009兲 The response of three coupled FitzHugh–Nagumo neurons, under high-frequency driving, to a subthreshold low-frequency signal is investigated. We show that an optimal amplitude of the highfrequency driving enhances the response of coupled excited neurons to a subthreshold lowfrequency input, and the chemical synaptic coupling is more efficient than the well-known electrical coupling 共gap junction兲, especially when the coupled neurons are near the canard regime, for local signal input, i.e., only one of the three neurons is subject to a low-frequency signal. The influence of additive noise and the interplay between vibrational and stochastic resonance are also analyzed. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3076396兴 In bistable systems, it has been shown that the role of noise in improving the quality of signal detection can be played by other types of driving, such as, a chaotic signal or a high-frequency periodic force. In the latter case, known as vibrational resonance (VR), the system is under the action of a two frequency signal. Such bichromatic signals are pervasive in different fields, including brain dynamics, where for instance bursting neurons may exhibit two widely different time scales, and telecommunications, where information carriers are usually highfrequency waves modulated by a low-frequency signal that encodes the data. Two-frequency signals are also of interest in several other fields, such as, laser physics, acoustics, and neuroscience. Recent studies have shown that, in extended arrays of neurons, coupling noticeably enhances the stochastic coherence effect. This arrayenhanced stochastic coherence has been reported so far, to our knowledge, only in the case of linear diffusive electrical coupling, mediated by gap junctions between the neurons. But another very important means of signal transmission between neurons is via chemical synapses, which provide a nonlinear pulsed coupling only when the presynaptic neuron is excited. It is thus of interest to examine the effect of this kind of nonlinear coupling on the VR effect described above. Our numerical results, detailed below, show that chemical synapses are more efficient at enhancing coherence than gap junctions. I. INTRODUCTION
The external influence can considerably affect the signal detection by nonlinear system. Stochastic resonance 共SR兲 where the response of a nonlinear system to a weak deterministic signal is enhanced by external random fluctuation1 is the most relevant example of this fact. Recently, Ullner et al. gave a detailed description of several new noiseinduced phenomena in the FitzHugh–Nagumo 共FHN兲 neuron
in Ref. 2. They have investigated the Canard-enhanced SR,3 the effect of noise-induced signal processing in systems with complex attractors,4 and a new noise-induce phase transition from a self-sustained oscillatory regime to an excitable behavior.5 They also showed that optimal amplitude of highfrequency driving enhances the response of an excitable system to a low-frequency subthreshold signal.6 In the latter case, known as vibrational resonance 共VR兲, the system is under the action of the two frequency signal. Such bichromatic signals are pervasive in many different fields, including brain dynamics,7 where, for instance, bursting neurons may exhibit two widely different time scales. However, most of the relevant studies only considered the single neuron3,5,6 or neurons with linear electrical coupling 共EC兲 共Ref. 4兲 and omitted another important case, chemical coupling 共CC兲. Recently, Balenzuela et al. showed that a substantial increase in the coherence resonance 共CR兲 of chemical coupled Morris–Lecar models can be observed, in comparison with the electrical coupled ones.8 As investigated in Ref. 9 when coupled neurons are under a common Gaussian white noise, chemical synaptic coupling is more efficient than electrical coupling for local signal input, i.e., only one of the coupled neurons is subject the period signal. Considering these, the VR of coupled neurons is investigated in this paper. We make comparisons of the response to external low-frequency signal between chemical coupled and electrical coupled neurons, which are located near the canard regime and are subject to a common high-frequency driving. The effect of noise on the VR in coupled neurons is also studied. The contents of this paper are arranged as follows: In Sec. II, VR in single FHN neurons with different bifurcation parameters is introduced briefly. The comparisons of VR between the two kinds of coupling are made and the effect of noise on VR in coupled neurons is given in Sec. III. Finally, conclusions and discussions are made in Sec. IV. II. VIBRATIONAL RESONANCE IN THE SINGLE FHN MODEL
a兲
Electronic mail:
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1054-1500/2009/19共1兲/013117/6/$25.00
In the presence of two harmonic signals, the FHN model10 is defined by the following equations: 19, 013117-1
© 2009 American Institute of Physics
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dx x3 = x − − y, 3 dt
共1兲
dy = x + a + A cos共t兲 + B cos共⍀t兲, dt
共2兲
where x共t兲 represents the membrane potential of the neuron and y共t兲 is the related to the conductivity of the potassium channels existing in the neuron membrane. The value of the time scale ratio = 0.01 is chosen so that the activator x共t兲 evolves much faster than the inhibitor y共t兲. The terms A cos共t兲 and B共⍀t兲 stand for the low- and high-frequency components of the external signal, respectively. In what follows we will choose A = 0.01, so that the system is below the excitation threshold, and ⍀ Ⰷ , in particular ⍀ = 5 and = 0.1. In Eq. 共2兲 we have considered no phase shift between the two driving signals, but it can be checked that the existence of an arbitrary phase shift does not alter the results that follow. The parameter a determines the behavior of the system. For a ⬎ 1.0 the FHN model is excitable, and for a ⬍ 1.0 it shows an oscillatory behavior. At the bifurcation a = 1.0 the stability of the only fix point will be changed.11 Between these two cases an intermediate behavior can appear. For values of the parameter a slightly beyond the bifurcation point, small oscillations near the unstable fix point exist instead of large spikes and these are the so-called canard oscillations.12 An important fact of the treatment of canard oscillation is that a very small change in the parameter a leads to a large different in the trajectories. This change in the parameter a can be caused by some instantaneous influence of noise as investigated in Refs. 3 and 11 or external high-frequency driving as investigated in Ref. 6. To evaluate the amplitude of the input frequency in the output signal, we calculate the Fourier coefficient Q for the input frequency . We use the Q parameter instead of the power spectrum because we are interested in the transport of the information encoded in the frequency . For this task the Q parameter is a much more compact tool than the power spectrum,1,13 Qsin =
Qcos =
2n
冕
2n
冕
2n/
2x共t兲sin共t兲dt,
0 2n/
2x共t兲cos共t兲dt,
0
Q = 冑Q2sin + Q2cos , where n is the number of periods 2 / covered by the integration time. The maximum of Q shows the best phase synchronization between input signal and output firing. It is to be noted that in the case of phase synchronization one could expect a response of the Q measure but not vice versa and the phase synchronization between the input signal and the output of the neuron can be seen from the following figures of time series. Also, as information in the neuron system is carried through large spikes instead of subthreshold oscillations, we are more interested in the frequency of spikes. So following Ref. 4, we set the threshold Vs = 0 in the calcula-
FIG. 1. 共Color online兲 Response Q of the FHN neuron with different values of parameter a at the low frequency vs the amplitude B of the highfrequency input signal. 共a兲 a = 1.01; 共b兲 a = 1.03; 共c兲 a = 1.05.
tion of Q. If V ⬍ Vs, V is replaced by the value of the fixed point V f 共here V f = −1兲; otherwise, V remains the same. We fix the amplitude of the low-frequency signal component and increase the high-frequency amplitude. Figure 1 shows the VR in FHN neurons with different values of parameter a. In Figs. 1共a兲–1共c兲, the value of parameter a is chosen as a = 1.01, a = 1.03, and a = 1.05, respectively. The dependence of the neuron’s response on the amplitude of the high-frequency driving displays a resonant form with clearly defined maximums at the optimal values of B, similar to what happens in SR. The staircase form of this dependence is caused by the abrupt discrete appearance of new spikes in the spike train as the forcing amplitude changes. The closer a is to the bifurcation value 共a = 1.0兲, the smaller the optimal value of B. The first peak in Fig. 1共a兲 is caused by the canard oscillations near the fix point with small amplitude compared with the big spike as shown in Fig. 2共a兲. Figure 2共b兲 shows the case of the second peak in Fig. 1共a兲.
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Vibrational resonance
Isyn i =
兺
gsyns j共Vi − Vsyn兲,
共6兲
j苸neigh共i兲
where gsyn is the synaptic coupling strength and Vsyn is the synaptic reversal potential that determines the type of synapse. For the excitatory synapse considered in this paper, Vsyn = 0. The dynamics of the synapse variable s j is governed by V j. s j is defined by s˙ j = ␣共V j兲共1 − s j兲/ − s j/syn ,
␣0 , ␣共V j兲 = 1 + exp共− V j/Vshp兲
FIG. 2. 共Color online兲 Time series of x共t兲, high-frequency input 共green line, the amplitude is 10 times higher than that in the model兲, and low-frequency input 共red line, the amplitude is 50 times higher than that in the model兲. The parameter is a = 1.01; 共a兲 B = 0.012; 共b兲 B = 0.026.
III. VIBRATIONAL RESONANCE IN COUPLED NEURONS
Now we consider three bidirectional coupled FHN neurons subject to a common high-frequency driving and noise described by
x3 dxi = xi − i − y i − Isyn i , 3 dt
dy i = xi + a + Ai cos共t兲 + B cos共⍀t兲 + C共t兲, dt
共3兲
共4兲
where i = 1 , 2 , 3 index the neurons, = 0.01 as used in Sec. II. Ai cos共t兲 and Bi cos共⍀t兲 are low- and high-frequency driving, respectively. A set of different phase differences have been tested 共the figures are not given in this paper兲 and the main effects remain independent of the phase between each of the high-frequency signals, so we let the neurons be subjected to a common high-frequency driving in the model. is Gaussian white noise with zero mean and intensity C for is the synaptic current through neuron i. each neuron. Isyn i For the electrical coupling, Isyn i =
兺
gsyn共Vi − V j兲,
j苸neigh共i兲
where gsyn is the conductance of the synaptic channel. For the chemical coupling,14
共5兲
共7兲
where synaptic decay rate syn is equal to 1 / ␦. The synaptic recovery function ␣共V j兲 can be taken as the Heaviside function. When the neuron is in the silent state 共V ⬍ 0兲, s is slowly decreasing and the first equation of Eq. 共7兲 can be taken as s˙ j = −s j / syn; while in the other case, s jumps fast to 1 and acts on the postsynaptic neurons. Note that in this coupling case the neuron is coupled only when its presynaptic neuron is active, which is quite different from the continuous connection between electrical coupled neurons. The parameters used in Eqs. 共3兲–共7兲 are Vsyn = 0, ␣0 = 2, Vshp = 0.05, ␦ = 1.2, and gsyn = 0.1 which is large enough to synchronize the neurons for both the cases of electrical and chemical coupling when the coupled neurons are subjected to a common threshold stimuli. The rest parameters are given in each case. First we consider the noise-free case C = 0. For the purpose of investigating the information propagation in coupled neurons under high-frequency driving, we study VR in three coupled neurons with local stimulus, that is, only one element is subject to external low-frequency signal and we only examine the response of the second neurons to external input instead of the mean field which means that the V2 is used to calculate Q 共to use V3 can get the same results because of the symmetry兲. The parameters of input periodic signal are taken as A1 = 0.01, A2 = 0, A3 = 0, = 0.1, and ⍀ = 5 so that there is no spiking for all the neurons in the absence of highfrequency driving. Note that the value of is much smaller than the two internal frequencies of neurons. As discussed in Ref. 8 chemical synapses only act while the presynaptic neurons is spiking, whereas electrical coupling connects neurons at all times. The canard oscillation will not process through chemical coupling, so the first peak in Fig. 1共a兲 disappears in Fig. 3共a兲. Chemical coupling enables small oscillatory neurons to be free from each other and gives more opportunities for them to fire. Once one spikes, it will stir the others to spike synchronously. While for electrical coupling, strong synchronization between subthreshold oscillatory neurons results in the decrease of oscillatory amplitude and thus the increase of the threshold for firing 关Fig. 3共c兲兴. Though the common high-frequency input may act as a synchronizing force on all oscillators and has the most impact during the subthreshold dynamics in the case of chemical coupling, it does not decrease the amplitude of canard oscillatory so that the neurons have more opportunities to fire. When the parameter a is slightly beyond the bifurcation value, the subthreshold oscillatory 共Canard兲 will
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FIG. 3. 共Color online兲 Response Q of the second FHN neuron coupled via different types of coupling at the low frequency vs the amplitude B of the high-frequency input signal. The stimulus is A1 = 0.01, A2 = 0, A3 = 0, and the parameter of a is 共a兲 a = 1.01; 共b兲 a = 1.03; 共c兲 a = 1.05.
be suppressed through the electrical coupling, so it is hard for the neuron coupled via electrical coupling to respond to the input signal, as shown in Figs. 3共a兲 and 3共b兲. As we can see in Fig. 3, for local stimulus A1 = 0.01, A2 = 0, A3 = 0, chemical coupling is more efficient than electrical coupling for signal processing. The other two cases with local stimuli A1 = 0.01, A2 = 0, A3 = 0.01, and global stimuli A1 = 0.01, A2 = 0.01, A3 = 0.01 are displayed in Fig. 4 and Fig. 5, respectively. As we can see, for local stimulus, chemical coupling is more efficient than electrical coupling for signal processing 共Fig. 4兲, but for global stimulus, chemical coupling is not as efficient as the electrical one 共Fig. 5兲. In the global stimulus case, the continuous connection in electrical coupled neurons lead to high synchronization and can make better control of the firing rate than the selective connection in chemical coupled neurons.
FIG. 4. 共Color online兲 Response Q of the second FHN neuron coupled via different types of coupling at the low frequency vs the amplitude B of the high-frequency input signal. The stimulus is A1 = 0.01, A2 = 0, A3 = 0.01, and the parameter of a is 共a兲 a = 1.01; 共b兲 a = 1.03; 共c兲 a = 1.05.
However, the global input is not common in real neural systems. In fact, local input is a more ubiquitous case rather than a more restricted case. In neural systems with a large amount of neurons, it is unnecessary and impossible to add external signals to all the involved individuals. Only weak and local input is reasonable and guarantees the low energy consumption in large neural networks. This may be relevant to the fact that chemical coupling is more universal in mammals than electrical coupling. So far we have not considered the influence of noise in the VR of the coupled neurons. In order to study the interplay of VR and SR in coupled neurons, we now change the
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FIG. 6. 共Color online兲 Response of the second neuron coupled via 共a兲 chemical coupling and 共b兲 electrical coupling at the low-frequency signal in the presence of additive noise. The parameter of a is 1.05.
and 6共b兲 demonstrate the chemical coupling and electrical coupling cases, respectively. IV. CONCLUSION
FIG. 5. 共Color online兲 Response Q of the second FHN neuron coupled via different types of coupling at the low frequency vs the amplitude B of the high-frequency input signal. The stimulus is A1 = 0.01, A2 = 0.01, A3 = 0.01, and the parameter of a is 共a兲 a = 1.01; 共b兲 a = 1.03; 共c兲 a = 1.05.
intensity C of the additive noise in the system and the stimulus is A1 = 0.01, A2 = 0, A3 = 0, and the numerical integrations of the system with noise are done by the explicit Euler– Maruyama algorithm,15 with a time step 0.005. Figure 6 shows that by adding noise to coupled neurons the response dependence is shifted to the left and decreased. Hence, with increasing noise the maximum of response is achieved for a smaller value of the amplitude B of high-frequency driving. This fact could be relevant for efficient information processing, because natural fluctuations or noise are able to replace a fraction of the high-frequency driving and help to reduce the necessary energy. But if the noise intensity is too large, VR disappears and the signals fail to process. Figures 6共a兲
In conclusion, we have studied the dynamical response of coupled neurons to biochromatic signals with two very different frequencies and made comparisons of the response to the external signal between chemical coupled and electrical coupled FHN neurons. In the local input case, chemical coupling is more effective for subthreshold low-frequency signal propagation due to its selective coupling. This is very important in a practical system. As in neural systems with a large amount of neurons, only weak and local input is reasonable and guarantees the low energy consumption in signal processing. While in the global input case, the contiguous synchronization of the electrical coupled neurons can control the frequent firing rate and thus behave better 共VR兲 than the chemical coupled one, but the global input may be not common in real neural systems. We have also studied the effect of noise of VR in coupled neurons and demonstrated that noise can enhance VR in coupled neurons; but if the noise intensity is too large, VR may disappear, i.e., the signal processing of the low-frequency signal which encodes the information has been destroyed. We expect our findings will be relevant for different fields in neuroscience including deep brain stimulation, where also high frequency signal are used. Given the ubiquity of two-frequency signals in neuron as-
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semblies and an optimal strength of high-frequency driving may enhance the transmission of information. ACKNOWLEDGMENTS
This work is supported by the Key National Natural Science Foundation of China 共Grant No. 50537030兲, the National Natural Science Foundation of China 共Grant No. 50707020兲, and the Postdoctoral Science Foundation of China 共Grant No. 20070410756兲. 1
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L. Gammaitoni, P. Hänggi, P. Jung, and F. Marachesoni, Rev. Mod. Phys. 70, 223 共1998兲. 2 E. Ullner, dissertation, Institute of Physics, Potsdam University, 2004. 3 E. I. Volkov, E. Ullner, A. A. Zaikin, and J. Kurths, Phys. Rev. E 68, 026214 共2003兲.
4
E. I. Volkov, E. Ullner, A. A. Zaikin, and J. Kurths, Phys. Rev. E 68, 061112 共2003兲. 5 E. Ullner, A. Zaikin, J. García-Ojalvo, and J. Kurths, Phys. Rev. Lett. 91, 180601 共2003兲. 6 E. Ullner, A. Zaikin, R. Báscones, J. García-Ojalvo, and J. Kurths, Phys. Lett. A 312, 348 共2003兲. 7 G. M. Shepherd, Foundations of the Neuron Doctrine 共Oxford University Press, New York 1990兲. 8 P. Balenzuela and J. García-Ojalvo, Phys. Rev. E 72, 021901 共2005兲. 9 X. Li, J. Wang, and W. Hu, Phys. Rev. E 76, 041902 共2007兲. 10 R. FitzHugh, Biophys. J. 1, 445 共1961兲. 11 A. Pikovksy and J. Kurths, Phys. Rev. Lett. 78, 775 共1997兲. 12 P. Glendinming, Stability, Instability and Chaos 共Cambridge University Press, Cambridge, 1994兲. 13 A. A. Zaikin, J. García-Ojalvo, L. Schimansky-Geier, and J. Kurths, Phys. Rev. Lett. 88, 010601 共2001兲. 14 J. Drover, J. Rubin, J. Su, and B. Ermentrout, SIAM J. Appl. Math. 65, 69 共2004兲. 15 D. J. Higham, SIAM Rev. 43, 525 共2001兲.
CHAOS 19, 013118 共2009兲
Generalized synchronization of chaotic systems: An auxiliary system approach via matrix measure Wangli Hea兲 and Jinde Caob兲 Department of Mathematics, Southeast University, Nanjing 210096, China
共Received 5 November 2008; accepted 8 January 2009; published online 17 February 2009兲 In this paper, generalized synchronization of two chaotic systems is investigated. The auxiliary system approach, which is suggested by H. Abarbanel, N. Rulkov, and M. Sushchik 关Phys. Rev. E 53, 4528 共1996兲兴, is used to detect and study generalized synchronization. Based on the Lyapunov method and matrix measure, some less restrictive criteria are obtained to guarantee the asymptotical stability of the error system between the response system and the auxiliary system, which indicates the drive-response systems are synchronized in a general sense. It is shown that the feedback gain can be reduced by means of the matrix measure approach, compared to the norm method. All theoretical results are illustrated by analytical and numerical examples. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3076397兴 Since 1990, chaos synchronization has been a hot topic, which was intensively investigated in the past decade. Many tapes of synchronization have been reported due to its potential application in secure communication, chemical reactions, biological networks, etc. Among them, generalized synchronization is of significant importance owing to its robustness with respect to parameter mismatch in practice. This paper proposes a general theory for generalized synchronization via the auxiliary system approach, which is easy to verify whether the drive and response systems are synchronized in a general sense when identical synchronization fails. Furthermore, the use of matrix measure can reduce the feedback gain. This is good for practical applications. Finally, numerical simulations are provided to verify the theoretical results. I. INTRODUCTION
Chaos synchronization has attracted considerable attention in many fields, such as physics, secure communication, automatic control, chemical, and biological systems, etc.1–4 Usually, two coupled systems are called synchronized if they evolve according to exactly the same dynamics. The primary configuration of chaos synchronization, proposed by Pecora and Carrol,5 consists of two systems: driving system and response system. The idea is to use the signals of the driving system to control the response system so that they oscillate in a synchronized state. Until now, many types of synchronization have been presented, such as, complete 共identical兲 synchronization, lag synchronization, anticipated synchronization, phase synchronization, generalized synchronization, etc.5–20 Generalized synchronization 共GS兲 means the existence of some smooth vector function between the states of drive-response system, and it can exhibit much richer dynamics than identical synchronization. At the same time, due to its robustness, GS may be wilder or more practical than a兲
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those of identical synchronization. In fact, GS is a robust phenomenon,14,15 which has been verified by several authors. It means parameter mismatch does not always destroy the GS, they will remain as GS even after a change in their parameters. This extends its application ranges because in practice we can never construct two absolutely identical systems. The functional relation between driving and response systems in GS can be very complicated. Detecting the presence of the transformation between driving and response systems seems important. Some numerical methods were proposed to detect the existence of the transformation between the states of the coupled systems. Later, analytical approaches were developed. In Ref. 15, Kocarev and Parlitz proved that GS occurred if the response system was asymptotically stable. In their paper, they also mentioned that Rulkov 共private communication兲 suggested a simple way for detection of GS by plotting the variables of the response system versus the same variables of a second identical response system starting from different initial conditions. In fact, it was the idea of the auxiliary system approach. Later, Abarbanel, Rulkov, and Sushchik expressed the auxiliary system approach in their formal paper.16 The idea is to construct an auxiliary system, which is identical to the response system. If the synchronization manifold of the response system and the auxiliary system are stable, then we can conclude that GS occurs. However, in this case, we know nothing about the transformation. With a given linear transformation, Yang and Chua17 studied GS by means of the error system between the response system and the transformation of the driving system. The requirement of the linear part was a significant restriction. Recently, Meng and Wang18 investigated GS via nonlinear control, where a nonlinear controller was designed to achieve GS. The introduced controller relaxed the conditions on system parameters of the response system itself. However, the total use of the matrix norm made the obtained feedback gain restrictive. Also, the response system is with-
19, 013118-1
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out delay, which limits its application range. Inspired by Refs. 16 and 18, a more general model is proposed to investigate GS between the driving system and the response system via the auxiliary system approach. Moreover, matrix measure is introduced to deal with the inequalities. It can have positive as well as negative values, whereas a norm can assume only non-negative values. Because of these special properties, the results obtained via matrix measure usually are less restrictive than the ones using the norm.19 It is shown that the feedback gain decreases by means of a matrix measure. In the simulations, we show that the rich dynamics and robustness of the GS, also the advantage of the matrix measure. The remainder of this paper is organized as follows: in Sec. II, the problem formulations are presented for GS. In Sec. III, based on Lyapunov theory and matrix measure, three theorems are obtained for GS. Also, some corollaries and remarks are given to show the advantage of this paper. In Sec. IV, examples are given to show the validity and effectiveness of the proposed approach. In Sec. V, we give our conclusions. II. MODEL DESCRIPTION AND PRELIMINARIES
The equation of the drive system is 共1兲
x˙ = h共x兲,
where x共t兲 = 共x1共t兲 , x2共t兲 , . . . , xm共t兲兲T 苸 Rm denotes the state variable of drive system, and h共·兲 is a continuous vector function. The response system is given by y˙ 共t兲 = Ay共t兲 + Bf共y共t兲兲 + Cf共y共t − 兲兲 + K共y − 共x兲兲,
共2兲
where y共t兲 = 共y 1共t兲 , y 2共t兲 , . . . , y n共t兲兲T 苸 Rn is the state variable of the response system. A, B, and C are system matrices with proper dimension. f共·兲 is a continuous vector function. is time delay and satisfies ⬎ 0. K is the coupling matrix or the feedback gain matrix. 共·兲 is the transformation from Rm to Rn, which is differentiable. For function f共·兲, we have the following assumption: Assumption 1: There exists constant l ⬎ 0, for any z1, z2 苸 Rn such that 储f共z1兲 − f共z2兲储 艋 l储z1 − z2储.
共3兲
Remark 1: The response system in this paper unifies several well-known chaotic systems, such as, the Ikeda oscillator, Chua’s circuits, and delayed neural networks, etc. It extends the model in Ref. 18. Therefore, the results are more general in this paper. Now, we introduce the concept of matrix measure. Definition 1: 共Ref. 21兲 The matrix measure of a real square matrix A = 共aij兲n⫻n is as follows:
p共A兲 = lim
→0+
储I + A储 p − 1 ,
where 储 . 储 p is an induced matrix norm on Rn⫻n, I is the identity matrix, and p = 1 , 2 , ⬁. When the matrix norm
n
储A储1 = max 兺 兩aij兩, j
储A储2 = 冑max共ATA兲,
i=1 n
储A储⬁ = max 兺 兩aij兩, i
j=1
再 再
冎 冎
we can obtain the matrix measure
1共A兲 = max a jj + j
n
兺 兩aij兩 i=1,i⫽j n
⬁共A兲 = max aii + i
兺
,
1 2共A兲 = max共AT + A兲, 2
兩aij兩 .
j=1,j⫽i
Next, we introduce two lemmas, which are needed in the proof of our main results. Lemma 1: 共Ref. 21兲 The matrix measure p共·兲 defined in Definition 2 has the following properties: 共i兲
− 储A储 p 艋 p共A兲 艋 储A储 p,
共ii兲
p共␣A兲 = ␣ p共A兲,
共iii兲
p共A + B兲 艋 p共A兲 + p共B兲,
∀ A 苸 Rn⫻n;
∀ ␣ ⬎ 0,
∀ A 苸 Rn⫻n; ∀ A,B 苸 Rn⫻n .
Lemma 2: 共Ref. 22兲 Let k1 and k2 be constants with k1 ⬎ k2 ⬎ 0, and y共t兲 is a non-negative continuous function defined on 关t0 − , + ⬁兴 which satisfies the following inequality for t 艌 t 0: D+y共t兲 艋 − k1y共t兲 + k2¯y 共t兲, where ¯y 共t兲 supt−艋s艋ty共s兲. Then y共t兲 艋 ¯y 共t0兲e−r共t−t0兲 , where r is a bound on the exponential convergence rate and is the unique positive solution of r = k 1 − k 2e r . Here the upper-right Dini derivative D+y共t兲 is defined as + ¯ D+y共t兲 = lim h→0+ y共t + h兲 − y共t兲 / h, where h → 0 means that h approaches 0 from the right-hand side. III. MAIN RESULTS
In this section, we consider the generalized synchronization of systems 共1兲 and 共2兲. Our approach is based on an auxiliary system which is introduced by Rulkov et al.16 The auxiliary system is identical to the response system with different initial conditions and driven by the same signal of the drive system. In their paper, they addressed that GS occurred if the synchronization manifold of the response system and the auxiliary system was stable. First, we construct the auxiliary system z˙共t兲 = Az共t兲 + Bf共z共t兲兲 + Cf共z共t − 兲兲 + K共z − 共x兲兲,
共4兲
which is a replica of the response system 共2兲. In order to show that Eqs. 共1兲 and 共2兲 are GS, define e = y − z. Based on Eqs. 共2兲 and 共4兲, the error system is given by
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e˙共t兲 = Ae + B共f共e共t兲 + z共t兲兲 − f共z共t兲兲兲 + C共f共e共t − 兲 + z共t − 兲兲 − f共z共t − 兲兲兲 + Ke.
共5兲
Theorem 1: Under Assumption 1, if the coupling matrix K satisfies
p共A + K兲 ⬍ − l储B储p − l储C储 p,
Let
g共e共t − 兲兲 = f共e共t − 兲 + z共t − 兲兲 − f共z共t − 兲兲.
共7兲
V共e共t兲兲 = 储e共t兲储 p .
Then Eq. 共5兲 can be written as e˙共t兲 = 共A + K兲e + Bg共e共t兲兲 + Cg共e共t − 兲兲.
共6兲
Then the upper right-hand derivative of V共e共t兲兲 with respect to time along the solution 共6兲 is as follows:
储e共t + h兲储 p − 储e共t兲储 p h
= limh→0+
储e共t兲 + he˙共t兲 + o共h兲储 p − 储e共t兲储 p h
= limh→0+
储e共t兲 + h共共A + K兲e共t兲 + Bg共e共t兲兲 + Cg共e共t − 共t兲兲兲兲 + o共h兲储 p − 储e共t兲储 p h
艋 limh→0+
储e共t兲 + h共A + K兲e共共t兲兲储 p − 储e共t兲储 p + 储Bg共e共t兲兲储 p + 储Cg共e共t − 兲兲储 p h
艋 limh→0+
储I + h共A + K兲储 p − 1 储e共t兲储 p + 储B储 p储g共e共t兲兲储 p + 储C储 p储g共e共t − 兲兲储 p . h
共8兲
p共K兲 ⬍ − p共A兲 − l储B储 p − l储C储 p,
Based on Assumption 1, one has 储g共e共t兲兲储 p 艋 l储e共t兲储 p,
p = 1,2,⬁,
then systems 共1兲 and 共2兲 achieve GS. Proof: Choose the following Lyapunov function:
g共e共t兲兲 = f共e共t兲 + z共t兲兲 − f共z共t兲兲,
D+V共e共t兲兲 = limh→0+
储C储 p ⬎ 0,
储g共e共t − 共t兲兲兲储 p 艋 l储e共t − 共t兲兲储 p .
储C储 p ⬎ 0,
共11兲
共9兲
Substituting inequalities 共9兲 into the right-hand side of inequality 共8兲 yields D+V共e共t兲兲 艋 p共A + K兲储e共t兲储 p + l储B储 p储e共t兲储 p + l储C储 p储e共t − 共t兲兲储 p 艋 共 p共A + K兲 + l储B储 p兲储e共t兲储 p + l储C储 p sup 储e共s兲储 p .
p = 1,2,⬁,
then systems 共1兲 and 共2兲 will achieve GS. Proof: From Eq. 共11兲, one has p共A兲 + p共K兲 ⬍ −l储B储 p − l储C储 p. It is easy to obtain p共A + K兲 ⬍ −l储B储 p − l储C储 p based on Lemma 1. According to Theorem 1, we can say that systems 共1兲 and 共2兲 obtain GS. Theorem 2: Let Assumption 1 hold and C = 0, if the coupling matrix K satisfies
t−艋s艋t
共10兲 Let k1 = − p共A + K兲 − l储B储 p and k2 = l储C储 p. From Eq. 共7兲, we have k1 ⬎ k2 ⬎ 0. By Lemma 2, it follows that 储e共t兲储 p 艋
sup 储e共s兲储 pe−r共t−t0兲 ,
p共A + K兲 + l储B储 p ⬍ 0,
p = 1,2,⬁.
共12兲
Then systems 共1兲 and 共2兲 obtain GS. Proof: Consider the same Lyapunov function as Theorem 1. Based on Eq. 共10兲, one has
t0−艋s艋t0
where r is the unique positive solution of r = k1 − k2er = − p共A + K兲 − l储B储 p − l储C储 per . Therefore, e共t兲 converges exponentially to zero with a convergence rate of r. It shows that the synchronization manifold of the response system and the auxiliary system is stable. Therefore, systems 共1兲 and 共2兲 are synchronized in a general sense. This completes the proof. Corollary 1: Under Assumption 1, if the coupling matrix K satisfies
D+V共e共t兲兲 艋 p共A + K兲储e共t兲储 p + l储B储 p储e共t兲储 p 艋 共 p共A + K兲 + l储B储 p兲储e共t兲储 p = − ␣V共e共t兲兲,
共13兲
where ␣ = −共 p共A + K兲 + l储B储 p兲. From Eq. 共12兲, ␣ ⬎ 0, thus V共t兲 艋 V共0兲e−␣t. When t goes to infinity, V共t兲 converges to zero, that is, limt→⬁储e共t兲储 p = 0. Therefore, the error system 共6兲 is asymptotically stable. Systems 共1兲 and 共2兲 achieve GS. Corollary 2: Let Assumption 1 hold and C = 0, if the coupling matrix K satisfies
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p共K兲 ⬍ − p共A兲 − l储B储 p,
p = 1,2,⬁.
共14兲
Then systems 共1兲 and 共2兲 obtain GS. Proof: It is easy to prove according to Theorem 2 and Lemma 1. Here we omit it. Remark 2: From Theorems 1 and 2, we can see that whether GS occurs in the drive-response systems in this paper mainly depends on the structure of the response system and the feedback gain matrix K. With a given response system and K, even the drive system changes, they can also achieve GS. The different drive systems and transformations 共x兲 just change the shape of the manifold of synchronized motions. This will be illustrated in our simulations. Remark 3: If the drive system and response system are identical, according to Theorems 1 and 2, even if parameter mismatch occurs, the manifold of synchronized motions will not be destroyed if we properly choose the matrix K. The robustness of GS makes it very powerful. That is because in practice we can never construct two absolutely identical systems. Theorems 1 and 2 guarantee systems 共1兲 and 共2兲 are GS, but we have no idea about the transformation function between the drive system and response system. It should be noticed that in Theorems 1 and 2, the function 共x兲 just shows that the drive signal couples with the response system through a transformation. It cannot be concluded that they are GS with y = 共x兲. If we expect they are GS with y = 共x兲, nonlinear control must be used to guarantee systems 共1兲 and 共2兲 are in a state of generalized synchronization with y = 共x兲. The response system can be described by y˙ 共t兲 = Ay共t兲 + Bf共y共t兲兲 + Cf共y共t − 兲兲 + K共y − 共x兲兲 + u. 共15兲 Theorem 3: Under Assumption 1, if the coupling matrix K satisfies
p共A + K兲 ⬍ − l储B储p − l储C储 p,储C储 p ⬎ 0,
p = 1,2,⬁
共16兲
and the controller u is designed as
共19兲
p = 1,2,⬁,
and the controller u is designed as u = D共x兲h共x兲 − A共x兲 − Bf共共x共t兲兲兲.
共20兲
Then systems 共1兲 and 共2兲 will obtain GS. Proof: The similar proof is as in Theorem 3, omitted here. Remark 4: Matrix measure can have positive as well as negative values, whereas a norm can assume only nonnegative values. It is sign-sensitive in that 共−A兲 ⫽ 共A兲 in general, whereas 储−A储i = 储A储i. Because of these special properties, the results based on matrix measure usually are more precise and less restrictive. In our paper, the feedback gain is reduced by means of matrix measure, compared with the results in Ref. 18. It proves that the matrix measure approach is really effective. In fact, matrix measure can also be applied to study synchronization in networks. There are many papers concerning synchronization in complex networks.23–26 Only a few of them are mentioned here. For more detail, refer to the introduction of Suykens et al.27 and the references therein. In Refs. 25 and 26, some less conservative synchronization criteria are presented based on the concept of matrix measure. Furthermore, comparison with other existing literature is reported in detail in the remarks, which also shows the advantage of matrix measure. IV. NUMERICAL SIMULATIONS
In this section, we will give three examples to illustrate our theoretical results. Example 1: Consider the Rössler system28 as the drive system, dx1 = − 共x2 + x3兲, dt
dx2 = x1 + 0.2x2 , dt 共21兲
dx3 = 0.2 + x3共x1 − 5.7兲. dt The response system is a three-dimensional CNN model29 as
u = D共x兲h共x兲 − A共x兲 − Bf共共x共t兲兲兲 − Cf共共x共t − 兲兲兲, 共17兲 then systems 共1兲 and 共2兲 achieve GS. Proof: The synchronization error between the drive system and the response system is defined as e共t兲 = y共t兲 − 共x共t兲兲, then the error dynamical system can be described as e˙共t兲 = y˙ − D共x兲x˙
dy = Ay共t兲 + Bf共y共t兲兲 + K共y − 共x兲兲, dt
共22兲
where y共t兲 = 共y 1共t兲,y 2共t兲,y 3共t兲兲T , f共x共t兲兲 = 共f共y 1共t兲兲, f共y 2共t兲兲, f共y 3共t兲兲兲T ,
= 共A + K兲e共t兲 + B共f共y共t兲兲 − f共共x共t兲兲兲兲 + C共f共y共t − 兲兲 − f共共x共t − 兲兲兲兲,
p共A + K兲 + l储B储 p ⬍ 0,
共18兲
where D is the Jacobian of . Using the similar proof as in Theorem 1, we can easily get the error system 共18兲 as exponentially stable. Namely, systems 共1兲 and 共2兲 can exponentially achieve generalized synchronization with the relationship y = 共x兲. Theorem 4: Let Assumption 1 hold and C = 0, if the coupling matrix K satisfies
A=
冤
−1
0
0
0
−1
0
0
0
−1
冥 冤 ,
1.25 − 3.2 − 3.2
B = − 3.2 − 3.2
1.1 4.4
冥
− 4.4 , 1.0
f共y兲 = 21 共兩y + 1兩 − 兩y − 1兩兲. K is the coupling matrix. First, choose the map as 共x兲 = 共x1 + x2 + x3 , x2 + x3 , x3兲T.
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1 5
6 1
4 2
0 5
2
x2
x2
0 0
4 0 5
6 8
1
10 12 10
5
0
x1
5
10
15
1 5
2
1
0 x1
1
2
FIG. 1. 共Color online兲 Chaotic attractors of the Rössler system 共21兲 and CNN 共22兲.
The above two chaotic systems are totally different. Figure 1 illustrates the chaotic trajectories of systems 共21兲 and 共22兲. In order to show they can be synchronized in the generalized sense, we introduce the auxiliary system
dz = Az共t兲 + Bf共z共t兲兲 + K共z − 共x兲兲, dt
共23兲
where z共t兲 = 共z1共t兲 , z2共t兲 , z3共t兲兲T. It is identical to the response system. By properly choosing the coupling matrix K, the synchronized manifold y共t兲 = z共t兲 can be stable. It can be easily verified that 储B储2 = 7.0099 and l = 1. Based on Theorem 2, let K = diag兵−6.1, −6.1, −6.1其, then 2共A + K兲 + l储B储2 = −0.0901⬍ 0. Systems 共21兲 and 共22兲 can achieve GS. The initial values of the drive system, response system, and the auxiliary system are chosen as 共x1共0兲 , x2共0兲 , x3共0兲兲 = 共1.5, 2.0, 3.0兲 , 共y 1共0兲 , y 2共0兲 , y 3共0兲兲 = 共−0.001, 0.01, 0.2兲 and 共z1共0兲 , z2共0兲 , z3共0兲兲 = 共0.4, −0.3, 0.6兲. Also, the additive noises, randomly chosen from 关0, 0.5兴, are added in the response system and auxiliary system, independently. Figure 2 depicts the response and auxiliary systems are identical synchronization. Figure 3 shows that oscillations in the drive and response systems are not identical and thus these systems are synchronized in the generalized sense. Second, if we choose another as 共x兲 = 共x1x3 + x2 , x22 + x1 , x3x2 + 2兲T, systems 共21兲 and 共22兲 can also be GS. The additive noises, randomly chosen from 关0, 0.5兴, are added in the response system and auxiliary system. Figure 4 shows the response and auxiliary systems are identical synchronization. The changes of only influences the shape of the GS. The relationships of corresponding variables of driveresponse systems are reported in Fig. 5. The synchronized motions in Fig. 3 and Fig. 5 are different. Third, we choose Chua’s circuit30 as the drive system,
dx1 = 10共− x1 + x2 − g共x1兲兲, dt 共24兲 dx2 = x1 − x2 + x3 , dt
dx3 = − 18x2 , dt
where g共x1兲 = bx1 + 0.5共d − b兲共兩x1 + 1兩 − 兩x1 − 1兩兲, d = −4 / 3, b = −3 / 4. Figure 6 shows it has a chaotic attractor. The response system is the same as Eq. 共22兲. Let remain unchanged, that is, 共x兲 = 共x1x3 + x2 , x22 + x1 , x3x2 + 2兲T. The initial values of the drive system are chosen as 共x1共0兲 , x2共0兲 , x3共0兲兲 = 共−1 , 0.2, 0.5兲T. Figures 7 and 8 show that systems 共24兲 and 共22兲 are GS with different synchronized motions. In the simulations, we can see that the main factor to determine GS between drive and response system lies on the structure of the response system and the coupling matrix K. The drive system and the transformation heavily influence the synchronized shape. If we want to achieve GS with a given relationship, some controller must be added to obtain GS in the designed manner. Example 2: In order to show that the matrix measure approach is effective, we choose the same example as Ref. 18. The drive system is Eq. 共21兲, and the response system is also a Rössler system with the controller u,
dy = Ay共t兲 + Bf共y共t兲兲 + K共y − 共x兲兲 + u, dt where
f共y兲 =
冢
冢
0 0 y 1y 3 + 0.2
0 −1
−1
冣
,
冣
0 , A = 1 0.2 0 0 − 5.7
冢 冣 0 0 0
B= 0 0 0 . 0 0 1
共25兲
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FIG. 2. 共Color online兲 The projection of the chaotic attractor generated by systems 共22兲 and 共23兲 onto the plane 共y 1 , z1兲, 共y 2 , z2兲, 共y 3 , z3兲. It shows that systems 共22兲 and 共23兲 are synchronized.
25
20
18 16
20 15
14
15 12
10
10 5
y3
5
y2
y1
10
8 6
0 0
4
5 2
5 10
15 10
0
0
x1
10
20
10 20
10
x2
0
2
10
0
10
x3
20
30
FIG. 3. 共Color online兲 The projection of the chaotic attractor generated by systems 共21兲 and 共22兲 onto the plane 共x1 , y 1兲, 共x2 , y 2兲, 共x3 , y 3兲. We can see that the oscillations in the drive and response systems are not identical; they are synchronized in the general sense.
10
10
10
5
0
z3
15
z2
15
z1
15
5
0
5
y1
10
0
15
5
0
5
y2
10
15
0
0
5
y3
10
15
FIG. 4. 共Color online兲 The projection of the chaotic attractor generated by systems 共22兲 and 共23兲 onto the plane 共y 1 , z1兲, 共y 2 , z2兲, 共y 3 , z3兲.
120
100
35
30
100
80 25
80 60
20
60
y3
y2
y1
15 40
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40 20
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20 0 0
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20 10
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x1
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20
20 20
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x2
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10
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x3
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30
FIG. 5. 共Color online兲 The projection of the chaotic attractor generated by systems 共21兲 and 共22兲 onto the plane 共x1 , y 1兲, 共x2 , y 2兲, 共x3 , y 3兲.
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Chaos 19, 013118 共2009兲
Generalized synchronization
1
drive-response systems without and with the controller. The generalized synchronization error between drive and response systems are also depicted in Fig. 10. Example 3: In this case, we choose a time-delay system to illustrate the effectiveness of our theoretical results. Theorem 1 in Ref. 18 cannot be applied in this case. The drive system is also chosen as the Rössler system 共21兲, and the response system is the delayed neural network,31
08
06
04
x2
02
0
02
y˙ 共t兲 = Ay共t兲 + Bw共y共t兲兲 + Dw共y共t − 兲兲 + u,
04
06
08
4
3
2
1
0 x1
1
2
3
4
where w共x兲 = 共tanh x1 , tanh x2兲T. The map is defined as 共x兲 = 共x31 + 0.5, x23 − 0.3兲T,
FIG. 6. 共Color online兲 Chaotic attractor of Chua’s circuit.
The map is the = 共−2x21 + 1 , x22 , 2x3兲T, and
D共x兲 =
冢
0
− 4x1
same
0
as
Ref.
A=
共x兲
18,
0
冣
D=
The controller u is defined as Eq. 共20兲, where h共x兲 = Ax共t兲 + Bf共x共t兲兲. Here we use norm-1 to verify Theorem 4: 储B储1 = 1, l = 1. Let K = diag兵−2.3, −2.3, 0其, then 1共A + K兲 + l储B储1 = −0.1⬍ 0. According to Theorem 4, systems 共21兲 and 共25兲 are generalized synchronization with y = 共x兲. If we use the Theorem in Ref. 18, the absolute value of diagonal entries K must be larger than 6.7897, but in ours, we only need the first and second diagonal entries K are larger than 2.2. The third diagonal entry can be any non-negative value. The matrix K must be diagonal in Ref. 18, in fact, in our case, if one chooses
K=
冢
− 1.3 −1 0
1
1
冣
− 1.3 0 , 0 0
冉 冉
−1
0
0
−1
冊
2.0
− 0.1
− 5.0
4.5
冉
− 1.5 − 0.1 − 0.2
−4
冊
,
冊
,
= 1.
0
冊
Then D共x兲 =
冉
3x21 0 0
0 2x3
共27兲
.
The controller u is designed according to Eq. 共18兲. Figure 11 shows that system 共26兲 has a chaotic attractor. It can easily be verified that l = 1, 储B储2 = 6.9099, 储D储2 = 4.0094. Let K 0兲 = 共 0−10 −10 , 2共A + K兲 = −11⬍ −l储A储2 − l储D储2 = −10.9193. According to Theorem 3, systems 共21兲 and 共26兲 can achieve synchronization with y = 共x兲. Figures 12 and 13 show they are synchronized in a designed manner.
GS is an extension of identical synchronization and can give much richer dynamics than identical ones. Therefore, it has wider applications. Due to its robustness with respect to parameter mismatches in the physical world, its significance is of practical importance in communication systems. It has
2 1 8
0 8
1 6
1 6
0 7
1 4
1 4
0 6
1 2
1 2
0 5
1
z3
2 1 8
z2
1 0 9
1
0 4
0 8
0 8
0 3
0 6
0 6
0 2
0 4
0 4
0 1
0 2
0 2
0
0
0
,
V. CONCLUSIONS
1共A + K兲 + l储B储1 ⬍ 0 also holds. The eigenvalue of K is only 1.9209. In this case, matrix measure is better than norm which can reduce the feedback gain. Figure 9 shows the relationships between the corresponding variables of the
z1
B=
2x2 0 . 0 2
0
共26兲
0 5 y1
1
0
1 y2
2
0
0
1 y3
2
FIG. 7. 共Color online兲 The projection of the chaotic attractor generated by systems 共22兲 and 共23兲 onto the plane 共y 1 , z1兲, 共y 2 , z2兲, 共y 3 , z3兲.
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W. He and J. Cao 5
4
5
3 4
0 2
3 1
10
2
y3
0
y2
y1
5
1 1 2
15
0 3
20
5
0 x1
4
5
1
0 x2
1 10
1
0 x3
10
FIG. 8. 共Color online兲 The projection of the chaotic attractor generated by systems 共22兲 and 共24兲 onto the plane 共x1 , y 1兲, 共x2 , y 2兲, 共x3 , y 3兲.
50
100
0
y3
0
y2
0.04 −50
−200 0
x1
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20
−100 −20
0.02
−10
x2
0
0
10
50
120
50
0
100
40
−50
80
y2
−100 −150
−300 −10
0
x1
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20
0 −20
0
10
x3
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30
20
30
20 10
20
−250
10
30
60 40
−200
0
y3
y1
0.06
−100 −150
y1
0.08
50
−50
−250 −10
0.1
−10
x2
0
10
0
x3
FIG. 9. 共Color online兲 The projection of the chaotic attractor generated by systems 共21兲 and 共25兲 onto the plane 共x1 , y 1兲, 共x2 , y 2兲, 共x3 , y 3兲. The first three ones are without the controller, and they are generalized synchronization with unknown transformation. The second are with the controller, and they are synchronized in the sense of y = 共x兲.
4
0 5
3 5
0
0 5 0 0 5
0 5
3
1 1
2 5
1 5
e
3
2
e2
e
1
1 5 2
2 5
2 1 5
3
2 5
3 5
1 3
4
0 5
3 5
0 0
5 t
10
4
4 5
0
5 t
10
5
0
5 t
FIG. 10. 共Color online兲 The generalized synchronization errors of drive system 共21兲 and response system 共25兲.
10
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Chaos 19, 013118 共2009兲
Generalized synchronization 8
6
4
x2
2
0
2
4
6 0.8
0.6
0.4
0.2
0
0.2 x1
0.4
0.6
0.8
1
1.2
FIG. 11. 共Color online兲 Chaotic attractor of delayed Hopfield neural network. 1500
500
1000
400 300
y2
y1
500 0
200 100
500
0
1000 10
5
0
5
x1
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100
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0
5
10
0
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10
x3
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20
25
15
20
25
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1500
500
1000
400
y2
y1
500 0
300 200 100
500
0
1000 10
5
0
5
x1
10
100
15
x3
FIG. 12. 共Color online兲 The projection of the chaotic attractor generated by systems 共21兲 and 共26兲 onto the plane 共x1 , y 1兲, 共x3 , y 2兲. The first three ones are without the controller. The second are with the controller. 0 2
0 5 0 4
0
0 3 0 2
0 2
0 4
e2
e1
0 1 0 0 1 0 6
0 2 0 3
0 8
0 4 1
0
2
4
t
6
8
10
0 5
0
5
10 t
FIG. 13. 共Color online兲 The generalized synchronization errors of systems 共21兲 and 共26兲.
15
20
013118-10
proven that using a suitable choice of the transformation function, the quality of the received signal can be improved. This paper has proposed a general model to study GS via the auxiliary system method. Matrix measure is used in the analysis, and it proves that the matrix measure is effective, which reduces the feedback gain. It is good for practical applications. Moreover, we also verify the robustness of the GS by numerical simulation. The theoretical results may give some light on its application in communication systems and networks. ACKNOWLEDGMENTS
This work was jointly supported by the National Natural Science Foundation of China under Grant No. 60874088, the 333 Project of Jiangsu Province, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20070286003, and the Foundation for Excellent Doctoral Dissertation of Southeast University YBJJ0806. 1
Chaos 19, 013118 共2009兲
W. He and J. Cao
A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences 共Cambridge University Press, Cambridge, 2001兲. 2 S. Sundar and A. A. Minai, Phys. Rev. Lett. 85, 5456 共2000兲. 3 T. Yang and L. O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 44, 976 共1997兲. 4 G. Chen and X. Dong, From Chaos to Order: Methodologies Perspectives and Application 共World Scientific, Singapore, 1998兲. 5 L. M. Pecora and T. L. Carrol, Phys. Rev. Lett. 64, 821 共1990兲.
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S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, Phys. Rep. 366, 1 共2002兲. 7 D. Huang and R. Guo, Chaos 14, 152 共2004兲. 8 W. Lin and Y. He, Chaos 15, 023705 共2005兲. 9 N. J. Corron, J. N. Blakely, and S. D. Pethel, Chaos 15, 023110 共2005兲. 10 M. Rsenblum, A. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 共1996兲. 11 W. He and J. Cao, Phys. Lett. A 372, 408 共2008兲. 12 W. Yu and J. Cao, Chaos 16, 023119 共2006兲. 13 X. Huang and J. Cao, Nonlinearity 19, 2792 共2006兲. 14 N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, Phys. Rev. E 51, 980 共1995兲. 15 L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 共1996兲. 16 H. Abarbanel, N. Rulkov, and M. Sushchik, Phys. Rev. E 53, 4528 共1996兲. 17 T. Yang and L. O. Chua, Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, 215 共1999兲. 18 J. Meng and X. Wang, Chaos 18, 023108 共2008兲. 19 W. He and J. Cao, Nonlinear Dyn. 55, 55 共2009兲. 20 Q. Liu and J. Cao, Math. Comput. Modell. 43, 423 共2006兲. 21 M. Vidyasagar, Nonlinear System Analysis 共Prentice-Hall, Englewood Cliffs, 1993兲. 22 A. Halanay, Differential Equations 共Academic, New York, 1996兲. 23 I. V. Belykh, V. N. Belykh, and M. Hasler, Chaos 16, 015102 共2006兲. 24 R. Amritkar and C. K. Hu, Chaos 16, 015117 共2006兲. 25 M. Chen, IEEE Trans. Circuits Syst., II: Express Briefs 53, 1185 共2006兲. 26 M. Chen, IEEE Trans. Circuits Syst., I: Regul. Pap. 55, 1335 共2008兲. 27 J. A. K. Suykens and G. V. Osipov, Chaos 18, 037101 共2008兲. 28 O. E. Rössler, Phys. Lett. 57A, 397 共1976兲. 29 G. Chen, J. Zhou, and Z. Liu, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 2229 共2004兲. 30 L. O. Chua, Arch. Elektrotech. 共Berlin兲 46, 250 共1992兲. 31 H. Lu, Phys. Lett. A 298, 109 共2002兲.
CHAOS 19, 013119 共2009兲
Power grid vulnerability: A complex network approach S. Arianos,1,2,a兲 E. Bompard,1,b兲 A. Carbone,2,c兲 and F. Xue1,d兲 1
Dipartimento di Ingegneria Elettrica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 2 Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
共Received 2 December 2008; accepted 12 January 2009; published online 20 February 2009兲 Power grids exhibit patterns of reaction to outages similar to complex networks. Blackout sequences follow power laws, as complex systems operating near a critical point. Here, the tolerance of electric power grids to both accidental and malicious outages is analyzed in the framework of complex network theory. In particular, the quantity known as efficiency is modified by introducing a new concept of distance between nodes. As a result, a new parameter called net-ability is proposed to evaluate the performance of power grids. A comparison between efficiency and net-ability is provided by estimating the vulnerability of sample networks, in terms of both the metrics. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3077229兴 Technological infrastructures are of vital importance for contemporary societies. As a consequence of the world wide growing interconnections, the security of networks such as world-wide-web, transport, power systems, is becoming a priority in the agenda of policy-makers, industrial and academic researchers. In recent years several blackouts occurring in USA and Europe have drawn a lot of attention to security problems in electric power transmission systems. In these scenarios, it is convenient to go beyond the traditional deterministic bottom-up description in favor of a statistical top-down approach. Also the specific area of power systems has attracted the physicists community interested in the applications of complex network theory. In this paper, we investigate the topological structure and resilience of power grids by adopting a complex network description. We notice that the geodesic distance, used in complex network metrics, can be generalized to account for the flow capacity between nodes. Based on this new concept of distance, a metric called net-ability is introduced to estimate the performance and resilience of power networks upon line removal. I. INTRODUCTION
Modern states and societies can only function if the necessary infrastructures are continuously available and fully operative. Critical infrastructures are organizations or facilities of key importance to public interest whose failure or impairment could result in detrimental supply shortages, substantial disturbance to public order or economic impact. The theory of complex networks is increasingly being exploited to tackle those sorts of issues. For a comprehensive review on complex networks we refer to Ref. 1. Examples of applications include facilities for electricity generation, transmission and distribution, oil and gas production, telecommunia兲
Electronic mail:
[email protected]. Electronic mail:
[email protected]. Electronic mail:
[email protected]. d兲 Electronic mail:
[email protected]. b兲 c兲
1054-1500/2009/19共1兲/013119/6/$25.00
cation, water supply, agriculture 共food production and distribution兲, public health 共hospital, ambulances兲, transportation systems 共fuel supply, railways, airports, harbors兲, financial and security services.2–20 Due to their importance, a crucial issue is learning how to improve the tolerance of critical infrastructure to failures and attacks. A line of research investigating issues of flow and transportation in complex networks is under active development.21–30 A major threat for the proper functioning of power networks is that of large blackouts that may involve big cities or even portions of states. Traditionally such occurrences were caused by accidental faults and thus were quite rare; however, in recent years, power systems, as well as other critical infrastructures, have become a potential target for intentional attacks. The main difference is that malicious attacks may not be random but rather directed specifically to the most sensitive parts of the system, in terms of the impact they can cause. Thus, most of the applications of complex network concepts to power systems are aimed at understanding the behavior of power grids both in case of accidental failures and of malicious attacks.31–44 The tolerance of a network to failures is normally intended as the ability of the network to maintain its connectivity properties after the deletion of a fraction of its nodes or lines. In this way the problem can be mapped into a standard percolation problem, of the type extensively studied in statistical physics.45,46 However, a pure connectivity approach, which may be suitable, for instance, in the case of the World Wide Web, does not seem to catch most of the crucial features of a power network. In general a power network can indeed undergo severe damages even without any inverse percolation taking place; on the contrary, it can happen that some less important nodes become isolated, thus changing the connectivity of the network, without strongly affecting its global performance. A parameter introduced to evaluate the tolerance of complex networks to outages is the efficiency.47 In the present paper we further develop this concept and propose a new parameter to evaluate the performance of a power grid,
19, 013119-1
© 2009 American Institute of Physics
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which we name net-ability. The new definition takes into account some peculiar features of electrical networks, such as, the flow limits and the power flow allocation through the network, due to the inherent physical laws. The paper is structured as follows: In Sec. II we review the definition of efficiency pointing out its meaning in relation to power grids. In Sec. III we introduce the definition of net-ability. In Sec. IV we provide examples of the application of net-ability to evaluate the static tolerance to outages of a few sample networks. Comparison between efficiency and net-ability is finally provided. General conclusions and comments are provided in Sec. V. Finally, in the Appendix we recall some basic notions about power systems analysis used in the paper. II. EFFICIENCY AND VULNERABILITY
As a preliminary step, let us briefly recall the definition of geodesic distance commonly used in the literature on complex networks. Let us start considering an unweighted graph: the number of lines in a path connecting nodes i and j is called the length of the path. A geodesic path 共or shortest path兲 between i and j is the path connecting these nodes with minimum length. The length of the geodesic path is the geodesic distance dij between i and j. If one is dealing with a weighted graph, the length of a path is the sum of the weights of the lines constituting that path. The global efficiency E of a network was first introduced by Latora and Marchiori47 as follows: E=
1 1 , 兺 N共N − 1兲 i⫽j dij
共1兲
where N is the number of nodes of the network and dij is the geodesic distance between the nodes i and j; the sum is taken over all pairs of nodes of the network. The global efficiency is a measure of the performance of the network, under the assumption that the efficiency for sending information between two nodes i and j is proportional to the reciprocal of their distance. In many networks it happens that some nodes and lines are more important than others. While naively one would say that the most important nodes are those with the highest degree,48 for large networks it is often nontrivial to find out which are the components that are actually most critical for the performance of the network. Since the efficiency has been associated with the performance of the network, a natural way to find critical components of a network is by looking for the nodes or lines whose removal causes the biggest drops in efficiency. The vulnerability VE共l兲 of a line l can be defined as the drop in the performance when the line l is removed from the network,49 VE共l兲 =
E − El , E
共2兲
where E is the global efficiency of the network and El is the global efficiency after the removal of the line l. When a node is removed, all the lines attached to the node are removed as well.
A definition of network vulnerability is the maximum vulnerability of all its nodes,50 V = max V共l兲. l
共3兲
The general definition of vulnerability, Eq. 共2兲, as a drop in the efficiency can be usefully applied also to power networks. However, when applied to power grids, some problems arise with the definition of efficiency given by Eq. 共1兲. Specifically, the efficiency defined by Eq. 共1兲 shows three main problems when applied to power grids: 共1兲 In electrical circuits power does not flow from a node i to another node j along a single specific path 共for instance, the geodesic path兲, but rather along all the paths connecting i to j according to the power flow; see the Appendix for a simplified method to solve the power flow equations. Therefore the classical idea of geodesic distance is not suited for power grids and a different concept of distance needs to be introduced. 共2兲 In Eq. 共1兲 the sum is taken over all pair of nodes. However in electrical circuits power flows only from generation to load nodes, so only distances between generatorload pairs should be taken into account. 共3兲 For each pair 共i , j兲 of generation and load nodes, the network has a different transfer capability Cij in transmitting power. Suppose we increase the power injection at node i until the first line reaches its line flow limit; Cij is equal to the power injection in that moment.
III. FROM EFFICIENCY TO NET-ABILITY
In the same spirit of the efficiency described in the previous section, the net-ability of a power transmission grid is defined as a measure of its performance under normal operating conditions. The function of a power transmission network is to transmit a time dependent amount of power from generation nodes to load nodes in the most convenient technical and economic way. The economic issues are related to transmission costs and economic efficiency 共social surplus兲 of the underlying market, while the technical issues refer to losses, voltage drop, and stability. The actual ability of a power transmission grid to perform properly depends on its topological structure and on the impedance and flow limits of its lines. The concept of distance dij may be explained as the difficulty to transfer the relevant quantity between the nodes 共i , j兲 of a network. Distance in general depends on the path that one follows and thus should be defined as a function of the characteristics of the lines along the path. The economic and technical difficulties for transmission of electrical power through a path depend on both the power flow through the lines and on their impedance: with the same impedance, higher power flow increases costs; with the same power flow, higher impedance increases costs. Consequently, the distance from node i to node j along path k is related not only to the impedance of each line of the path but also to the power flows through the lines of the path. As a result, we define the electrical distance as
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Power grid vulnerability I i =1
Z ij
I i =1
i zii −z ij
U ij
I i =1
i
Ui
zij −z jj I j =−1
j
j I j =−1
I j =−1
dkij = 兺 f lkZl ,
共4兲
l苸k
where is the power transmission distribution factor of line l in path k and Zl its impedance 关see Eq. 共A9兲 for details兴. On account of Eq. 共4兲, we propose the following definition for the net-ability of a power transmission grid: 1 1 兺 兺 Cij 兺 pk , NGND i苸G j苸D k苸Hij ij dkij
共5兲
where G and D are the sets of generator and load nodes, respectively, while Hij is the set of paths from generator i to load j; likewise NG and ND are the total numbers of generators and loads, respectively. Finally, pkij is the power share of path k in transmitting power from i to j. Let us stress that the definition of distance given in Eq. 共4兲 is referred to a specific path. There is not the concept of geodesic distance or shortest path here, in principle all the paths are to be taken into account separately. Let us call Zij the equivalent impedance of the circuit whose ends are node i and node j; Uij is the voltage between i and j and Ii is the current injected at node i and extracted at node j 共Ii = −I j兲. As shown in Fig. 1 the equivalent impedance is defined as Zij =
j
FIG. 1. The computation scheme of equivalent impedance.
Uj
f lk
A=
i
Uij . Ii
Furthermore, let Ii = 1, I j = −1, and Ih = 0 ∀ h ⫽ i , j 共meaning that a unit current is injected at node i and extracted at j, while no current is extracted nor injected in other nodes兲, then the computation of equivalent impedance is sketched in Fig. 1 and amounts to
is considered as the current, dkij for any involved path between i and j is equal to the equivalent impedance of the circuit whose ends are i and j, ∀ k.
dkij = Zij
共7兲
Again, the expression of the distance between i and j given in Eq. 共7兲 is not related to a specific path, since it takes into account all the existing paths between i and j. Substituting Eq. 共7兲 into Eq. 共5兲 and keeping in mind that
兺
pkij = 1,
k苸Hij
we obtain A=
1 C 兺 兺 ij . NGND i苸G j苸D Zij
共8兲
In analogy with the expression of the vulnerability given by Eq. 共2兲, we define the vulnerability of line l as the netability drop caused by an outage 共cut兲 of the line l, VA共l兲 =
A − Al . A
共9兲
IV. CASE STUDY
In this section we use the two definitions, given by Eqs. 共2兲 and 共9兲, to estimate the line vulnerability of IEEE sample networks,51 made of 30 and 57 nodes. For each case, the results obtained from efficiency and net-ability are compared with the overload rate, defined in the following.52 The overload in an electrical network is given by P=
兩P 兩
兺 liml , l苸L P
共10兲
l
Zij =
Uij = Uij ⇒ Zij = Ui − U j Ii = 共zii − zij兲 − 共zij − z jj兲 = zii − 2zij + z jj ,
共6兲
where zij is the ith, jth element of the impedance matrix, see the Appendix. In the following, electrical networks are analyzed using a dc model. For a discussion of the reasons to choose a dc rather than an ac model, see the Appendix. In a dc power flow model the distance dkij defined in Eq. 共4兲 is equal to the variation of the voltage angle between nodes i and j, when 1 unit of active power is injected at i and collected at j. Since in the dc power flow model the variation of voltage angle is considered as the equivalent dc voltage and the active power
where Pl is the power flow through line l calculated by the dc load flow model 共see the Appendix兲, Plim l is the flow limit of line l and the sum is taken over the set L of lines in the network. We are interested in the sensitivity of P to line outages. Therefore, we call P共l兲 the real power performance parameter of the network upon cutting line l. The overload rate is then defined as W共l兲 =
P共l兲 − P . P
共11兲
In Fig. 2, we have plotted the vulnerability VE 关Eq. 共2兲兴, the vulnerability VA 关Eq. 共9兲兴 and the overload rate 关Eq. 共11兲兴 versus line removal for the IEEE test cases with 30 and 57 nodes, respectively. A few comments on the overload rate are appropriate. As we have shown, the overload rate is obtained
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TABLE II. Variances 2 of the curves of efficiency drop, net-ability drop, and overload rate; correlation coefficients between efficiency/overload, net-ability/overload, and overload/overload for the IEEE 30- and 57-nodes test cases plotted in Fig. 2.
2 共30兲 2 共57兲 共30兲 共57兲
(a)
(b) FIG. 2. 共Color online兲 Vulnerabilities and overload rate vs line removal for a 30 node, 41 lines IEEE test case 共above兲; for a 57 node, 78 lines IEEE test case 共below兲.
by computing the power flow through each line of the network in the dc approximation. For a given network, the value of the dc power flow through each line is a nonlinear function of power injections 共withdraws兲 at the generators 共loads兲. On the other hand, those values are not taken into account in the definitions of efficiency and net-ability. Furthermore, in the IEEE test cases, several generators produce pure reactive power, namely they are assigned a real power output equal to zero. On one side this means that these are not treated as generators in a dc flow model; on the other side these nodes are considered as generators both by the efficiency and net-ability algorithms. In order to overcome this limit, we have chosen to assign arbitrary values of active power output to the generators which are purely reactive. In Table I, we explicitly show these changes: in particular we keep the IEEE numeration for the generators, Pg indicates the IEEE real power output, while Pg⬘ indicates the real power output assigned here. In conclusion, one cannot expect a complete match between the results based on efficiency or net-ability and those based on the dc power flow. However, it appears from Fig. 2 TABLE I. Real power conversion for the IEEE 30- and 57-node generators. Case 30 Node Node Node Node Node Node
1 2 5 8 11 13
Pg
Pg⬘
260.2 40 0 0 0 0
260.2 40 210 130 95 78
Case 57 Node Node Node Node Node Node Node
1 2 3 6 8 9 12
Pg
Pg⬘
128.9 0 40 0 450 0 310
128.9 120 40 55 450 230 310
Efficiency
Net-ability
Overload
1.94 0.81 0.08 0.13
30.55 10.42 0.43 0.76
24.05 17.17 1 1
that in each of the sample cases under consideration the methods based on net-ability and overload rate computation can show evidence of a few highly critical lines; on the contrary, the plots obtained by the efficiency method are much smoother, without any sharp peak. In order to quantify this difference, in Table II we report the variances 2 of the curves plotted in Fig. 2. Moreover, the correlation coefficients between efficiency/overload, net-ability/overload, and overload/overload are reported. We observe that the variances of the net-ability and overload curves are of the same order of magnitude, while those obtained from the efficiency curve are about one order of magnitude smaller. Likewise the correlation coefficients between net-ability/overload are significantly larger than those between efficiency/overload. V. CONCLUSIONS
In this paper, a new network metric called net-ability is proposed, to evaluate the global performance of electric power grids. Our aim was to estimate the impact of line outages on the network performance in order to identify the most critical lines. In this respect we have analyzed sample networks taken from the IEEE database.51 For each system, three different methods to evaluate the impact of line outages have been used: 共1兲 the method based on efficiency; 共2兲 the new method based on net-ability; 共3兲 the computation of line overloads by dc power flow. Since the latter is the approach which takes into account the specific details of power grids, it can be considered as the reference method. From this point of view, the net-ability is capable of identifying some of the most critical lines. However, we stress how the line overloads depend nonlinearly upon the values of power injections/withdraws at the nodes. In real power networks such values are not constant, as the demand and production of electrical power vary considerably in time, for instance, depending on the period of the year and the hours of the day. On the other hand, both the efficiency and the net-ability approach are based essentially on the topological features of the network and do not take into account the actual values of power injections and withdraws. In order to check the validity of these topological approaches for real networks, one should compare their results with those obtained from overload computation, after performing a sort of time integration of the latter. At present, such kind of time integration looks difficult to be implemented in an algorithm. However, it is actually performed by direct observation by those companies which are in charge of the management and control of power grids in each country
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and therefore have monitored each country network for years. We have investigated the data of the Italian power grid from Terna-Rete Elettrica Nazionale S.p.A.53 in terms of netability, in order to find the most critical lines in the network. Although explicit results are confidential for obvious security reasons, we can say that a good match has been found between the results obtained by the net-ability algorithm and the experimental measurements collected by Terna. ACKNOWLEDGMENTS
This work was supported by the Next Generation Infrastructures Foundation 共NL兲 and by SiTI–Higher Institute on Innovation Territorial Systems 共ITA兲. APPENDIX: LINEARIZED POWER SYSTEMS MODELS
Here we provide a brief review of the main issues and tools of power system analysis used in this work; for a comprehensive treatment we refer to Ref. 54. A power transmission system can be schematically represented as a grid whose lines are electrical transmission lines, while nodes are the points where electrical power can be injected, withdrawn or redistributed. Accordingly, in a power grid one can distinguish three types of nodes: generation nodes 共generators or power plants兲, load nodes 共consumers兲, and transmission nodes. Each line in a power network has its own maximum power flow capability, which is the maximum amount of power flow that the line can sustain. Power transmission systems operate in a sinusoidal steady state. For a circuit made of N + 1 buses operating in the ac regime the nodal equations are written as
冢冣冢 I1 I2 ]
=
IN
Y 11 Y 12 ¯ Y 1N Y 21 Y 22 ¯ Y 2N ]
]
]
Y N1 Y N2 ¯ Y NN
冣冢 冣
,
共A1兲
UN
Y ij = y ijeı␥ij
Under these approximations, the power flow through line l, connecting nodes i and j, is given by f l = Pi = − P j =
UiU j sin共␦i − ␦ j兲 ␦i − ␦ j = . xij xij
共A4兲
In a general circuit made of N nodes and L lines, where ¯P is the vector of real power injections, ¯␦ is the vector of phase angles and ¯f is the vector of power flows we have N
兺 fl = 兺 l:i→∀j j=1
␦i − ␦ j xij
共A5兲
,
¯P = B¯␦ ,
共A6兲
where B is the N ⫻ N admittance matrix, Bij = −
U i = U ie ı␦i
are complex quantities. In matrix notation Eq. 共A1兲 writes I = YU,
共A3兲
• reactive power balance equations are ignored; • line losses are ignored, that is, the resistance of each line is set to zero, so only the reactance 共imaginary part of the impedance兲 is considered: y ii = y ij = y jj = 1 / xij 共xij is the reactance of the line connecting i to j兲, ␥ii = ␥ jj = − / 2 and ␥ij = / 2; • all voltage magnitudes are identically set to one per unit, Ui = 1 ∀ i; • all voltage angles are assumed to be small, ␦i → 0 ∀ i.
where I i = I ie ı
Uk*Y ki* = Pi + ıQi; 兺 k=1
P and Q are called real and reactive power, respectively; see Ref. 54 for a thorough explanation. Since the quantities involved in an ac system show a sinusoidal behavior, solving a full ac power flow model means that one has to solve a system of nonlinear equations, which is widely known to be a formidable task. The most common method to reduce the power flow problem to a set of linear equations is called the dc power flow. The dc power flow approach is based on a number of approximations:
Pi =
U1 U2 ]
N
S i = U iI i* = U i
共A2兲
where I is the vector of current sources, Y is the line admittance matrix, and U is the vector of node voltages. Node 0 is selected as the reference node 共ground兲, and node voltages Ui are defined with respect to node 0. The elements of the matrix Y are formed as follows: • diagonal elements Y ii: sum of the admittances of the lines connected to node i; • off-diagonal elements Y ij: minus the sum of the admittances of the lines connecting nodes i and j. The complex power Si flowing through a node i is defined as the product of the voltage Ui and the complex conjugate of the current Ii,
1 xij
for i ⫽ j,
Bii = 兺
j⫽i
1 . xij
In terms of the vector of power flows we have ¯f = H¯␦ ,
共A7兲
where H is the L ⫻ N transmission matrix, Hli = − Hlj =
1 , xij
Hlk = 0
∀ k ⫽ i, j.
The admittance matrix B is singular since the sum of the elements of each row is equal to zero, 兺Ni=1Bij = 0 ∀ j. This means that the total injection power is equal to zero, N
Pi = 0 ⇒ Pi = − 兺 P j . 兺 i=1 j⫽i To avoid this redundancy, a slack node, for instance, node N, is chosen and set ␦N = 0. Thus one can eliminate the corresponding terms in power vectors and matrices without losing
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information. In this spirit, B⬘ and H⬘ are submatrices, obtained, respectively, from B and H by deleting the row and column 共only the column in case of H兲 corresponding to the slack node N, while ¯␦⬘ and ¯P⬘ are, respectively, the vector of phase angles and vector of node power injections without the slack node N. The matrix B⬘ can be inverted and thus one can rewrite Eqs. 共A6兲 and 共A7兲 in terms of the modified vectors and matrices as ¯␦⬘ = B⬘−1¯P⬘ ,
共A8兲
¯f = H⬘B⬘−1¯P⬘ = AP ¯ ⬘.
共A9兲
The power transmission distribution factors 共PTDF兲 of the circuit are the entries of the matrix A in Eq. 共A9兲. 1
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CHAOS 19, 013120 共2009兲
Pinning synchronization of delayed dynamical networks via periodically intermittent control Weiguo Xia and Jinde Caoa兲 Department of Mathematics, Southeast University, Nanjing 210096, China
共Received 9 September 2008; accepted 23 December 2008; published online 23 February 2009兲 This paper investigates the synchronization problem for a class of complex delayed dynamical networks by pinning periodically intermittent control. Based on a general model of complex delayed dynamical networks, using the Lyapunov stability theory and periodically intermittent control method, some simple criteria are derived for the synchronization of such dynamical networks. Furthermore, a Barabási–Albert network consisting of coupled delayed Chua oscillators is finally given as an example to verify the effectiveness of the theoretical results. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3071933兴 Synchronization in complex dynamical networks has been intensively investigated during the last decade. In order to realize network synchronization, many control techniques have been introduced, such as impulsive control, adaptive control, pinning control, and so on. On the other hand, intermittent control is an effective and widely used approach in the engineering fields. In this paper, some periodical intermittent controllers are added to partial nodes to drive the network to synchronize, and some novel network synchronization criteria are derived based on rigorous mathematical analysis. A numerical example is presented to verify the theoretical results.
I. INTRODUCTION
Complex networks have recently received increasing attention from various fields of science and engineering. Networks are everywhere in nature and our daily life, such as ecosystems, the internet, World Wide Web, social networks, neural networks, and so on. A complex network can be described by a set of nodes and edges interconnecting these nodes together. Since Pecora and Carroll1 found chaos synchronization in the laboratory in 1990, chaos synchronization has been intensively studied over the last few years due to its potential applications in many different areas, such as secure communication, biological systems, information science, etc. Especially, the synchronization of all the dynamical nodes in complex networks has become a focal research topic recently, and much work has been devoted to this issue.2–16 In Ref. 2, some basic research methods were introduced in studying the synchronization of an array of coupled dynamical systems. Based on this, some novel synchronization criteria for the coupled neural networks with or without delayed coupling were given in Refs. 3 and 4. In Ref. 5, Yu et al. further investigated the global synchronization of hybrid coupled networks with delays. Zhou et al. established some less conservative criteria for global synchronization of dea兲
Electronic mail:
[email protected];
[email protected].
1054-1500/2009/19共1兲/013120/8/$25.00
layed dynamical network with impulsive effects in Ref. 6. In Ref. 7, Belykh et al. applied the connection graph method to investigate global stability of coupled chaotic systems. In the case where the network cannot synchronize by itself, many control techniques including continuous feedback and discontinuous feedback have been developed to drive the network to synchronize.8–16 In Ref. 8, Zhang et al. presented some criteria for network synchronization via impulsive control. In Ref. 9, Zhou et al. discussed the synchronization of an uncertain dynamical network by adding an adaptive controller to each node. But in practice, it is too costly and impractical to add controllers to all nodes in a large-scale network. To reduce the number of controlled nodes, pinning control, in which controllers are only applied to partial nodes, is introduced. In Ref. 17, pinning control of spatiotemporal chaos, and later in Ref. 18, global and local control of spatiotemporal chaos in coupled map lattices, were studied. In Refs. 10 and 11, both intentional and random pinning control schemes were investigated. Furthermore, in Ref. 12, the minimum number of nodes needed to be pinned to realize network synchronization was studied. More recently, Yu et al. derived some criteria to choose the kind of pinning schemes, the types of controllers, and the coupling strength to achieve network synchronization in Ref. 13. In Ref. 14 based on the master stability function approach, Sorrentino et al. defined the pinning controllability of a network in terms of the spectral properties of an extended network topology. In Refs. 15 and 16, Belykh et al. studied the synchronization in a new model of small-world networks with blinking interactions. On the other hand, intermittent control is an engineering approach that has been widely used in engineering fields, such as manufacturing, transportation, and so on. In Ref. 19, Zochowski introduced intermittent control to nonlinear dynamical systems. In Refs. 20–22, the stabilization problems of chaotic systems with or without delays by periodically intermittent control were discussed. Huang et al. discussed the synchronization of coupled chaotic systems with delay by using intermittent state feedback in Ref. 23. In this paper, based on a given complex delayed dynamical network, using
19, 013120-1
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pinning control and periodically intermittent control methods, some novel criteria for synchronization of such complex networks are derived. Numerical simulations are also presented to show the effectiveness of the proposed method. The rest of the paper is organized as follows: In Sec. II, a general model of complex delayed dynamical networks and a definition for network synchronization are given. In Sec. III, some pinning synchronization criteria for the general complex delayed dynamical networks are established. Section IV applies the proposed network synchronization criteria to a Barabási–Albert 共BA兲 共Ref. 24兲 network consisting of coupled delayed Chua oscillators, and numerical results illustrate the effectiveness of our method. Finally, conclusions are drawn in Sec. V.
II. PRELIMINARIES
Consider a generally controlled complex delayed dynamical network consisting of N identical nodes with linearly diffusive couplings described by the following equations:
Now, a mathematical definition for network synchronization is introduced as follows: Definition 1: Let xi共t ; t0 , X0兲共1 艋 i 艋 N兲 be a solution of the controlled network 共1兲, where X0 = 共x01 , x02 , . . . , x0N兲 苸 Rm⫻N. Assume that f : R+ ⫻ ⍀ ⫻ ⍀ → Rm is continuously differentiable, ⍀ 債 Rm. If there is a nonempty subset ⌫ 債 ⍀, with x0i 苸 ⌫共1 艋 i 艋 N兲, such that xi共t ; t0 , X0兲 苸 ⍀共1 艋 i 艋 N兲 for all t 艌 t0, and lim 储xi共t;t0,X0兲 − s共t;t0,x0兲储2 = 0,
t→⬁
where x0 苸 ⍀, then the controlled network 共1兲 is said to achieve network synchronization and ⌫ ⫻ ¯ ⫻ ⌫ is called the region of synchrony for the complex delayed dynamical network 共1兲. If we define error states as ei共t兲 = xi共t兲 − s共t兲 共1 艋 i 艋 N兲, then we can derive the following error dynamical system: e˙i共t兲 = f共t,xi共t兲,xi共t − 兲兲 − f共t,s共t兲,s共t − 兲兲 N
+ c 兺 gijAe j共t兲 + vi ,
共3兲
j=1
N
x˙i共t兲 = f共t,xi共t兲,xi共t − 兲兲 + c 兺 gijAx j共t兲 + vi,
1 艋 i 艋 N,
1 艋 i 艋 N,
j=1
共1兲 where xi共t兲 = 共xi1共t兲 , . . . , xim共t兲兲T 苸 Rm is the state variable of the ith delayed dynamical node, f : R+ ⫻ Rm ⫻ Rm → Rm is a continuously vector-valued function, node dynamics is x˙共t兲 = f共t , x共t兲 , x共t − 兲兲, vi 苸 Rm are the control inputs. Here, c is the coupling strength, A 苸 Rm⫻m is the inner-coupling matrix, and G = 共gij兲N⫻N 苸 RN⫻N is the coupling configuration matrix. If there is a link from node i to node j, then gij ⬎ 0共j ⫽ i兲; otherwise, gij = 0. It is assumed that G is irreducible and satisfies the diffusive coupling connection, 兺Nj=1gij = 0. Note that the coupling configuration matrix G and inner coupling matrix A are not assumed to be symmetric. Throughout the paper, we always assume that f共t , x共t兲 , x共t − 兲兲 satisfies the uniform Lipschitz condition with respect to the time t, i.e., for any x共t兲 = 共x1共t兲 , . . . , xm共t兲兲T, y共t兲 = 共y 1共t兲 , . . . , y m共t兲兲T 苸 Rm, there exist constants kij ⬎ 0 satisfying 兩f i共t,x共t兲,x共t − 兲兲 − f i共t,y共t兲,y共t − 兲兲兩 m
艋 兺 kij共兩x j共t兲 − y j共t兲兩 + 兩x j共t − 兲 − y j共t − 兲兩兲,
共2兲
j=1
where 1 艋 i 艋 m. Remark 1: It is easy to check that almost all the wellknown chaotic systems with delays or without delays, such as the Lorenz system, Rössler system, Chen system, Chua circuit, as well as the delayed Hopfield neural networks and delayed CNNs, and so on 共see Refs. 6 and 25, and the references therein兲 satisfy the form of Eq. 共2兲. Let s共t兲 = s共t ; t0 , x0兲 苸 Rm with x0 苸 Rm, be a solution of the node dynamics x˙共t兲 = f共t , x共t兲 , x共t − 兲兲. Here, s共t兲 may be an equilibrium point, a periodic orbit, or even a chaotic attractor.
where 1 艋 i 艋 N. In the following, assume that 储A储2 = ␣ ⬎ 0 and denote by min the minimum eigenvalue of the matrix 共A + AT兲 / 2. Let ˆ +G ˆ T兲 / 2 and 艌 艌 ¯ 艌 be the eigenvalues of ˆ s = 共G G 1 2 N ˆ is a modified matrix of G via replacˆ s, where G the matrix G ing the diagonal elements gii by 共min / ␣兲gii. Note that genˆ does not possess the property of zero row sums. erally G Moreover, there does not exist a definite relationship beˆ for the general tween the eigenvalues of G and those of G matrix G. The notation X ⬎ Y where X and Y are symmetric matrices, means that X − Y is positive definite. Some useful lemmas are given in Appendix A.
III. PINNING SYNCHRONIZATION OF A DELAYED DYNAMICAL NETWORK VIA INTERMITTENT CONTROL
In order to realize synchronization of the complex delayed dynamical network by pinning periodically intermittent control, some controllers are added to partial nodes of the network. Without loss of generality, let the first l nodes be selected and pinned, and the controllers vi共1 艋 i 艋 N兲 can be described by
冦
− kei共t兲, 1 艋 i 艋 l,
vi = 0, 0,
t 苸 关nT,nT + h兲,
l + 1 艋 i 艋 N, 1 艋 i 艋 N,
t 苸 关nT,nT + h兲;
t 苸 关nT + h,共n + 1兲T兲,
冧
共4兲
where k ⬎ 0 is a positive constant, T ⬎ 0 denotes the control period, 0 ⬍ h ⬍ T, and n = 0 , 1 , 2 , . . .. Thus the error system 共3兲 can be rewritten as, when t 苸 关nT , nT + h兲,
013120-3
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Synchronization via intermittent control N
e˙i共t兲 = f共t,xi共t兲,xi共t − 兲兲 − f共t,s共t兲,s共t − 兲兲 + c 兺 gijAe j共t兲 − kei共t兲,
1 艋 i 艋 l,
j=1
共5兲
N
e˙i共t兲 = f共t,xi共t兲,xi共t − 兲兲 − f共t,s共t兲,s共t − 兲兲 + c 兺 gijAe j共t兲,
l + 1 艋 i 艋 N;
j=1
when t 苸 关nT + h , 共n + 1兲T兲,
trol gain k can be sufficiently large and there exist positive constants a1 and a2, such that
e˙i共t兲 = f共t,xi共t兲,xi共t − 兲兲 − f共t,s共t兲,s共t − 兲兲 N
+ c 兺 gijAe j共t兲,
1 艋 i 艋 N.
共6兲
j=1
Our objective is to design suitable T, h, and k such that the delayed dynamical network can realize synchronization. The main results are stated as follows. Theorem 1: Suppose that 艋 h and 艋 T − h. Let h = R1T and = R2T. If there exist positive constants a1, a2, and k, such that Q = 共p +
1 2 a1
兲IN + c␣Gˆs − D ⬍ 0,
p − 21 a2 + c␣1 艋 0, 共7兲
a1 ⬎ q,
␥共R1 − R2兲 − 共a2 + q兲共1 − R1兲 ⬎ 0, where D = diag共k, . . . ,k,0, . . . ,0兲, l
1
˜ = diag共k, . . . ,k兲, D l
˜ is obtained by removing the 1 , 2 , . . . , l row-column and Q pairs of matrix Q. Using the Schur complement,26 it is easy ˜ ⬍ 0 and k ⬎ 共E to verify that13 Q ⬍ 0 is equivalent to Q max −1 T ˜ − BQ B 兲. If the control gain k can be sufficiently large, ˜ ⬍ 0. Note that Q ˜ = 共p then Q ⬍ 0 is equivalent to Q ¯ ¯ 1 ˆ s, where G ˆ s =G ˆs for i , j = 1 , . . . , N − l. + a 兲I + c␣G ij
p − 21 a2 + c␣1 艋 0, 共8兲
a1 ⬎ q,
␥共R1 − R2兲 − 共a2 + q兲共1 − R1兲 ⬎ 0, then the synchronous solution S共t兲 of controlled network 共1兲 is asymptotically stable under the periodical intermittent con1 m 共2k2 trollers 共4兲, where p = max1艋i艋m pi = max1艋i艋m 2 兺s=1 is 2共1−兲 m 2共1−兲 + ksi 兲, q = max1艋i艋mqi = max1艋i艋m兺s=1ksi , ␥ ⬎ 0 is the smallest real root of the equation a1 − ␥ − q exp兵␥其 = 0. One ˜ −1BT兲. can simply select k ⬎ max共E − BQ For the system 共1兲 without time-delay term x共t − 兲, i.e., = 0, the pinning periodical controllers 共4兲 can also be applied to the network. Thus we can derive the following theorem. Theorem 2: Consider the network 共1兲 without time delay. If there exist positive constants a1, a2, and k, such that ˆ s − D ⬍ 0, ¯ = 共¯p + 1 a 兲I + c␣G Q 2 1 N
N−l
2共1−兲 m p = max1艋i艋m pi = max1艋i艋m 2 兺s=1 共2k2 兲, q is + ksi m 2共1−兲 ksi , and ␥ ⬎ 0 is the small= max1艋i艋mqi = max1艋i艋m兺s=1 est real root of the equation a1 − ␥ − q exp兵␥其 = 0. Then the synchronous solution S共t兲 of controlled network 共1兲 is asymptotically stable under the periodical intermittent controllers 共4兲. A proof is given in Appendix B. ˆ s − D = 共 E−D˜ B 兲, where E and B Let Q = 共 p + 21 a1兲IN + c␣G ˜ BT Q are matrices with appropriate dimensions.
2 1 N−l
¯ˆ s p + 21 a1 + c␣max共G 兲 ⬍ 0,
l+i,l+j
Thus we can derive the following corollary. Corollary 1: Suppose that 艋 h and 艋 T − h. If the con-
¯p − 21 a2 + c␣1 艋 0,
共9兲
a1h − a2共T − h兲 ⬎ 0, where D = diag共k, . . . ,k,0, . . . ,0兲, l
N−l
2共1−兲 m ¯p = max1艋i艋m¯pi = max1艋i艋m 21 兺s=1 共k2 兲. Then the is + ksi synchronous solution S共t兲 of controlled network 共1兲 is asymptotically stable under the periodical intermittent controllers 共4兲. The proof of this theorem is similar to that of Theorem 1. Thus it is omitted here. Remark 2: The above theorems are available for the network coupled through full state variables. But if the network is coupled through only one or few state variables, i.e., the inner-coupling matrix A = diag共a1 , . . . , as , 0 , . . . , 0兲 is a diagonal matrix with a1 , . . . , as ⫽ 0, it is difficult to find appropriate controllers 共4兲 to satisfy the sufficient conditions in Theorems 1 and 2. Some other technique and analysis approach may be employed to deal with this case.
013120-4
Chaos 19, 013120 共2009兲
W. Xia and J. Cao
1
x
2
0.5 0 −0.5
FIG. 1. Chaotic behavior of time delayed Chua attractor.
−1 10 5
5 0
0
−5 −10
x
3
−5
x
1
兩f 1共t,xi共t兲,xi共t − 兲兲 − f 1共t,s共t兲,s共t − 兲兲兩
IV. NUMERICAL SIMULATION
In this section, a numerical example is given to verify the effectiveness of the proposed network synchronization criteria. Consider the controlled delayed dynamical network 共1兲 consisting of 100 identical Chua oscillators with time delayed nonlinearity, which is described by 100
x˙i共t兲 = f共t,xi共t兲,xi共t − 兲兲 + c 兺 gijAx j共t兲 + vi,
1 艋 i 艋 100,
1 艋 关␦共1 + b兲 + 2 ␦共b − a兲兴兩xi1共t兲 − s1共t兲兩 + ␦兩xi2共t兲 − s2共t兲兩,
兩f 2共t,xi共t兲,xi共t − 兲兲 − f 2共t,s共t兲,s共t − 兲兲兩 艋 兩xi1共t兲 − s1共t兲兩 + 兩xi2共t兲 − s2共t兲兩 + 兩xi3共t兲 − s3共t兲兩, and 兩f 3共t,xi共t兲,xi共t − 兲兲 − f 3共t,s共t兲,s共t − 兲兲兩 艋 兩xi2共t兲 − s2共t兲兩 + 兩xi3共t兲 − s3共t兲兩
j=1
+ ⑀兩xi1共t − 兲 − s1共t − 兲兩.
共10兲 where A = diag共1 , 1.2, 1兲, G = 共gij兲100⫻100 is a symmetrically diffusive coupling matrix with gij = 0 or 1共j ⫽ i兲, and the coupling strength c = 35. The dynamics of the Chua oscillator is given by27,28 x˙i共t兲 = f共t,xi共t兲,xi共t − 兲兲 = Cxi共t兲 + g1共xi共t兲兲 + g2共xi共t − 兲兲, 共11兲 where 1 艋 i 艋 100, xi共t兲 = 共xi1共t兲 , xi2共t兲 , xi3共t兲兲T 苸 R3, g1共xi共t兲兲 = 共− 21 ␦共a − b兲共兩xi1共t兲 + 1兩 − 兩xi1共t兲 − 1兩兲 , 0 , 0兲T 苸 R3, g2共xi共t − 兲兲 = 共0 , 0 , −⑀ sin共xi1共t − 兲兲T 苸 R3,
C=
冢
− ␦共1 + b兲
␦
0
1
−1
1
0
− −
冣
1
,
and ␦ = 10, = 19.53, = 0.1636, a = −1.4325, b = −0.7831, = 0.5, ⑀ = 0.2, and = 0.02. Figure 1 shows the chaotic behavior of the Chua attractor. In addition, it is easy to verify that
冦
− kiei共t兲,
vi = 0,
0,
Let k11 = ␦共1 + b兲 + 21 ␦共b − a兲 = 5.416, k12 = ␦ = 10, k13 = 0, k21 = k22 = k23 = 1, k31 = ⑀ = 1.953, k32 = = 19.53, k33 = = 0.1636, and = 21 . One can easily get that p = 11.1142, and q = 30.53. Here we assume that the network structure of Eq. 共10兲 obeys the scale-free distribution of the BA model.24 The parameters of the BA model are given by m0 = m = 5, N = 100. In the simulations, we add the periodical intermittent controllers to the first 20 nodes. In addition, the values of the parameters for the controllers 共4兲 are taken as T = 0.2 and h ¯ˆ s 兲 = 0.16. Since = 0.9067, ␣ = 储A储 = 1.1, and 共G
ki = qieTi 共t兲ei共t兲, 1 艋 i 艋 l,
t 苸 关nT,nT + h兲,
l + 1 艋 i 艋 100, 1 艋 i 艋 100,
2
max
= −1.6414, then if we choose a1 = 98 and a2 = 92.5, it is easy to verify that Eq. 共8兲 in Corollary 1 is satisfied. Our objective is to seek an appropriate control gain k for which synchronization happens. In Corollary 1, the control ˜ −1BT兲兴, gain is required to be large enough 关k ⬎ max共E − BQ but it may be much larger than the needed value. So the adaptive control approach is adopted here.9,12,13,25 We applied the following controllers to the network:
t 苸 关nT,nT + h兲;
t 苸 关nT + h,共n + 1兲T兲.
冧
共12兲
Chaos 19, 013120 共2009兲
Synchronization via intermittent control
90
100
80
80
70
60
60
40 ei3(t), 1 ≤ i ≤ 100
ei1(t), 1 ≤ i ≤ 100
013120-5
50 40 30
20 0 −20
20
−40
10
−60
0
−80
−10
0
0.5
t
1
1.5
−100
0
0.5
t
1
1.5
FIG. 2. Synchronous errors ei1共t兲 共1 艋 i 艋 100兲 of the controlled delayed network for 0 艋 t 艋 1.5.
FIG. 4. Synchronous errors ei3共t兲 共1 艋 i 艋 100兲 of the controlled delayed network for 0 艋 t 艋 1.5.
Though we have not found a rigorous proof for the synchronization of network 共10兲 with controllers 共12兲, the method is very useful in this example. The initial conditions of the numerical simulations are as s共0兲 follows: xi共0兲 = 共−8 + 0.5i , −5 + 0.5i , −10+ 0.5i兲T, = 共2 , 0.2, 0.3兲T, where 1 艋 i 艋 100, and k1共0兲 = ¯ = k20共0兲 = 1, q1 = ¯ = q20 = 0.04. The synchronous errors ei共t兲 are illustrated in Figs. 2–4 and the values for the control gains after synchronization satisfy ki 艋 k16 = 68.74, 1 艋 i 艋 20, which illustrate that the adaptive control approach can obtain a more applicable control gain.
novel synchronization criteria for such dynamical network are derived. In addition, this method could also be applied to the networks without time delay. Finally, numerical simulations have verified the effectiveness of the presented method. ACKNOWLEDGMENTS
The authors thank the referees and the editor for their valuable comments and suggestions. This work was jointly supported by the National Natural Science Foundation of China under Grant No. 60874088, and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20070286003.
V. CONCLUSIONS
In this paper, the synchronization issue of a general model of complex delayed dynamical networks is studied. The periodically intermittent control scheme is introduced to drive the network to achieve synchronization. Based on the Lyapunov stability theory and pinning control method, some
Lemma 1: 共Schur complement26兲 The following linear matrix inequality 共LMI兲:
冉
冊
Q共x兲 S共x兲 ⬎ 0, S共x兲T R共x兲
where Q共x兲 = Q共x兲T, R共x兲 = R共x兲T, is equivalent to one of the following conditions:
100
共i兲 Q共x兲 ⬎ 0,R共x兲 − S共x兲TQ共x兲−1S共x兲 ⬎ 0;
80
ei2(t), 1 ≤ i ≤ 100
APPENDIX A: LEMMAS
共ii兲 R共x兲 ⬎ 0,Q共x兲 − S共x兲R共x兲−1S共x兲T ⬎ 0.
60
Lemma 2: 共Ref. 29兲 Let : 关 − , ⬁兲 → 关0 , ⬁兲 be a continuous function such that
40
˙ 共t兲 艋 − a共t兲 + b max t is satisfied for t 艌 . If a ⬎ b ⬎ 0, then
20
共t兲 艋 关max 兴exp兵− ␥共t − 兲其,
0
−20
t 艌 ,
where max t = supt−艋艋t 共兲, and ␥ ⬎ 0 is the smallest real root of the equation 0
0.5
t
1
1.5
FIG. 3. Synchronous errors ei2共t兲 共1 艋 i 艋 100兲 of the controlled delayed network for 0 艋 t 艋 1.5.
a − ␥ − b exp兵␥其 = 0. Lemma 3: Let : 关 − , ⬁兲 → 关0 , ⬁兲 be a continuous function such that
013120-6
Chaos 19, 013120 共2009兲
W. Xia and J. Cao
˙ 共t兲 艋 a共t兲 + b max t
共s兲 艋 max +
is satisfied for t 艌 . If a ⬎ 0, b ⬎ 0, then
冕
t
It follows that
共t兲 艋 max t 艋 关max 兴exp兵共a + b兲共t − 兲其,
t 艌 ,
max t = sup 共s兲 艋 max +
where max t = supt−艋艋t共兲. Proof: Note that
t−艋s艋t
t
共a + b兲max d .
共t兲 艋 max t 艋 关max 兴exp兵共a + b兲共t − 兲其,
Integrating both sides of this inequality from to t, one has
冕
冕
From the well-known Gronwall–Bellman inequality, one has
˙ 共t兲 艋 共a + b兲max t .
共t兲 艋 共兲 +
− 艋 s 艋 t.
共a + b兲max d,
t 艌 .
APPENDIX B: PROOF OF THEOREM 1
Construct the following Lyapunov function: N
t
1 V共t兲 = 兺 eTi 共t兲ei共t兲. 2 i=1
共a + b兲max d .
Thus for a fixed t and t 艌 , when 艋 s 艋 t, we have 共s兲 艋 max + 兰t 共a + b兲max d. When − 艋 s 艋 , we have 共s兲 艋 max . As a consequence, we can get
N
N
i=1
i=1
共B1兲
Then the derivative of V with respect to time t along the solutions of Eqs. 共5兲 and 共6兲 can be calculated as follows: When nT 艋 t ⬍ nT + h, for n = 0 , 1 , 2 , . . .,
N
N
l
V˙共t兲 = 兺 eTi 共t兲e˙i共t兲 = 兺 eTi 共t兲共f共t,xi共t兲,xi共t − 兲兲 − f共t,s共t兲,s共t − 兲兲兲 + c 兺 兺 gijeTi 共t兲Ae j共t兲 − 兺 keTi 共t兲ei共t兲 m
N
艋兺兺 i=1 r=1
再
i=1 j=1
m
m
1 1 2共1−兲 2 ksr eir共t − 兲 兺 共2krs2 + ksr2共1−兲兲e2ir共t兲 + 2 兺 2 s=1 s=1
N
N
N
1 艋 p 兺 eTi 共t兲ei共t兲 + q 兺 eTi 共t − 兲ei共t − 兲 + c 兺 2 i=1 i=1 i=1
冎
i=1
N
N
+ c兺 兺
l
gijeTi 共t兲Ae j共t兲
i=1 j=1
− 兺 keTi 共t兲ei共t兲 i=1
N
兺 ␣gij储ei共t兲储2储e j共t兲储2 j=1
j⫽i N
+ c兺
l
giimineTi 共t兲ei共t兲
i=1
− 兺 keTi 共t兲ei共t兲 i=1
ˆ − D兲e共t兲 + 1 qeT共t − 兲e共t − 兲 = eT共t兲共pIN + c␣G 2 = eT共t兲
冉冉
冊
冊
1 ˆ s − D e共t兲 − 1 a eT共t兲e共t兲 + 1 qeT共t − 兲e共t − 兲 p + a1 IN + c␣G 1 2 2 2
艋 − a1V共t兲 + qV共t − 兲,
共B2兲
m where e共t兲 = 共储e1共t兲储2 , 储e2共t兲储2 , . . . , 储eN共t兲储2兲T, pi = 21 兺s=1 共2k2 is 2共1−兲 2共1−兲 m + ksi 兲, qi = 兺s=1 ksi , p = max1艋i艋m pi, and q = max1艋i艋m qi. The first inequality in Eq. 共B2兲 is a result of the Cauchy inequality. The first matrix inequality in Eq. 共7兲 yields the last inequality in Eq. 共B2兲. Since a1 ⬎ q, it follows from Lemma 2 that
V共t兲 艋
max
nT−艋艋nT
V共兲exp兵− ␥共t − nT兲其,
共B3兲
where ␥ is the smallest real root of the equation a1 − ␥ − q exp兵␥其 = 0.
Similarly, when nT + h 艋 t ⬍ 共n + 1兲T, one has V˙共t兲 = eT共t兲
冉冉
冊
冊
1 ˆ s e共t兲 p − a2 IN + c␣G 2
1 1 + qeT共t − 兲e共t − 兲 + a2eT共t兲e共t兲 2 2 1 1 艋 a2eT共t兲e共t兲 + qeT共t − 兲e共t − 兲 2 2 = a2V共t兲 + qV共t − 兲.
共B4兲
The inequality in Eq. 共B4兲 follows from the inequality p − 21 a2 + c␣1 艋 0. Thus it follows from Lemma 3 that
013120-7
Chaos 19, 013120 共2009兲
Synchronization via intermittent control
V共t兲 艋
max
nT+h−艋艋nT+h
V共兲exp兵共t − nT − h兲其,
共B5兲
where  = a2 + q. Now, we estimate V共t兲 based on Eqs. 共B3兲 and 共B5兲. For 0 艋 t ⬍ h, V共t兲 艋 max−艋艋0 V共兲exp兵−␥t其. For h 艋 t ⬍ T,
Note that nT + h 艋 t ⬍ 共n + 1兲T and substitute h = R1T and = R2T into Eq. 共B7兲, one obtains V共t兲 艋 max V共兲exp兵t − 共n + 1兲R1T −艋艋0
− 共n + 1兲T共␥R1 − ␥R2兲其 艋 max V共兲exp兵t − R1t − t共␥R1 − ␥R2兲其 −艋艋0
艋 − max V共兲exp兵− 关␥共R1 − R2兲 − 共1 − R1兲兴t
V共t兲 艋 max V共兲exp兵共t − h兲其
−艋艋0
h−艋艋h
+ ␥共R1 − R2兲h其.
艋 max V共兲exp兵共t − h兲 − ␥共h − 兲其. −艋艋0
Therefore, for any t 艌 0, we have
For T 艋 t ⬍ T + h,
V共t兲 艋 max V共兲exp兵− 关␥共R1 − R2兲 − 共1 − R1兲兴t −艋艋0
V共t兲 艋 max V共兲exp兵− ␥共t − T兲其
+ ␥共R1 − R2兲h其.
T−艋艋T
艋 max V共兲exp兵− ␥共t − T兲 + 共T − h兲 − ␥共h − 兲其. −艋艋0
For T + h 艋 t ⬍ 2T,
共B8兲
As V共t兲 = 21 兺Ni=1eTi 共t兲ei共t兲, one has 储e共t兲储2 艋 共2 max V共兲兲1/2 −艋艋0
V共t兲 艋
max
T+h−艋艋T+h
再
V共兲exp兵共t − T − h兲其
⫻exp −
艋 max V共兲exp兵共t − T − h兲 + 共T − h兲 −艋艋0
+
− 2␥共h − 兲其. By induction, we can derive the following estimation of V共t兲 for any integer n. For nT 艋 t ⬍ nT + h,
− n␥共h − 兲其.
共B6兲
Note that nT 艋 t ⬍ nT + h and substitute h = R1T and = R2T into Eq. 共B6兲, one obtains
V共t兲 艋 max V共兲exp兵n共1 − R1兲T − n␥共R1 − R2兲T其 −艋艋0
艋 max V共兲exp兵共1 − R1兲t + ␥共R1 − R2兲共− t + h兲其 −艋艋0
= max V共兲exp兵− 关␥共R1 − R2兲 − 共1 − R1兲兴t −艋艋0
+ ␥共R1 − R2兲h其. For nT + h 艋 t ⬍ 共n + 1兲T,
V共t兲 艋 max V共兲exp兵共t − nT − h兲 + n共T − h兲 −艋艋0
− 共n + 1兲␥共h − 兲其.
共B7兲
共B9兲
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−艋艋0
冎
␥共R1 − R2兲h . 2
As ␥共R1 − R2兲 − 共1 − R1兲 ⬎ 0 and based on Definition 1, we can draw the conclusion. The proof is thus completed. 1
V共t兲 艋 max V共兲exp兵− ␥共t − nT兲 + n共T − h兲
关␥共R1 − R2兲 − 共1 − R1兲兴 t 2
013120-8
A. L. Barabási and R. Albert, Science 286, 509 共1999兲. J. Zhou, T. Chen, and L. Xiang, Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 2923 共2006兲. 26 S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory 共SIAM, Philadelphia, 1994兲. 24 25
Chaos 19, 013120 共2009兲
W. Xia and J. Cao 27
X. Wang, G. Zhong, K. Tang, K. Man, and Z. Liu, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 48, 1151 共2001兲. 28 H. Huijberts, H. Nijmeijer, and T. Oguchi, Chaos 17, 013117 共2007兲. 29 A. Halanay, Differential Equations: Stability, Oscillations, Time Lags 共Academic, New York, 1966兲.
CHAOS 19, 013121 共2009兲
The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles? Carlo Bianca1,a兲 and Lamberto Rondoni2,b兲 1
Dipartimento di Matematica ed Informatica, Universitá di Catania, Viale Andrea Doria 6, 95125 Catania, Italy 2 Dipartimento di Matematica and CNISM, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
共Received 4 August 2008; accepted 27 January 2009; published online 3 March 2009兲 It is known that the nonequilibrium version of the Lorentz gas 共a billiard with dispersing obstacles 关Ya. G. Sinai, Russ. Math. Surv. 25, 137 共1970兲兴, electric field, and Gaussian thermostat兲 is hyperbolic if the field is small 关N. I. Chernov, Ann. Henri Poincare 2, 197 共2001兲兴. Differently the hyperbolicity of the nonequilibrium Ehrenfest gas constitutes an open problem since its obstacles are rhombi and the techniques so far developed rely on the dispersing nature of the obstacles 关M. P. Wojtkowski, J. Math. Pures Appl. 79, 953 共2000兲兴. We have developed analytical and numerical investigations that support the idea that this model of transport of matter has both chaotic 共positive Lyapunov exponent兲 and nonchaotic steady states with a quite peculiar sensitive dependence on the field and on the geometry, not observed before. The associated transport behavior is correspondingly highly irregular, with features whose understanding is of both theoretical and technological interests. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3085954兴 The theory of billiards subjected to external forces and to deterministic thermostats, i.e., nonequilibrium billiard models of transport of matter,1,2 is much less developed than the theory of equilibrium billiards, i.e., billiards whose particles move under the action of no forces, except for the collisions with scatterers. Indeed, the phenomenology of nonequilibrium billiards has been little investigated. In particular, nondispersing nonequilibrium billiards have been studied only in a few works, such as Refs. 3 and 4, and it is not known whether nonequilibrium billiards with flat obstacles are chaotic. In Ref. 3 it was conjectured that the nonequilibrium Ehrenfest gas is not chaotic, at least at small fields and dissipation. This observation was suggested by numerical evidence and by the fact that collisions with the obstacles do not defocus trajectories, while the external fields and the corresponding dissipative forces have a focusing effect. Exponential separation of phase space trajectories, if any, may then only be produced by the singularities of the dynamics, i.e., by a set of zero Liouville measure. No evidence of chaos was found in Ref. 3. However, by refining the study of the dependence of the steady states on field and on the shape of the billiard table, we have found that the asymptotic state of the system could be chaotic, in certain circumstances. Furthermore, we observe an extremely sensitive and peculiar dependence of the steady states on all parameters. I. INTRODUCTION
The Ehrenfest model of diffusion 共named after the Austrian-Dutch physicists Paul and Tatiana Ehrenfest兲 was proposed in the early 1900s in order to illuminate the statisa兲
Electronic mail:
[email protected]. Electronic mail: lamberto
[email protected].
b兲
1054-1500/2009/19共1兲/013121/10/$25.00
tical interpretation of the second law of thermodynamics and to study the applicability of the Boltzmann equation. In the Ehrenfest wind-tree model,5 the pointlike 共“wind”兲 particles move on a plane and collide with randomly placed fixed square scatterers 共“tree”兲. This model has been recently reconsidered in Ref. 6 to prove that microscopic chaos is not necessary for Brownian motion. A one-dimensional version of this model has been considered in Ref. 7 to investigate the origin of diffusion in nonchaotic systems. In Ref. 7 the authors identify two sufficient ingredients for diffusive behavior in one-dimensional nonchaotic systems: 共a兲 a finite-size algebraic instability mechanism and 共b兲 a mechanism that suppresses periodic orbits. A nonequilibrium modification of the model, with regularly placed scatterers, has been proposed in Ref. 3 to test the applicability of the so-called fluctuation relation8–11 to nonchaotic systems. This modified model was chosen under the assumption that, at small and vanishing fields at least, it must be nonchaotic since collisions with flat boundaries do not lead to exponential separation of nearby phase space trajectories. However the question of whether such a model can have positive Lyapunov exponents, as functions of the field, is open. Indeed, the techniques so far developed, e.g., by Chernov12 and Wojtkowski,13 rely on the dispersing nature of the billiard obstacles. In this paper, the dynamical properties of the nonequilibrium version of the Ehrenfest gas are considered as functions of the field and of the parameters, which determine the billiard table. Numerical tests are performed to find chaotic attractors and to compute the Lyapunov exponents. The construction of a sort of bifurcation diagram of the attractor as a function of the electric field and of the geometry is attempted. The result turns out to be quite peculiar: chaotic regimes with an extremely sensitive dependence on the pa-
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共Rmc,nc , s兲 with s 苸 A, while the hexagon side with a triple 共Hmc,nc , s兲 = 共mc , nc , s兲 with s 苸 B. The rhombi lying in the y-axes have R0,i = 共0 , i兲 with i 苸 ZO and the ones lying in the x-axes have R j,0 = 共j , 0兲 with j 苸 ZE. The geometry of the model is determined by the side L of the hexagonal Wigner–Seitz cell, so that xL = 冑3 / 2 and y L = 3L / 2. Let R0 = 共0 , 0兲 be the rhombus with the center in the origin of the Cartesian coordinates, where l is its side length, and sx and sy are the half lengths of the major and minor diagonals, respectively, so that l = 冑s2x + s2y . To prevent the overlap of rhombi, the side length of the rhombus inside one hexagon has to verify FIG. 1. The modified Ehrenfest gas. In the present paper, the side of the elementary cell is set to 1.291, while the semiaxes of the rhombus are chosen to be 1.1 and 0.7573.
rameters appear possible, although not easy to establish rigorously. If this model can be taken to approximate the transport of matter in microporous membranes, our results confirm the sensitive dependence of microporous transport on all relevant parameters observed, e.g., in Refs. 14 and 15. Indeed, the current of the nonequilibrium Ehrenfest gas is proportional to the sum of the Lyapunov exponents 关cf. Eq. 共11兲 below兴, which varies with the parameters as irregularly as the attractors do. II. THE MODEL
The billiard table consists of rhombi of side length l with distances along the x and y directions between the centers of two nearest neighboring rhombi given by xL and y L, respectively. The centers of the rhombi are fixed on a triangular lattice in a plane and have coordinates
冉冊
xc = m cl 1 + n cl 2, yc
mc,nc 苸 Z,
where l1 = 共xL , 0兲 and l2 = 共0 , y L兲 are the lattice vectors. If all the pairs 共mc , nc兲 are selected, the billiard is invariant under the group of spatial translations generated by l1 and l2. Accordingly, the whole lattice can be mapped onto a so-called Wigner–Seitz cell, with periodic boundary conditions 共Fig. 1兲. The elementary Wigner–Seitz cell of the triangular lattice is a hexagon of length side L and area AWS = 兩l1 ⫻ l2兩. The centers of all other cells are identified by the pairs 共mc , nc兲 苸 ZE ⫻ ZE or 共mc , nc兲 苸 ZO ⫻ ZO, where ZE = 兵n 苸 Z : 兩n兩 is even其 and ZO = 兵n 苸 Z : 兩n兩 is odd其. Because of the bijective correspondence between rhombi and pairs 共mc , nc兲, one may label a generic rhombus by Rmc,nc and the corresponding hexagon by Hmc,nc. Further, a label can be put on the sides of the rhombi and of the hexagons, introducing an alphabet A = 兵r1 , r2 , r3 , r4其, starting from the right vertical side and oriented clockwise, for the sides of the rhombi and an alphabet B = 兵h1 , h2 , h3 , h4 , h5 , h6其 for the hexagon sides, starting from the right vertical side and oriented clockwise. In this alphabet, the sides of a generic rhombus of the lattice can be labeled by a triple
0ⱕlⱕ
冑7 2
L,
which implies 0 ⱕ sx ⱕ xL. The case with l = 共冑7 / 2兲L corresponds to a billiard table that was recently considered in Refs. 14 and 15. Take l 苸 共0 , 共冑7 / 2兲L兲, hence sx ⬍ xl. The horizon of the billiard depends on the quantity sy and in particular on the difference y L − 2sy. If sy ⱖ y L / 2, the horizon is finite; if sy ⬍ y L / 2, it is infinite. The infinite horizon case allows collision-free trajectories, parallel to the x-axis. When the dynamics is followed within the Wigner–Seitz cell, the position of the point particle of mass M must be supplemented by the couple 共mc , nc兲 in order to determine its actual position in the infinite plane. The space between the rhombi forms the two-dimensional domain, in which the particle moves with velocity v during the free flights, while collisions with the sides of the rhombi obey the law of elastic reflection. In order to drive the model out of equilibrium, an external electric field parallel to the x-axis, E = ⑀xˆ , is applied. If there were no interaction with a thermal reservoir, any moving particle would be accelerated by the external field, on average, leading to an indefinite increase in energy in the system, and there would be no stationary state. Therefore, in Ref. 3 the particle has been coupled to a Gaussian thermostat. The resulting model, with periodically distributed scatterers, has been called the nonequilibrium Ehrenfest gas. Its phase space has four coordinates 共x , y , px , py兲 and its equations of motion are given by x˙ = px/M,
p˙x = ⑀ − ␣共p兲px ,
y˙ = py/M,
p˙y = − ␣共p兲py
with ␣共p兲 = − ⑀ px ,
共1兲
where ⑀ is the electric field and ␣ is the Gaussian thermostat. The quantity −div共p˙ , q˙ 兲 = ␣ = ⑀ px is known as the phase space contraction rate. Because of the Gaussian thermostat, p2x + p2y is a constant of motion; hence there are only three independent variables, and one may replace px and py by the angle 苸 共− , 兴 that p = p共cos , sin 兲 forms with the x-axis. For sake of simplicity, we set M = 1 and p = 1. Then, if 共xt , y t兲 denotes the position at time t and the velocity angle t, measured with respect to the x-axis, the trajectory between two collisions reads1,2
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3
3
2.5
2
2 1
1.5 0
1
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-1
0 -2
-0.5
-3
-1
0
1
2
3
4
-1
-1
0
1
2
3
4
5
6
FIG. 2. The closed period-two orbit ⍀o2 = 共R−1,−1 , r4兲共R1,1 , r3兲 for = / 11 and xl = 5冑3 / 3 共left panel兲. The open periodic orbit ⍀o2 = 共R0,0 , r4兲共R1,1 , r3兲 ⫻共R2,0 , r4兲 for xl = 2, sx = 0.8, and sy = 0.5 共right panel兲.
1 sin t xt = x0 − ln , ⑀ sin t0 1 y t = y 0 − 共t − t0兲, ⑀ tan
共2兲
t t = exp共− ⑀共t − t0兲兲tan 0 , 2 2
where t0 is the time of the previous collision, while the collision map C is given by xt⬘ = xt , y t⬘ = y t ,
共3兲
t⬘ = − t ⫾ 2 , where t is the incidence angle, 共xt , y t兲 is the collision point, and is the angle that the side of the rhombus makes with the x-axis. The sign ⫾ depends on the side on which the bounce occurs. Hence C is piecewise linear in t. Considering the dynamics as a geodesic flow on a Riemann manifold, the appropriate metric for this system is16–18 ds2 = e−2⑀x共dx2 + dy 2兲, which implies that the quantities y = e−⑀x py and t = t + ⑀y t = t0 + ⑀ are conserved. Also, the path length L共P0 , Pt兲 between P0 = 共x0 , y 0 , t0兲 and Pt = 共xt , y t , t兲 turns out to be 1 L共P0, Pt兲 = e−⑀x0 sin t0兩cot t0 − cot t兩. ⑀
共4兲
III. PERIODIC ORBITS AND STABILITY MATRICES
Using the symbols introduced above, a trajectory segment ⍀N that consists of N collisions can be labeled by a finite symbolic sequence such as 共Ri1,j1 , s1兲 ⫻共Ri2,j2 , s2兲 ¯ 共RiN,jN , sN兲, with 共ic , jc兲 苸 ZE ⫻ ZE or 共ic , jc兲
苸 ZO ⫻ ZO and sc 苸 A for c = 1 , . . . , N. Periodic trajectories will be labeled by sequences that are infinitely many copies of a fundamental finite sequence. There are two types of periodic orbits; those that are periodic in the plane, i.e., that return to the initial point in the plane 共they are closed: ⌬xi = 0 or ⌬y i = 0兲 and those whose periodicity relies upon the periodicity of the triangular lattice and return to the same relative position in a different cell 共they are open: ⌬xi ⫽ 0 or ⌬y i ⫽ 0兲. The velocity vectors of the closed orbits with two collisions have to be orthogonal to both sides of the rhombi where collisions occur. This implies 0 = / 2 + and 0⬘ = / 2 − , where 0 is the outgoing velocity angle and 0⬘ is the incoming angle, because the absolute value of the velocity angle decreases and preserves the sign during the free flight.2 Then, the closed period-two orbits fly between rhombuses in the same line parallel to the y-axes and have period and length L given by19
冉 冊 冉 冊
+ 2 4 2 = ln , ⑀ − tan 4 2 tan
2 L = e−⑀x0 sin . ⑀
Now, fix the geometry of the model through the parameters L ⬎ 0, sx ⬎ 0, and sy ⬎ 0 and take the initial conditions x0,y 0 = ⫾ y l, 0 = ⫾
. 2
It is easy to show that the periodic orbits 共R0,0 , r4兲 ⫻共R0,2 , r3兲 共Fig. 2, left panel兲 and 共R0,0 , r3兲共R0,−2 , r4兲, with electric field
⑀=
− + tan ln cos , ⫿y 0 + mx0 + sy
exist if and only if19
共5兲
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2 2 ⬍⑀⬍ . 3L 3L − 2sy
共6兲 J+,− =
For open orbits with two collisions in the finite horizon case, simple algebra shows that the following symbolic representations 共Ri,j , r4兲共Ri−1,j+1 , r2兲共Ri+2,j , r4兲 and 共Ri,j , r1兲共Ri+3,j+1 , r3兲共Ri+2,j , r1兲 cannot be realized. However, orbits with symbolic representation 共Ri,j , r4兲共Ri+1,j+1 , r3兲 ⫻共Ri+2,j , r4兲 and its symmetric counterpart 共Ri,j , r3兲 ⫻共Ri+1,j−1 , r4兲共Ri+2,j , r3兲 共Fig. 2, right panel兲 do exist. Their period is given by
2⑀ tan tan ⬘0
共tan 0⬘ + tan 兲tan 0
tan 0⬘ + tan
1 tan 0 − tan ⬘0 − ⑀ tan 0 tan 0⬘ + tan
tan ⬘0 − tan tan 0⬘ + tan
冣
.
共8兲
Similarly, the flights from a side with negative slope to a side with positive slope yield
J−,+ = 2 = ln ⑀
冢
共tan 0 + tan 兲tan 0⬘
1 . − tan 4
冉 冊
冢
−
共tan 0 − tan 兲tan 0⬘
2⑀ tan tan ⬘0
共− tan 0⬘ + tan 兲tan 0 − tan ⬘0 + tan
1 tan ⬘0 − tan 0 − ⑀ tan 0 tan − tan ⬘0
tan ⬘0 + tan tan 0⬘ − tan
冣
,
共9兲
and those from one side to a parallel one yield The open periodic orbits with four collisions 共Fig. 5, left panel兲 and symbolic sequence 共R0,0 , r4兲共R−1,1 , r2兲共R1,1 , r3兲 ⫻共R0,0 , r1兲共R2,0 , r4兲 and its symmetric image 共R0,0 , r3兲 ⫻共R−1,−1 , r1兲共R1,−1 , r4兲共R0,0 , r2兲共R2,0 , r3兲 do exist if19
再
1 sin ⬘0 max − sx,− xl + ln ⑀ sin 0
再
冎
冎
1 sin ⬘0 ⬍ x0 ⬍ min − xl + sx + ln ,0 . ⑀ sin 0
共7兲
To compute the Lyapunov exponents for these orbits and any other trajectory, consider the stability matrix JS for a trajectory as the product of free flight stability matrices JF and collision stability matrices JC, n
JS = 兿 JC共i兲JF共i兲. i=1
The number of degrees freedom for the billiard map is 2, and the variables that we will use are 共x0 , 0兲. Thus
JC =
冢 冣 ⬘ 0
⬘ x0
x0⬘ x0⬘ . 0 x0
=
冉 冊 −1 0 0
1
.
The free flight matrix JF depends on the side which the trajectory leaves and on the one which it reaches. There are two different types of side, the ones with positive slope, of the equation y = tan x + c, and the ones with negative slope, of the equation y = −tan x + d, where c and d are real numbers. Let us compute the free flight matrix J+,− of a trajectory, which goes from a side with positive slope to a side with negative slope. Let 共x0 , y 0 , 0兲 and 共x0⬘ , y 0⬘ , 0⬘兲 be the initial condition on a side with positive slope and the final condition on a side with negative slope, respectively. By using the equations of the trajectory and by the implicit function theorem,19,20 we obtain the Jacobian matrix of the free flight,
J⫾,⫾ =
冢
共tan 0 ⫿ tan 兲tan 0⬘ 共tan 0⬘ ⫿ tan 兲tan 0
0
1 tan 0 − tan ⬘0 1 − ⑀ tan 0 tan 0⬘ ⫿ tan
冣
.
共10兲
If 1 and 2 are the eigenvalues of the stability matrix JS for a periodic orbit of period , the two Lyapunov exponents are i = 共1 / 兲log兩i兩, i = 1 , 2, and one obtains j=
共1 + 2兲 ⌬x =− , ⑀
共11兲
where j is the current and ⌬x is the corresponding displacement in the direction of the field.2 Both the Lyapunov exponents of the closed periodic orbits, with period two, vanish. Indeed, consider that this periodic orbit has ⌬x = 0; hence 1 + 2 = 0. Furthermore, the stability matrix of these periodic orbits, which is JS = JCJ−,+JCJ+,−, is given by
JS =
冢
4 tan2 − 共1 − tan2 兲 4⑀ tan 共tan2 − 1兲 共1 + tan2 兲2 共1 + tan2 兲2 4⑀ tan 共tan2 − 1兲 共1 + tan2 兲2
−
4 tan2 − 共1 − tan2 兲 共1 + tan2 兲2
冣
,
whose determinant is 1, while its trace vanishes. This implies that both Lyapunov exponents vanish. IV. NUMERICAL ESTIMATES OF LYAPUNOV EXPONENTS
In this paper, a system is called chaotic when it has at least one positive Lyapunov exponent. We note that the boundary of our system is not defocusing and the external field has a focusing effect, so the overall dynamics should not be chaotic in general, although it is not obvious that this is the case for all values of the electric field ⑀. In this section we examine the stationary state and the Lyapunov exponents, obtained by using the algorithm developed by Benettin et al.,21 for different values of ⑀, ranging from small to large fields.
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λ1
0.15 0.1 0.05 0
FIG. 3. The largest Lyapunov exponent 1 for electric fields between 0.02 and 1, for trajectories with 107 collisions, and random initial condition.
−0.05 −0.1 −0.15 −0.2 −0.25
ε 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
A. Chaos for large electric fields
As already noted in Ref. 3, the doubt is that, starting from a generic initial condition, convergence to the steady state might be too slow to be discovered, for reasons which had not been investigated. Indeed, even in cases in which convergence is observed, the particle often appears to peregrinate in a sort of chaotic quasisteady state for very long times, before eventually settling on a periodic or quasiperiodic steady state. Reference 3 suggested that this might always be the case. Therefore, we extended the simulations of the cases with one apparently positive Lyapunov exponent, up to 1.5⫻ 108 collisions. The behavior of the system still appears to be nontrivial, in cases such as that of ⑀ = 0.374. Furthermore, plotting the last 104 iterates of a trajectory with length of 5 ⫻ 107 collisions, we cover a large fraction of the phase space, which appears quite close to that covered by the last 104 iterates of a trajectory of 1.5⫻ 108 collisions 共Fig. 4兲.
Numerical simulations of the model starting with random initial conditions and electric field in the range of 关0.02, 1兴 have been initially performed for a trajectory with length of n = 107 collisions. The parameters chosen for the geometry are L = 1.291, sx = 0.7573, and sy = 1.1, which correspond to a case in which the angles of the rhombi are irrational with respect to ; then, according to a conjecture by Gutkin,22 the equilibrium version of this model 共i.e., the ⑀ = 0 case兲 should be ergodic. For simulations of 107 collisions, Fig. 3 shows that the fields that appear to lead to one positive Lyapunov exponent cover a range larger than those that appear to correspond to two negative exponents. However, do 107 collisions suffice for a generic trajectory to characterize the stationary state? For the cases with two negative exponents, the answer is affirmative since the trajectory is clearly captured by an attracting periodic orbit. But the cases with one apparently positive exponent are not equally clear.
η
1
η
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
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FIG. 4. The last 104 iterates of the bounce map for ⑀ = 0.374 out of a trajectory of 5 ⫻ 107 collisions 共left panel兲 and for a trajectory of 1.5⫻ 108 collisions, starting from the last phase space point of the previous trajectory 共right panel兲. The distribution appears to be the same, suggesting that the system is in an apparently stationary state. Here, = cos , with as the angle between the outgoing velocity and the side of the rhombus, and r is the perimetral distance of the collision point from the right corner of the rhombus 共 and r are called Birkhoff coordinates兲.
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FIG. 5. The open period-four orbit 共R0,0 , r3兲共R−1,−1 , r1兲共R1,−1 , r4兲共R0,0 , r2兲共R2,0 , r3兲 in the plane for ⑀ = 0.374, sy = 0.7573, sx = 1.1, L = 1.291, 0 = −2.0619, and x0 = −0.4885 共left panel兲. The four points of this periodic orbit are inflated to show their embedding in the attractor 共right panel兲. The coordinates are as in Fig. 4.
⑀ = 0.374. The values of ⑀, of the initial conditions, and of the Lyapunov exponents are reported in Tables I and II.
The conclusion that a chaotic stationary state has been reached seems reasonable in this case, as the computed positive Lyapunov exponent also indicates, having apparently converged to 0.144 with three digits of accuracy, after only 106 collisions. To strengthen these results, we have looked for an unstable periodic orbit embedded in the attractor, and we have found one periodic orbit of period four, which apparently lies in the attractor and has a positive Lyapunov exponent 共Fig. 5, right panel兲. The other possibility is that the orbit is isolated and is separated from the attractor by such a small neighborhood that is numerically impossible to resolve. Another interesting example is given by ⑀ = 0.5: the last 7000 points of a trajectory with length of 5 ⫻ 107 collisions are compared with the last 104 points of the trajectory with length of 2 ⫻ 108 共Fig. 6兲. Also in this case the stationary state seems to have been reached, and a periodic orbit of period four with one positive Lyapunov exponent seems to be embedded in the attractor, similarly to the case of
η
1
B. Chaos for small electric fields
Numerical simulations of the model were performed starting with random initial conditions and considering the electric field in the interval of 关0.002, 0.1兴, for trajectories with length of n = 107. Looking at Fig. 7 共left panel兲 we find more cases with one apparently positive Lyapunov exponent than with two negative exponents. Plotting the last 104 points, in trajectories with length of 1.5⫻ 108, for the electric fields that produced apparently positive Lyapunov exponents, the situation is practically the same as for large fields: some cases with one exponent that appeared to be positive after 107 collisions eventually produce 共after 1.5⫻ 108 collisions兲 two negative exponents and a periodic or quasiperiodic steady state.3,23 We have further increased the number of
η
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
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-0.2
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-0.8
-1
0
1
2
3
4
r
5
-1
0
1
2
3
4
r
5
FIG. 6. The last 7000 points of the bounce map for ⑀ = 0.5 out of a trajectory with length of 5 ⫻ 107 collisions 共left panel兲 and the last 104 points of a trajectory with length of 2 ⫻ 108 with same ⑀ 共right panel兲. The coordinates are defined as in Fig. 4.
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TABLE I. Numerically computed Lyapunov exponents for different trajectories and fields. Field
Collisions
Lyapunov exponent
0.374 0.374 0.374 0.374 0.5 0.5
106 1.5⫻ 107 6.5⫻ 107 2.15⫻ 108 5 ⫻ 107 2 ⫻ 108
0.144 232 0.144 317 0.144 291 0.144 320 0.166 648 0.166 622
collisions up to 3 ⫻ 108, but the exponent 1 remained positive in most of the cases 共Fig. 7, right panel兲 and an apparently chaotic attractor was reached. C. A small basin of attraction
The behavior illustrated above is rather peculiar and calls for some explanation. How can it be that a steady state is so hard to reach in so many cases? Usually, convergence to an attracting periodic orbit occurs rather quickly, while doubts remain in some of the cases we considered even after 108 iterations of the bounce map. Therefore, we have investigated in greater detail the specific example with ⑀ = 0.087, L = 1.291, sx = 0.7573, and sy = 1.1. One finds that the largest time dependent Lyapunov exponent, 1, rapidly settles on a positive value, as if the trajectory had reached a chaotic attractor. However, for randomly chosen initial conditions, a striking and precise monotonic 1 / n behavior sets in for 1, after a critical, typically large time Nc 共cf. left panel of Fig. 8兲, as if the trajectory had eventually collapsed on an attracting periodic orbit. Indeed, the asymptotic value of 1 is −0.004 838 with an estimated error not larger than 10−6.24 The accuracy of this result is due to the fact that a sufficiently long simulation approaches an attracting periodic orbit to practically full numerical precision; the Lyapunov exponents can then be computed with analogous accuracy. This is particularly true in the present example, which turns out to have an attracting orbit made of only 19 points, whose initial condition, up to 12 digit accuracy, is given by xpo = 0.418 447 478 686, y po = 0.492 193 019 206, and po = 0.718 794 450 586 共cf. right panel of Fig. 8兲. For such a small number of points, numerical errors cannot appreciably affect the result. In particular, no doubts remain, in this case, about the negative sign of both exponents, hence about the attracting nature of the orbit. The question now arises as to the shape and size of the basin of attraction B of the asymptotic periodic orbit because a particle with random initial conditions wanders around almost all phase space before getting into this set by numerical noise.
Our analysis of the evolution of trajectories, with initial condition close to the attracting periodic orbit, shows that B is quite limited in size and particularly hard to reach because it contains a very small region around the right vertex of the rhombus. Furthermore, by varying the initial conditions around 共xpo , y po , po兲 and computing the Lyapunov exponents, the irregular shape of B is evidenced by the times Nc, which vary most irregularly from O共10兲 to O共108兲, O共108兲 being typical for random initial conditions. Nevertheless, the radius of B, i.e., the maximum distance between any two points of B, is not smaller than 10−4, which is quite small but well above the distances that can be accurately measured with double precision numerical simulations. We conclude that B lies at the border of a chaotic repeller25 and that it is quite small and of irregular shape. Hence, only after a sufficiently long peregrination in phase space, when the transiently chaotic trajectory gets sufficiently close to B, may numerical errors let the trajectory jump inside B. At this stage, the sudden 共exponential兲 convergence of the trajectory and the consequent 1 / n behavior of the Lyapunov exponents begin. The same qualitative behavior has been observed for ⑀ = 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.010, 0.022, 0.040, 0.041, 0.042, 0.043, 0.050, 0.064, 0.065, 0.073, and 0.087. V. BEHAVIOR OF THE ATTRACTOR
It is interesting to understand the behavior of the steady state with the field, i.e., to build a kind of bifurcation diagram, e.g., to compare with the one for the Lorentz gas, whose obstacles are defocusing.2 Our analysis reveals substantial differences from the case of Ref. 2, as well as from standard low-dimensional dynamics. A. Multifurcation as function of the electric field
In order to visualize the behavior of the attractor as a function of the electric field, we consider a projection of the billiard map phase space: the projection onto the -axis, which shows a sort of “multifurcation” scenario. When the field is varied, a series of dramatic changes in the dynamics occurs. For the geometry determined by L = 1.291, sx = 0.7573, and sy = 1.1, numerical simulations were performed for a random initial condition, ignoring the initial transient behavior, for the electric fields in the range of 关0.01, 1兴 with a step ⌬⑀ = 0.01. At the end of a trajectory of 107 collisions, the last 103 points were plotted 共Fig. 9 presents only the upper half projection onto the -axis; the other half of this diagram is trivially obtained from this, by reflection along the line = 0, as a result of the symmetry imposed on the system by the external field兲. In this way, many electric fields are found
TABLE II. Initial conditions and Lyapunov exponents of periodic orbits of period four. All data have been computed analytically. Field
x0
y0
0
1
2
0.374 0.5
−0.489 715 782 827 413 85 0.305 766 748 010 10
−0.388 673 760 583 447 5 0.655 865 016 755 433 7
−2.061 994 618 33 −0.318 739 956 935 00
0.108 316 0.155 569
−0.281 452 −0.384 674
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0.07
0.07
λ1
0.06
0.05
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
-0.01
ε 0
λ1
0.06
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
-0.01
ε 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
FIG. 7. The largest Lyapunov exponent, 1, for different lengths of trajectory: 107 共left panel兲 and 3 ⫻ 108 共right panel兲 for ⑀ 苸 关0.002, 0.1兴 and random initial condition.
whose dynamics samples most of the space, while only a few fields show a periodic 共or quasiperiodic兲 steady state. The apparently chaotic regions and the regular regions are finely interspersed with each other, in a way which we have not found elsewhere in the literature. We have performed simulations also in the range of 关1, 1.3兴 and have only plotted the last 103 points out of 107 collisions, as in the previous case. In this range, all attractors have been found to be periodic orbits, which are reached well before 107 collisions. In other words, the stationary state is rapidly reached with large electric fields as expected for the correspondingly high dissipations. How conclusive are these results? Again, when a periodic steady state is reached, the situation is clear, while doubts remain when the steady state looks chaotic. Therefore we have considered the range of 关0.77, 0.86兴: a rather wide range, apparently chaotic, but also characterized by high dissipation, which favors ordered dynamics. Running trajectoλ
ries of 108 collisions and plotting the last 103 points, the apparently weakly chaotic states survived, despite the high dissipation. This study combined with the analysis of the largest Lyapunov exponent makes chaos quite plausible for these fields, although we cannot exclude that the trajectories collapse on periodic orbits after much longer times. On the other hand, there is no purpose in pushing further this analysis since it is bound to remain uncertain. Here, it suffices to have uncovered a rather peculiar behavior, unexpected for simple dynamical systems. Indeed, in our model, the passage from low-period attractors to apparently chaotic steady states is rather abrupt and seems to be discontinuous. Moreover, the periodic attractors always seem to coexist with transiently chaotic states. Differently, standard bifurcation scenarios are characterized by a gradual growth of the period of the attracting orbits; the orbits are globally attracting and lead to fast convergence of the trajectories toward them. To further investi-
η
0.06
1 0.8 0.6
0.04
0.4
0.02
0.2 0
0
-0.2
-0.02 -0.4
-0.04
-0.6 -0.8
-0.06 10000
n 100000
1e+06
1e+07
1e+08
-1
1e+09
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
r
5.5
FIG. 8. Behavior of the finite time Lyapunov exponents 1,2 with the number of collisions n, for ⑀ = 0.087 共left panel兲. The logarithmic scale in n clearly separates the initial, intermediate, and asymptotic regimes. The exponents rapidly converge to one positive and one negative value, which persist for very long, up to a number Nc of collisions, dependent on the initial condition. After Nc, both exponents converge as 1 / n to the value of −0.004 838. For random initial conditions, Nc is of order O共108兲; for initial conditions close to the asymptotic periodic orbit 共right panel兲, Nc varies irregularly between O共10兲 and O共107兲. The right panel reports the last 104 points of 3 ⫻ 108 collisions, which testifies that the motion has settled on a periodic orbit of 19 points only. The coordinates are defined as in Fig. 4.
Chaos 19, 013121 共2009兲
Chaos in the Ehrenfest gas
3
3
2.5
2.5
2
2 Angle Theta
Angle Theta
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1.5
1.5
1
1
0.5
0.5
0
0 0
0.1
0.2
0.3
0.4
0.5 Electric Field
0.6
0.7
0.8
0.9
1
1
1.05
1.1
1.15
1.2
1.25
1.3
Electric Field
FIG. 9. The upper half of the projection of the multifurcation diagram onto the -axis, for ⑀ in the range of 关0.01, 1兴. The last 103 points of trajectories with length of 107 collisions have been plotted 共left panel兲. The multifurcation diagram in the 共⑀ ; 兲 plane, for ⑀ in the range of 关1, 1.3兴, from the last 103 collisions out of a trajectory of 107 collisions 共right panel兲.
gate this behavior, we have honed the range of 关0.77, 0.78兴 with a step of 0.001 simulating trajectories of 108 collisions, finding a 共quasi兲periodic orbit at ⑀ = 0.771 and another one at ⑀ = 0.777, with apparently chaotic trajectories in the middle. Thus, we have further honed the range of 关0.770, 0.771兴 with a step of 0.0001 to find that a jump from 共quasi兲periodic orbits to apparent chaos happens for a field variation of just 10−5. We have also performed simulations in the range of 关0.771, 0.772兴, namely, from the 共quasi兲periodic orbit at ⑀ = 0.771 to an apparently chaotic case at ⑀ = 0.772. Plotting the last 103 points of trajectories of 108 collisions, apparent chaos and 共quasi兲periodic orbits appear again to be finely intertwined. The range of 关0.18, 0.38兴 was similarly studied, and some of the cases that appeared to be chaotic after 107 collisions turned into 共quasi兲periodic after 108, but not all. Therefore, the duration of the transients is exceedingly long in all cases, which is a manifestation of either very small basins of attraction or of the smallness of the stable islands, whose size would then vary wildly with the field. Although this is not conclusive evidence, our analysis supports the idea that there could be a discontinuous transition between chaos and regular motion, in the nonequilibrium Ehrenfest gas.
words it seems that the effect of the electric field prevails on the geometry effects. To analyze more carefully this fact, we have honed the interval of 关0.09, 0.1兴 for sx, with a step of 0.0005 and for trajectories of 108 collisions. A sudden transition between apparent chaos and periodic orbits happens at sx = 0.099. For ⑀ = 0.014, sx was taken in 关0, 1兴 with a step of 0.01, for trajectories with length of 107 collisions. We found that the situation is slightly different from the small field case. The apparently chaotic steady states occupy a smaller region of the phase space than the steady states of the case of ⑀ = 0.374. For instance, the asymptotic state of ⑀ = 0.014 seems to fill almost completely the space , differently from the case of ⑀ = 0.374. Increasing the length of the trajectories up to 5 ⫻ 107 collisions, various apparently chaotic cases reduce to 共quasi兲periodic orbits, while other survive. Finally, we increased the number of collisions up to 108 for ⑀ = 0.014, again finding that some apparently chaotic cases turned into quasiperiodic cases, while other survived. We conclude that the dependence of the attractors on the parameter sx is qualitatively similar to the dependence on ⑀, although at times the dependence on ⑀ prevails. VI. CONCLUSION
B. Dependence on the geometry
In this section we outline the behavior of the attractors as functions of one of the parameters that determine the shape of the billiard: sx. We take two electric fields 共one large and one small兲 for which we had found apparently chaotic behaviors. Let ⑀ = 0.374 be the electric field, L = 1.291, and sy = 1.1; we study the attractor reliance on sx 苸 关0 , 1兴 with a step of 0.01. We considered trajectories with length of 107 collisions and looked at the last 103 points. We found a complicated scenario, analogous to the one for the dependence on ⑀, as sx was varied. In the chosen range, there are more cases with apparently chaotic attractors than with 共quasi兲periodic steady states. Increasing the number of collisions in each trajectory up to 5 ⫻ 107, we find that apparent chaos persists; in other
In this paper we have examined the nonequilibrium version of the Ehrenfest gas, which is a billiard model with an electric field and a Gaussian thermostat, whose point particle moves in the plane and undergoes elastic collisions with rhomboidal obstacles. The motivation was to understand the dynamics of this nonequilibrium model, whose obstacles have flat surfaces and hence do not defocus the trajectories, while its thermostat, which makes the dynamics dissipative, does focus them.26 We have shown that periodic orbits with one positive Lyapunov exponent embedded in what appear to be chaotic attractors do exist. Our numerical results have identified electric fields whose dynamics is strongly suggested to be chaotic, although conclusive results are out of reach at present because of a very peculiar phenomena. The stationary state, even if attracting and trivial, requires very
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long times to be reached because it coexists with a transiently chaotic state that covers most of the phase space. The dependence on the model parameters of the steady state behavior also presents peculiar features that, to the best of our knowledge, have not been observed before: a very irregular, possibly discontinuous dependence of the attracting orbit and/or of the size and shape of the stable islands, on ⑀ and sx. The peculiarity remains even if it is eventually proved that all steady states are periodic because of their coexistence with transiently chaotic states and because of the consequent irregular behavior of the convergence rates. As the sum of the Lyapunov exponents is proportional to the traveled distance 关cf. Eq. 共11兲兴, very irregular transport properties are obtained as well, perhaps as irregular as those conjectured for other low-dimensional dynamical systems.27 ACKNOWLEDGMENTS
We would like to thank M. P. Wojtkowski for his useful comments on a preliminary version of this paper and for enlightening discussions. We are indebted to O. G. Jepps and C. M. Monasterio for continuing encouragement and suggestions. C.B. was partly funded by the Lagrange Foundation, Torino, Italy. B. Moran and W. G. Hoover, J. Stat. Phys. 48, 709 共1987兲. J. Lloyd, M. Niemeyer, L. Rondoni, and G. Morriss, Chaos 5, 536 共1995兲. 3 S. Lepri, L. Rondoni, and G. Benettin, J. Stat. Phys. 99, 857 共2000兲. 4 G. Benettin and L. Rondoni, Math. Phys. Electron. J. 7, 1 共2001兲. 5 P. Ehrenfest and T. Ehrenfest, The Conceptual Foundations of the 1 2
Chaos 19, 013121 共2009兲 Statistical Approach in Mechanics 共Cornell University Press, Ithaca, NY, 1959兲, pp. 10–13. 6 C. P. Dettman, E. G. D. Cohen, and H. Van Beijeren, Nature 共London兲 401, 875 共1999兲. 7 F. Cecconi, D. del-Castillo-Negrete, M. Falcioni, and A. Vulpiani, Physica D 180, 129 共2003兲. 8 D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 共1993兲. 9 D. J. Evans and D. J. Searles, Phys. Rev. E 50, 1645 共1994兲. 10 G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 共1995兲. 11 G. Gallavotti and E. G. D. Cohen, J. Stat. Phys. 80, 931 共1995兲. 12 N. I. Chernov, Ann. Henri Poincare 2, 197 共2001兲. 13 M. P. Wojtkowski, J. Math. Pures Appl. 79, 953 共2000兲. 14 O. G. Jepps, C. Bianca, and L. Rondoni, Chaos 18, 013127 共2008兲. 15 O. Jepps and L. Rondoni, J. Phys. A 39, 1311 共2006兲. 16 M. P. Wojtkowski and C. Liverani, Commun. Math. Phys. 194, 47 共1998兲. 17 C. P. Dettmann and G. P. Morriss, Phys. Rev. E 53, 5502 共1996兲. 18 C. P. Dettmann and G. P. Morriss, Phys. Rev. E 54, 2495 共1996兲. 19 C. Bianca, “Chaotic and polygonal billiards as models of mass transport in microporous media,” Ph.D. thesis, Politecnico di Torino, 2008. 20 S. G. Krantz and H. R. Parks, The Implicit Function Theorem: History, Theory, and Applications 共Birkhauser, Boston, 2002兲. 21 G. Benettin, L. Galgani, A. Giorgilli, and J. M. Strelcyn, Meccanica 15, 9 共1980兲. 22 E. Gutkin, J. Stat. Phys. 83, 7 共1996兲. 23 As in Ref. 6, it is always difficult to decide whether these orbits are periodic or quasiperiodic. 24 Incidentally, as common in nonequilibrium billiards 共Ref. 5兲, the second Lyapunov exponent takes the same value as the first because the eigenvalues of the stability matrix are complex conjugate. 25 E. Ott and T. Tél, Chaos 3, 417 共1993兲. 26 The dissipation produced by the Gaussian thermostat sharply differentiates our dynamics from Hamiltonian dynamics, although both preserve the total energy. Indeed, in our case, the total energy equals the kinetic energy, which is a constant of motion. 27 R. Klages, Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics 共World Scientific, Singapore, 2007兲.
CHAOS 19, 013122 共2009兲
Node-to-node pinning control of complex networks Maurizio Porfiria兲 and Francesca Fiorilli Department of Mechanical and Aerospace Engineering, Polytechnic Institute of New York University, Brooklyn, New York 11201, USA
共Received 8 August 2008; accepted 20 January 2009; published online 3 March 2009兲 In this paper, we study pinning controllability of oscillator networks. We present necessary conditions for network pinning controllability based on the spectral properties of the oscillator network and the individual oscillator dynamics. We define a performance metric for pinning-control systems based on the location of pinned sites, the pinning-control gains, and the network topology. We show that for any network structure, uniform pinning of all the network nodes maximizes the pinningcontrol performance. We propose the node-to-node pinning-control strategy to optimize the control performance while avoiding to simultaneously control all the network sites. In this novel strategy, the pinning-control action rapidly switches from one node to another with the goal of taming the oscillator network dynamics to the desired trajectory. We illustrate our findings through numerical simulations on networks of Rössler oscillators. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3080192兴 Synchronization is a topic of great interest due to its observation in a large variety of phenomena of different natures. In many biological systems, synchronization plays an important role in self-organization of organisms’ groups. Examples of synchronization include animal grouping, opinion dynamics, molecular and cellular activity, and cardiac stimulation. In this paper, we consider the problem of synchronizing the dynamics of an oscillator network onto a desired trajectory generated by an exogenous oscillator. Control actions are applied only to a selected subset of the network oscillators, and information spreading throughout the network may allow for the synchronization of the whole network dynamics. For an opinion dynamics problem, the exogenous oscillator may be considered as an opinion leader who seeks to persuade the whole social network by interacting with a pool of well-connected individuals. In this paper, we show that the most efficient and performing way to control the network dynamics is to cyclically control each oscillator for a limited amount of time, that is, to go node to node.
I. INTRODUCTION
Pinning control has been proposed in the literature1–12 as a viable strategy to drive networks of coupled oscillators onto some desired common reference trajectory. The general idea behind pinning control is to use a feedback control input on just a limited subset of the whole dynamical system, that is, to actively control only a few network nodes. Specifically, a direct control action is active only on such pinned nodes, and it is propagated to the rest of the network through the coupling among the oscillators, represented by edges in the network. Criteria for local pinning controllability of complex network based on the master stability function 共MSF兲, see, for a兲
Electronic mail: mporfi
[email protected].
1054-1500/2009/19共1兲/013122/11/$25.00
example, Refs. 13–15, have been proposed in Ref. 9. In Ref. 9, it is shown that studying pinning controllability of a complex network is equivalent to analyzing the synchronizability of an augmented network. The augmented network includes an additional oscillator, whose trajectory corresponds to the reference trajectory to be tracked by the network, and directed edges linking this node to the pinned sites. A similar approach has been used in Ref. 6 to analyze pinning controllability in a discrete time setting. For a class of network oscillators, the results of the MSF approach may be extended to global and exponential network pinning controllability, see, for example, Refs. 3, 4, 10, and 12. In these works, Lyapunov stability theory is used to prove global exponential stability of the synchronization error dynamics under stricter conditions on the oscillator dynamics. For example, in Ref. 10 a simple and effective criterion for assessing global and exponential network pinning controllability in terms of the network connectivity and the fraction of pinned sites have been proposed. Lyapunov stability theory has also been used in Refs. 4, 5, and 11 to analyze adaptive pinning-control systems. Pinning sites and control gain selection for optimal synchronization have been analyzed in Refs. 3, 7, and 8. In Ref. 3, it is shown that random selection of pinned sites is effective in controlling random networks, whereas pinning nodes with high degree may facilitate the control of scale-free networks. In Ref. 7, it is shown that star-shaped networks are easily controllable when pinned at a large number of sites with small degree. In Ref. 8, the effects of degree distributions, degree correlations, and community structure on network controllability are studied. Based on extensive numerical results on a variety of network structures, in Ref. 8 it is conjectured that pinned sites have to be uniformly spread out in the network to enhance the synchronization performance. In this paper, we build on the framework proposed in Ref. 9 and on the findings in Refs. 3, 7, and 8 to 共i兲 provide general guidelines for optimally selecting pinned sites and pinning-
19, 013122-1
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M. Porfiri and F. Fiorilli
control gains, and ii兲 to introduce an effective and practically feasible pinning-control strategy that we call node-to-node pinning-control. We analyze the influence of the overall pinning-control gain, the number of pinned sites, the gain distribution, and the location of pinned sites on the network pinningcontrollability. We provide a sharp lower bound for the overall gain that is required to pinning control the network. Moreover, we establish a few necessary conditions for the gain and degree distributions and for the spectral properties of the oscillator network to guarantee pinning controllability. These results partially extend those presented in Ref. 16 for synchronization problems to pinning control. Our findings are derived by studying the spectral properties of the generalized Laplacian matrix that describes the augmented network topology. Further, we show that pinning-control performance is optimized by simultaneously pinning all the network nodes with the same pinning gain, regardless of the network topology. We propose to recreate this optimal scenario by cyclically pinning a single node in the network with a sufficiently fast-switching period. We refer to this technique as node-to-node pinning, in the sense that the pinningcontrol action switches from one node to another trying to tame the oscillator network dynamics to its own state. If the oscillator dynamics is sufficiently slower than the switching period, the oscillator network is virtually pinned at every site. In other words, the augmented network may be viewed as a time-varying switching network that under fastswitching conditions inherits the synchronization properties of its time-averaged counterpart, see, for example, Ref. 17. The correlation between the synchronizability of switching time-varying networks and the synchronizability of their time-average counterparts has been explored in Refs. 17–24. Local synchronization of switching time-varying networks is investigated in Refs. 17, 19, 20, and 24 using the MSF approach, perturbation methods, and Lyapunov stability tools. Global synchronization of two periodically or stochastically coupled oscillators in master-slave configuration is analytically and experimentally studied in Refs. 18, 22, and 23. Consensus problems over stochastic networks are analyzed in Ref. 21. In general, these works have shown that synchronization of a time-varying switching network is possible if its time-average network supports synchronization and if the switching period is sufficiently smaller than the oscillator individual dynamics. The paper is organized as follows. In Sec. II, we describe the pinning-controllability problem, we recall the key findings of Ref. 9, and we define a performance index for pinning control. In Sec. III, we analyze the effect of gain and node selection on pinning-control performance. We present necessary conditions for pinning control in terms of the control gains, the network degree distribution, and the largest eigenvalue of the network Laplacian matrix. We specialize these conditions to a representative set of networks to better clarify the role of control gains, location of pinned nodes, and network topology on the pinning-controllability performance. We optimize the network performance, and we show that node-to-node pinning control optimizes the control performance. In Sec. IV, we illustrate our findings through nu-
merical simulations on random and scale-free networks of Rössler oscillators. Section V is left for conclusions. Our notation throughout is standard. Z+ refers to the set of non-negative integers. 储 · 储 refers to the Euclidean norm in Rm or corresponding induced norm in Rm⫻m, with m 苸 Z+. The vector in Rm that consists of all unit entries is denoted 1m = 关1 , . . . , 1兴T. Im is the m ⫻ m identity matrix. The vector i is the vector whose ith entry is equal to 1 and all the em others are zero. The smallest and largest eigenvalues of a symmetric matrix A 苸 Rm⫻m are indicated with min共A兲 and max共A兲, respectively. The algebraic spectrum of A, say 兵i共A兲其m i=1, is ordered so that min共A兲 = 1共A兲 ⱕ 2共A兲 ⱕ ¯ ⱕ m−1共A兲 ⱕ m共A兲 = max共A兲. The unit norm eigenvector of A corresponding to i共A兲 is called vi共A兲. II. PINNING CONTROL A. Closed-loop dynamics of pinning-controlled networks
We consider a network of N mutually coupled oscillators. We assume that the communication network is time invariant, unity weighted, and bidirectional. We describe the communication network through a unity-weighted graph G = 共V , E兲. The vertex set V comprises the N oscillators’ set. The edge set E describes the communication channels, that is, 共i , j兲 苸 E with i , j 苸 V if the ith and jth oscillators are coupled. We algebraically describe the graph G through the graph Laplacian L, see, for example, Ref. 25, that is defined in terms of the degree matrix D and the adjacency matrix A by L = D − A. The degree matrix is a diagonal matrix whose diagonal elements are equal to the connectivity of the corresponding nodes. The entries of the adjacency matrix are 1 in correspondence to interconnected nodes and 0 otherwise. The minimum degree is denoted by ␦, and ⌬ is the maximum degree. The Laplacian matrix is positive semidefinite and has zero row sum.25 This implies that 1N 苸 Null共L兲
共1兲
and that min共L兲 = 0, where Null共L兲 is the null space of L. The smallest nonzero eigenvalue 2共L兲 is generally referred to as the graph algebraic connectivity and is different than zero if and only if the graph is connected.25 It has been shown that the algebraic connectivity is always less than or equal to the smallest degree connectivity, that is, 2共L兲 ⱕ ␦.25 The eigenvalues of the graph Laplacian have been extensively studied in the literature,25 and several sharp bounds for its eigenvalues have been recently established, see, for example, Refs. 26–31. For instance, in Ref. 26, it has been shown that the largest eigenvalue of the graph Laplacian of a connected graph is greater than or equal to ⌬ + 1 and that the equality holds if and only if ⌬ = N − 1. Sharp upper bounds for max共L兲 have been proposed in Refs. 27–29 and 31. Sharp lower bounds for max共L兲 have been proposed in Ref. 31. A comprehensive review of available lower and upper bounds for the algebraic connectivity has been presented in Ref. 30. Properties of the eigenvectors of L are discussed in Ref. 32. For a controlled network of N identical oscillators, the time evolution of the ith oscillator is described by
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Node-to-node pinning control
By block-diagonalizing equation 共4兲, we find N decoupled differential equations of the form
N
x˙i共t兲 = f共xi共t兲兲 − 兺 lijh共x j共t兲兲 + ui共t兲, j=1
xi共0兲 = xi0,
i = 1, . . . ,N,
共2兲
t ⱖ 0,
where xi共t兲 苸 Rn is the state of the ith oscillator, xi0 is the initial condition of the ith oscillator, f describes the oscillators’ individual dynamics, h is the inner linking function that characterizes the coupling between the states of coupled oscillators, ⬎ 0 is a control parameter that partially assigns coupling strength between oscillators, ui共t兲 is the control input to oscillator i, and scalars lij’s are the elements of the graph Laplacian L, that is, L = 关lij兴. Control inputs are applied only to a selected number r ⱕ N of network oscillators with the goal of taming the oscillators’ dynamics to a reference trajectory s共t兲. The reference trajectory is described by an independent pinner oscillator with unforced dynamics s˙共t兲 = f共s共t兲兲, acting as a master/ leader for the oscillator network. We define a state feedback control law as ui共t兲 = ki共h共s共t兲兲 − h共xi共t兲兲兲, where h共s共t兲兲 − h共xi共t兲兲 is a measure of the error between the reference dynamics and the dynamics of the ith oscillator. The positive gain ki quantifies the pinning-control effort at the ith node. Therefore, ki = 0 if the ith node is not pinned and the number of nonzero gains ki is equal to r. The overall cost of the pinning-control system can be quantified by the overall control gain C that is defined by N
C = 兺 ki .
共3兲
˙ i共t兲 = 关J f 共s共t兲兲 − i共M兲Jh共s共t兲兲兴i共t兲,
i = 1, . . . ,N, 共6兲
where i describes the error dynamics on the eigenspace of M related to i共M兲. For each i, the blocks in Eq. 共6兲 are the same. Therefore, we associate with the pinningcontrollability problem the following master equation:
˙ 共t兲 = 关J f 共s共t兲兲 − ␣Jh共s共t兲兲兴共t兲,
共7兲
where 共t兲 is a vector function in Rn and ␣ a positive real constant. By analyzing the linear stability of Eq. 共7兲 in terms of the parameter ␣, the network pinning controllability can be assessed. More specifically, one can construct the socalled MSF for any pair of functions f and h by computing the largest Lyapunov exponent of Eq. 共7兲 as a function of ␣. Thus, the network is pinning controllable if and only if the MSF is negative for ␣ = i共M兲, i = 1 , . . . , N. For most of the studied oscillator networks, see, for example, Ref. 15, it is found that the MSF is negative either within an unbounded region ␣ ⬎ ␣1 or in a bounded region 共␣1 , ␣2兲. In the case of an unbounded stability region, pinning controllability is possible if and only if min共M兲 ⬎ ␣1 .
共8兲
In the case of a bounded stability region, pinning controllability is possible if and only if Eq. 共8兲 holds and
i=1
We note that in the case f = 0, the pinning-controllability problem reduces to a consensus problem in the presence of group leaders, see, for example, Ref. 33. B. Performance index of pinning-control systems
Following Ref. 9, we assess the network pinning controllability by studying the linear stability of the synchronization manifold x1共t兲 = ¯ = xN共t兲 = s共t兲. By linearizing the network dynamics 共2兲 in the neighborhood of the reference trajectory, we find
i共t兲 = J f 共s共t兲兲i共t兲 − Jh共s共t兲兲 兺 Mij j共t兲,
i = 1, . . . ,N,
j=1
max共M兲 ⬍ ␣2 .
共9兲
We note that network pinning controllability may also be possible if the network is not synchronizable, that is, if the oscillators do not synchronize without the pinning-control system. Network synchronization is possible if and only if 2共L兲 ⬎ ␣1 and max共L兲 ⬍ ␣2, see, for example, Ref. 15. For instance, a network of isolated oscillators may be pinning controlled by pinning all the network nodes with a constant gain smaller than ␣2 and greater than ␣1. Based on these necessary and sufficient conditions, for a given network topology G, coupling strength , and overall control gain C, we define the following performance index for a pinning-control strategy:
共4兲 where J f and Jh are the Jacobians of the functions f and h and i共t兲 = s共t兲 − xi共t兲 is the error variation. The matrix M = 关Mij兴 is equal to M = L + K,
共5兲
where K = Diag关k1 , . . . , kN兴 is the gain matrix of the pinningcontrol system. The matrix M is symmetric and positive semidefinite, since both K and L are symmetric and positive semidefinite and is a positive quantity. Therefore its eigenvalues are real and non-negative. We note that the matrix M can be viewed as a generalized Laplacian of the graph G, see, for example, Refs. 25 and 32. The pinning-control actions may be viewed as weighted self-loops in the graph.
⌫共K兲 ⬅ min共M兲.
共10兲
We use the index 共10兲 to assess the performance of a given pinning-control strategy of total control authority gain C. More specifically, we seek for determining the distribution of control gains k1 , . . . , kN that maximizes the performance index 共10兲. For oscillators with unbounded stability region, pinning controllability is exclusively dictated by ⌫. On the other hand, in the case of oscillators with bounded stability region, it is also important to minimize the maximum eigenvalue of the matrix M. Further, we note that global pinning controllability of a class of networks is possible if min共M兲 is larger than a threshold value related to properties of the individual oscillator dynamics, see, for example, Ref. 10.
013122-4
Chaos 19, 013122 共2009兲
M. Porfiri and F. Fiorilli
A. Necessary conditions for pinning controllability
direct constraint on the network topology. Condition 共16b兲 is particularly relevant in the case only a selected small set of network nodes is pinned.
The extremal eigenvalues of the matrix M can be characterized in terms of the Rayleigh quotient using the wellknown Courant–Fisher theorem 共see, for example, Ref. 34兲,
B. Optimization of pinning-control performance: Node-to-node pinning control
III. PERFORMANCE ANALYSIS AND OPTIMIZATION OF PINNING-CONTROL SYSTEMS
min共M兲 =
max共M兲 =
min
xTMx
x苸RN,x⫽0
max x苸RN,x⫽0
T
x x xTMx x Tx
,
共11a兲
.
共11b兲
By substituting 1N for x in Eq. 共11a兲 and by accounting for Eq. 共1兲, we find the following bound on the performance index 共10兲: C ⌫共K兲 ⱕ . N
共12兲
This bound is independent of the network topology, including the algebraic connectivity and the distribution of the pinning-control gains. Similarly, by substituting for x in Eq. 共11b兲 the unit eigenvector of L associated with its maximum eigenvalue, we find max共M兲 ⱖ max共L兲 + QK共vN共L兲兲,
共13兲
where we defined the quadratic function QK共v兲 = vTKv. The function QK evaluated on unit norm vectors is smaller than kmax and larger than kmin, where kmin and kmax are the minimum and maximum values of the control gains k1 , . . . , kN, respectively. Additional necessary conditions can be derived by substituting for x in Eq. 共11b兲 the vector eiN, that is, max共M兲 ⱖ max 共di + ki兲 ⱖ i=1,. . .,N
再
⌬ + kmin ␦ + kmax .
冎
共14兲
By combining inequalities 共13兲 and 共14兲, we find the following lower bounds for the maximum eigenvalue of M: max共M兲 ⱖ
再
max共L兲 + kmin ␦ + kmax ,
冎
共15兲
where we used max共L兲 ⱖ ⌬ + 1 to exclude one of the inequalities in Eq. 共14兲, see, for example, Ref. 26. By combining inequalities 共8兲, 共9兲, 共12兲, and 共15兲, we obtain the following set of necessary conditions for pinning controllability: C ⬎ ␣1 , N
共16a兲
␦ + kmax ⬍ ␣2 ,
共16b兲
max共L兲 + kmin ⬍ ␣2 .
共16c兲
We note that condition 共16a兲 sets a minimum value for the average pinning-control gain. Condition 共16b兲 establishes a maximum value for the largest pinning-control gain in terms of the minimum graph degree. Inequality 共16c兲 imposes a
Equation 共12兲 provides an upper bound for the pinningcontrol performance in terms of the average pinning-control gain, that is, the ratio between the total control gain C and the number of network nodes N. This upper bound can be achieved by simultaneously pinning all the network nodes with the same control gain C / N. Indeed, M = L + 共C / N兲IN and for i = 1 , . . . , N, i共M兲 = i共L兲 +
C . N
共17兲
Hence, the extremal eigenvalues of M are given by min共M兲 =
C , N
max共M兲 = max共L兲 +
C . N
共18兲
Uniform pinning of the whole oscillator network allows for maximizing the performance index 共10兲, that is, for optimally managing the overall control gain. The largest eigenvalue of M is not generally minimized by the uniform pinning strategy. Indeed, QK共vN共L兲兲 in Eq. 共13兲 is bounded from below by C mini=1,. . .,N v2Ni共L兲 which yields Eq. 共18兲 only when mini=1,. . .,N v2Ni共L兲 = 1 / N. Nevertheless, as will be made clear in what follows through the analysis of representative networks, uniform pinning minimizes the maximum eigenvalue of M for some elementary networks and provides very good suboptimal solutions for a variety of complex networks. From conditions 共8兲 and 共9兲, it follows that for a bounded stability region, the oscillator network is pinning controllable by uniformly pinning all its nodes, if and only if
␣1N ⬍ C ⬍ ␣2N − max共L兲N.
共19兲
We note that the upper bound in Eq. 共19兲 goes to infinity if the stability region in the MSF is unbounded. Further, we note that disconnected networks can be pinning controlled by uniformly pinning all their nodes. In most applications, it may be practically difficult or even impossible to simultaneously pin the entire node set. Uniform pinning of the whole network may be alternatively achieved by cyclically pinning each network node at a high switching rate. In other words, uniform pinning of the whole network may be recreated through a single pinning-control action that is periodically applied at each network site for a fixed time duration. We refer to this pinning-control technique as node-to-node pinning. In this case, the control input applied to the ith oscillator is equal to ui共t兲 = ki共t兲共h共s共t兲兲 − h共xi共t兲兲兲, where the gain ki is a periodic function of time of period T. In the interval 关0 , T兲, the gain ki共t兲 equals C if t 苸 关共i − 1兲T / N , iT / N兲 and is zero otherwise. Indeed, the matrix M in Eq. 共5兲 is time-varying and its time-average is
013122-5
1 T
冕
T
0
Chaos 19, 013122 共2009兲
Node-to-node pinning control
共lij + ki共兲兲d = lij +
C . N
共20兲
Therefore, the time average of the generalized Laplacian is equal to L + 共C / N兲IN, that corresponds to uniformly pinning the whole node set. By referring to the claims and hypotheses presented in Refs. 17, 19, and 20, we find that if the network is pinning controllable by the uniform pinning, that is, inequalities 共19兲 are satisfied, then the node-to-node pinning-control tames the oscillators’ dynamics onto the reference trajectory when the switching period T is sufficiently small. The duration of the minimum switching period T required for guaranteeing that the average system properties are inherited by the periodically switching system can be estimated using the bounds in Ref. 17. We note that for oscillator networks with unbounded stability region, these results may be extended to global pinning controllability by combining the conditions on global pinning controllability presented in Ref. 10 with the global fast-switching results presented in Ref. 22. Moreover, the pinning node selection in node-to-node pinning control can also be driven by a random variable rather than be purely deterministic. In this case, the recent findings on synchronization of time-varying stochastic networks in Refs. 21, 23, and 24 can be used to estimate the minimum switching rate that guarantees node-to-node pinning controllability. For a positively weighted bidirectional network, see, for example, Ref. 21, the graph Laplacian remains a positive semidefinite matrix that has 1N in its null space. Thus, the performance index ⌫共K兲 in Eq. 共10兲 is still bounded from above by C / N and is optimized by simultaneously pinning all the network nodes with the same control gain. In other words, pinning-control performance is maximized by uniformly distributing the overall control gain throughout the oscillator network even if link heterogeneities are present in the network topology. Thus, node-to-node pinning control is optimal also for weighted networks. We note that the proposed pinning-control strategy is not directly applicable to directed networks since its optimality is based on the Courant–Fisher theorem for symmetric matrices. The analysis can be potentially extended to time-varying switching networks that change their topology sufficiently fast as compared to the individual oscillator dynamics.
C. Sample problems: Lattice, complete, and star networks
共21兲
where Circ共v兲 is the circulant matrix constructed from the vector v, see, for example, Ref. 34. If N is an even number, the largest Laplacian eigenvalue is equal to 4 and the ith entry of the corresponding eigenvector is 共−1兲i / 冑N. In this case, the maximum eigenvalue of the matrix M can be bounded using Eq. 共13兲 with QK共vN共L兲兲 = C / N. Therefore, for a lattice network with an even number of nodes, the uniform
共22兲
L = NIN − 1N1TN .
The maximum Laplacian eigenvalue is N and it has multiplicity equal to N − 1, that is, its eigenspace is equal to the orthogonal complement of the vector space generated by 1N. Therefore, the vector whose ith entry equals 共−1兲i / 冑N is an eigenvector related to the maximal Laplacian eigenvalue. Thus, the maximum eigenvalue of the matrix M can be bounded using Eq. 共13兲 with QK共vN共L兲兲 = C / N. This shows that the uniform pinning strategy minimizes the maximum eigenvalue of M for a fully connected network. The graph Laplacian of a star network whose Nth node is connected to all the other nodes is L=
冋
− 1N−1
IN−1 −
T 1N−1
N−1
册
共23兲
.
The maximum Laplacian eigenvalue is N and its corresponding eigenvector is vN共L兲 = 共1 / 冑N2 − N兲关−1 , . . . , −1 , N − 1兴. In this case, the maximum eigenvalue of the matrix M can be bounded using Eq. 共13兲 with QK共vN共L兲兲 = 共C + kN共N2 − 2N兲兲 / 冑N2 − N, which is in turn minimized by setting kN = 0. Therefore, for moderately large networks, that is, for N Ⰷ 1 the uniform pinning strategy approaches the lower bound predicted by Eq. 共13兲 with kN = 0. We further note that, when the central node of the star topology is pinned with the whole available gain C, we have 1 min共M兲 = 2 共C + N − 冑C2 + 2共N − 2兲C + N22兲,
共24a兲 max共M兲 = 21 共C + N + 冑C2 + 2共N − 2兲C + N22兲, 共24b兲 whereas when pinning all the peripheral nodes with a uniform gain C / 共N − 1兲, we have min共M兲 =
冉
1 C + N 2 N−1 −
max共M兲 =
The graph Laplacian of a lattice network of N nodes is L = Circ共关2,− 1,0, . . . ,0,− 1兴兲,
pinning control maximizes the performance index 共10兲 while minimizing the maximum eigenvalue of M. The graph Laplacian of a fully connected network is
冑
冉
1 C + N 2 N−1 +
冊
N−2 C2 C + N 2 2 , −2 共N − 1兲2 N−1
冑
冊
N−2 C2 C + N 2 2 . 2 −2 共N − 1兲 N−1
共25a兲
共25b兲
By comparing Eqs. 共24兲 and 共25兲, we observe that pinning the central node yields lower performance than pinning the peripheral nodes. Indeed, Eq. 共24a兲 is smaller than Eq. 共25a兲 and Eq. 共24b兲 is greater than Eq. 共25b兲. Moreover, pinning the peripheral nodes may decrease the largest eigenvalue of the matrix M as compared to uniform pinning in the case of moderately low overall pinning gain and small scale networks. Nevertheless, as the network size increases, the rela-
013122-6
Chaos 19, 013122 共2009兲
M. Porfiri and F. Fiorilli TABLE I. Key features of the studied networks.
Random network Scale-free network
2共L兲
max共L兲
␦
⌬
mini=1,. . .,N v2Ni共L兲
maxi=1,. . .,N v2Ni共L兲
11.02 0.92
45.57 157.49
12 1
43 156
0 0
0.80 0.98
tive difference between the maximal eigenvalues resulting from the two strategies approaches zero. IV. NUMERICAL ILLUSTRATION A. Pinning-controllability analysis of complex networks
We numerically analyze two different network topologies: an Erdos–Renyi random network and a scale-free network. Both networks comprise N = 1000 vertices and are created using the software PAJEK.35 The random network has average degree ¯d = 25.56. Therefore, the probability of a link to be present is ¯d / 2共N − 1兲 = 1.28%. The scale-free network is generated using the model proposed in Ref. 36. The baseline network comprises 100 nodes whose probability to be connected is uniform and equal to 30%. The maximum number of added edges at each step of the network growth is 1000. The probability of preferential attachment during the growth process is 60% that leads to a power law degree distribution whose exponent is 1 + 1 / ␣ = 2.67, see, Ref. 36. The average degree of the scale-free network is ¯d = 25.55. The numerical values of the algebraic connectivity and the Laplacian spectral radius, that is, 2共L兲 and max共L兲, are reported in Table I along with other relevant network properties. More specifically, Table I reports the minimum degree, the maximum degree, and the smallest and largest entries of the square of the Laplacian eigenvector vN共L兲. We note that both these networks are connected since the algebraic connectivity is nonzero. We further note that the largest Laplacian eigenvalue is relatively close to the maximum degrees. In addition, for both the studied networks the largest entry of the absolute value of vN共L兲 is attained in correspondence to the node with highest degree connectivity. We note that the maximum degree and consequently the largest Laplacian eigenvalue of the scale-free network are considerably larger than the corresponding quantities computed for the random network. This implies a more restricted gain selection range for synchronizability and pinning controllability of the scale-free network as compared to the random network, see, for example, Eq. 共16c兲. The full distributions of the eigenvalues of the Laplacian matrix for the two networks are illustrated for completeness in Fig. 1. The eigenvalue range is divided into 20 parts for better data presentation. The distribution of the components of the vector vN共L兲 is reported in Fig. 2. Figure 2 can be used for selecting the pinning nodes that minimize QK共vN共L兲兲, that is, for minimizing the bound in Eq. 共13兲. Global measures of the Laplacian eigenvectors have been introduced in Ref. 37. Here, we simply note that the entries of the eigenvectors associated with the largest Laplacian eigenvalues are zero almost everywhere but at a few network sites. Therefore,
pinning at these sites may increase the largest eigenvalue of M, thereby reducing the range of control gains for pinning control. For the studied networks, pinning at the site at which the absolute value vN共L兲 is maximized corresponds to the network selective pinning at the node with highest degree. This pinning strategy has been proposed in Refs. 3 and 9 as a viable strategy to enhance network controllability. Simulations of this pinning strategy for the two studied networks are reported in what follows. In order to assess the performance of the uniform pinning-control strategy for the two networks under examination, we analyze the extremal eigenvalues of the matrix M in terms of the overall pinning-control gain and total number of pinned nodes r. We assume that the control gain acting on each pinned node is equal to C / r and that the r pinned nodes are randomly selected with a uniform distribution. This pinning-control strategy is generally referred to as random pinning, see, for example, Ref. 9. We assume = 0.02 and we vary C from 0 to 1000 while changing r until the complete covering of the network. This parameter selection is motivated by the analysis of Rössler oscillators in Sec. IV B. We note that the spectral properties of M at a different value of the mutual coupling 쐓 can be retrieved from the data reported below at = 0.02 by noting that 共쐓L + K兲 = 共쐓 / 兲共L + 共 / 쐓兲K兲. Therefore, from a qualitative standpoint, increasing the mutual coupling corresponds to decreasing the total control gain; vice versa, decreasing the mutual coupling corresponds to increasing the overall control gain.
Random network 140
140
120
120
100
100
80
80
60
60
40
40
20
20
0
0
0.5
1
λ(L)/d
1.5
2
0
Scale−free network
0
0.5
1
1.5
λ(L)/d
FIG. 1. Laplacian eigenvalue distribution for the random and scale-free networks.
013122-7
Chaos 19, 013122 共2009兲
Node-to-node pinning control
1 0.5
0.5
0
0
−0.5
980
vN (L)
0.02
1
Random network
0.018
r = 1000
0.016 0.014
−1 1
100
200
300
400
500
600
1
700
800
900
Scale−free network
0.012 0.01
600
0.008
r
0.5
vN (L)
800
1000
0
400
0.006
−0.5
0.004 200
−1 1
100
200
300
400
500
600
700
800
900
1000
0.002
FIG. 2. Component distribution of the eigenvector vN共L兲 for the random and scale-free networks.
In Figs. 3 and 4, we show how the performance index 共10兲 changes as a function of r and C. These pictures report in a two-dimensional contour plot the minimum eigenvalue min共M兲 as the fraction of pinned nodes is varied from 0% to 98% along with a separate plot for min共M兲 against the overall control gain in the case all the network nodes are pinned. Figures 3 and 4 show that as the number of pinned nodes increases from 0% to 98% and C is held constant, the performance index monotonically increases. For the random network, significant changes are generally observed in the whole regions 0 ⬍ C ⬍ 1000 and 0 ⬍ r ⬍ 980. For the scalefree network, relevant changes are only limited to the regions 0 ⬍ C ⬍ 40 and 0 ⬍ r ⬍ 980 and in the regions 0 ⬍ C ⬍ 1000 and 0 ⬍ r ⬍ 40. Outside these regions the performance index is relatively constant as the fraction of pinned nodes is varied from 0% to 98%. By comparing the two networks in the range of 0 ⬍ r ⬍ 980, we evince that the random network has a better propensity to be pinning controlled as compared to the scale-free network. Indeed, as the fraction of pinned 0.35
1
0
r = 1000
600
C
800
1000
1000
0.3
0.25
45
800
800
40
700 0.2
600 0.15
500 400
0.1
300 200
0.05
100 0
200
400
C
600
800
1000
0
FIG. 3. 共Color online兲 Minimum eigenvalue min共M兲 as a function of the pinned node number r and total gain C for the random network.
35
600
r
700
0
50
900
900
r
400
nodes is varied from 0% to 98% and the control gain is varied from 0 to 1000, the largest values of min共M兲 reaches approximately 0.35 for the random network and 0.02 for the scale-free network. When all the network nodes are pinned, a drastic improvement in the pinning-control performance is observed for both networks. This improvement is particularly evident at moderately large gain levels 共C ⬎ 500 for the random network and C ⬎ 100 for the scale-free network兲 at which oscillator mutual coupling is dominated by pinning control. More specifically, min共M兲 approximately increases by a factor of 3 for the random network and 50 for the scale-free network as the fraction of pinned sites is varied from 98% to 100% and C is held at its maximum value of 1000. In Figs. 5 and 6, we show how the maximal eigenvalue of M changes as a function of r and C. As made clear by these graphs, the maximal eigenvalue of M is only marginally affected by the gain level C and the number of pinned
980
0
200
FIG. 4. 共Color online兲 Minimum eigenvalue min共M兲 as a function of the pinned node number r and total gain C for the scale-free network.
0.5 0
0
30
500
25
400
20
300
15
200
10
100
5
0
0
200
400
C
600
800
1000
0
FIG. 5. 共Color online兲 Maximum eigenvalue max共M兲 as a function of the pinned node number r and total gain C for the random network.
013122-8
Chaos 19, 013122 共2009兲
M. Porfiri and F. Fiorilli
1000 900
40
700
0.004
0.004
500
25
0.003
0.003
400
20
300
15
200
10
100
5 0
200
400
C
600
800
1000
0
FIG. 6. 共Color online兲 Maximum eigenvalue max共M兲 as a function of the pinned node number r and total gain C for the scale-free network.
nodes r if more than 10% of the network nodes are pinned. In this case, the maximum eigenvalue of M is close to its lower bound max共L兲. On the other hand, if only a few network nodes are pinned max共M兲 is relatively high, as the bound in Eq. 共15兲 predicts. The results illustrated in Figs. 3–6 show that the performances of random pinning are significantly lower than uniform pinning for both the studied network structures and especially for the scale-free network. Therefore, under fast-switching conditions, node-to-node pinning control is expected to outperform random pinning since it is capable of virtually recreating a uniform pinning scenario. To better illustrate the effect of node selection on the network pinning control, in Fig. 7, we report the maximal eigenvalue of M as a function of the total control gain C in case the networks are pinned only at the node with highest degree connectivity that also corresponds to the node at which the absolute value of vN共L兲 is maximized. Figure 7
Random network
1000
900
900
800
800
700
700
600
600
λmax (M)
1000
500 400
200
200
100
100
0
0
0
500
C
1000
Scale−free network
0.001
0
0.002
0.001
0
500
C
1000
0
0
500
C
1000
FIG. 8. Minimum eigenvalue min共M兲 as a function of the total gain C for the random and scale-free network in case the networks are pinned only at the node with highest degree.
shows that by selectively pinning the network node that corresponds to the largest entry of the absolute value of vN共L兲 the maximal eigenvalue of M is a strongly increasing function of the control gain. A similar behavior is observed for star networks when pinning the central node of the topology, see Eq. 共24b兲. As a consequence, pinning control of oscillator networks whose MSF has a bounded stability region becomes impractical. This drawback is not present in the uniform pinning strategy or the node-to-node pinning control since the maximal eigenvalue of M is approximately constant as C varies, see Figs. 5 and 6. Figure 8 reports the minimum eigenvalue of M for the same conditions of Fig. 7 and shows that for the studied networks selective pinning based on the highest node degree does not provide significant benefit to network controllability. B. Time evolution of the error dynamics
x˙i1共t兲 = − xi2共t兲 − xi3共t兲,
共26a兲
x˙i2共t兲 = xi1共t兲 − c1xi2共t兲,
共26b兲
x˙i3共t兲 = c2 + 共xi1共t兲 − c3兲xi3共t兲,
共26c兲
where xi1, xi2, and xi3 are the states of the ith oscillator. Following Ref. 8, we choose c1 = 0.165, c2 = 0.2, and c3 = 10, and we select a linear inner linking function h共xi兲 = Hxi, where
400 300
0.002
We specialize our results to a network of Rössler oscillators. The dynamics of the individual Rössler oscillator is
500
300
λmin (M)
30
λmin (M)
r
0.005
35
600
λmax (M)
Scale−free network
0.005
45
800
0
Random network
50
冤 冥 1 0 0
H= 0 0 0 . 0 0 1 0
500
C
1000
FIG. 7. Maximum eigenvalue max共M兲 as a function of the total gain C for the random and scale-free network in case the networks are pinned only at the node with highest degree.
This means that the oscillators are coupled through their first and third states. In this case, the parameters ␣1 and ␣2 that define the stability region in the MSF are approximately equal to 0.2 and 5.9, respectively. The MSF is estimated
013122-9
Chaos 19, 013122 共2009兲
Node-to-node pinning control 2
1000
2
900
1.8
900
1.8
800
1.6
800
1.6
700
1.4
700
1.4
600
1.2
600
1.2
500
1
500
1
400
0.8
400
0.8
300
0.6
300
0.6
200
0.4
200
0.4
100
0.2
100
0.2
0
0
200
400
C
600
800
1000
0
FIG. 9. 共Color online兲 Synchronization error as a function of the pinned node number r and control gain C for the random network.
following the approach presented in Ref. 13 and the Lyapunov exponents are computed using the algorithm proposed in Ref. 38. We note that the random network synchronizes in the absence of pinning control for the selected value of , see Table I. On the other hand, the scale-free does not synchronize in the absence of pinning control for any selection of , see Table I. Numerical integration is performed using the built-in MATLAB solver ODE45 with relative and absolute tolerances equal to 10−5 and a “refine” factor of 4. The initial conditions for the analysis are randomly chosen in the attractor region and are held constant for each network parametric analysis. From the necessary condition in Eq. 共16a兲 we find that pinning controllability of both networks may be achieved only if C ⬎ 200. Moreover, from conditions 共16b兲 and 共16c兲 and the data in Table I, we have that the random network may be pinning controllable only if kmax ⬍ 5.7 and kmin ⬍ 5.0. Similarly, for the scale-free network pinning controllability requires that kmax ⬍ 5.9 and kmin ⬍ 2.7. From Figs. 3 and 5, we have that the random network is pinning controllable for approximately C ⬎ 220 and r ⬎ 600. From Figs. 4 and 6, we have that the scale-free network is not pinning controllable for 0 ⬍ r ⬍ 980, since min共M兲 ⬍ ␣1. Both the networks are pinning controllable by the uniform pinning strategy if and only if 200⬍ C ⬍ 5000 and 200⬍ C ⬍ 2700 for the random and scale-free networks, respectively, see Eq. 共19兲. Numerical data reported below are limited to the case C ⬍ 1000; at higher gain levels, the error may grow unbounded in a finite time in the case only few nodes are pinned. We quantify the lack of synchronization of the oscillator network by the normalized error mean, defined by E=
1
␦t 兺 j=1 N
N
兺 储e 共0兲储 i=1 j
冕
t+␦t
储ei共兲储d .
共27兲
t
A similar error metric has been used in Ref. 8. Here, we choose t = 40 and ␦t = 1.
r
r
1000
0
0
200
400
C
600
800
1000
0
FIG. 10. 共Color online兲 Synchronization error as a function of the pinned node number r and control gain C for the scale-free network.
Figures 9 and 10 show the error measure E as a function of C and r. For C ⬍ 200, the oscillators do not generally synchronize. For gain levels larger than 200, the synchronization error decreases as the fraction of pinned nodes increases and is minimized when all the network nodes are pinned. By pinning all the network nodes, the error drastically reduces. Consistently with the data presented in Figs. 3 and 4, the synchronization performance of the random network is generally superior to that of the scale-free network in the case of randomly pinning a fraction of network nodes at a varying total control gain. The scale-free network shows sharper synchronization bounds than the random network, that is, the region in which the error measure decreases to one-tenth of its initial value at t = 0 is limited to approximately 860⬍ r ⬍ 1000 and C ⬎ 200. In contrast, the error measure for the random network reduces to one-tenth of its initial value approximately in the region 600⬍ r ⬍ 1000 and C ⬎ 360. By extending the integration time window, it is expected that the random network synchronizes in all the region of Fig. 3 in which the performance index is less than 0.2. In Figs. 11 and 12, we report the synchronization error for the node-to-node pinning control and the two studied networks for 0 ⬍ C ⬍ 400 and three different values of the switching period. The synchronization error for C ⬎ 400 is approximately constant and equal to 10−15 due to the finite precision of the numerical integration. The synchronization error is approximately constant for relatively low gain levels and sharply decreases as C passes the threshold value of 200 for both networks and for all the selected switching periods. The rate of decay of the synchronization error slightly increases as the switching period decreases. The numerical values of the error are very much consistent with the data in Figs. 9 and 10 for r = 1000. Therefore, node-to-node pinning control is able to recreate the optimal scenario illustrated in Figs. 9 and 10 even for relatively high switching rates, that is, T = 0.4. To better illustrate the effectiveness of node-tonode pinning control as compared to random or selective pinning, in Figs. 13 and 14, we compare the synchronization
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Chaos 19, 013122 共2009兲
M. Porfiri and F. Fiorilli
2
8
0
7
10 10
−2
10
Selective pinning: r = 100 Random pinning: r = 100 Node-to-node pinning: T = 0.004
6
−4
10
5
−6
E
E
10
−8
4
10
3
−10
10
T = 0.004
−12
10
2
T = 0.04
−14
10
1
T = 0.4
−16
10
0
50
100
150
200
C
250
300
350
0
400
FIG. 11. Synchronization error as a function of the node-to-node pinningcontrol gain C for the scale-free network.
error obtained through node-to-node pinning control with the synchronization error that results by pinning 10% of the network nodes that are either randomly chosen or selected in order of decreasing degree. The data for the random pinning are directly borrowed from Figs. 9 and 10. These results along with those reported in Figs. 9 and 10 show that nodeto-node pinning control is able to tame the oscillator dynamics onto the reference trajectory for gain levels considerably smaller than those required by selective or random pinning with the additional advantage of acting only on a single network site. V. CONCLUSIONS
We studied pinning controllability of complex networks. We determined necessary conditions for pinning controllability in terms of the pinning-control gains, location of pinned sites, and network structural properties. We found that pinning-controllability performance is optimized when all the network nodes are pinned at the same gain level. In other
0
200
400
C
600
800
1000
FIG. 13. Synchronization error as a function of the node-to-node pinningcontrol gain C for the random network and different control strategies.
words, we discovered that the optimal policy for managing a given control level is to equally distribute it throughout the network rather than focusing it at a few selected nodes. This finding is perfectly in line with the numerical simulations and conjectures proposed in Ref. 8. We introduced the concept of node-to-node pinning control to cope with the problem of optimizing the control performance while pinning only a limited set of network sites. In this approach, only one node is pinned at any instant in time and the node location is periodically modified to cover the entire network in a switching period. Under node-to-node pinning control all the network nodes are virtually pinned at the same time if the switching period is sufficiently fast. Numerical examples were used to illustrate the effect of the overall control gain and fraction of pinned sites on the control performance, as well as to assess the effectiveness of node-to-node pinning control. In particular, we considered a random and a scalefree network of identical Rössler oscillators.
8 2
10
Selective pinning: r = 100
7
0
10
−2
6
−4
5
10 10
−6
Node-to-node pinning: T = 0.004
4
E
E
10
Random pinning: r = 100
−8
10
3
−10
10
2
T = 0.004
−12
10
T = 0.04
−14
10
1
T = 0.4
0
−16
10
0
50
100
150
200
C
250
300
350
400
FIG. 12. Synchronization error as a function of the node-to-node pinningcontrol gain C for the random network.
0
200
400
C
600
800
1000
FIG. 14. Synchronization error as a function of the node-to-node pinningcontrol gain C for the scale-free network and different control strategies.
013122-11
ACKNOWLEDGMENTS
This research was supported by the National Science Foundation under Grant No. CMMI-0745753. The authors would like to thank the anonymous reviewers for careful reading of the manuscript and for giving useful suggestions that have helped improve the work and its presentation. R. Grigoriev, M. Cross, and H. Schuste, Phys. Rev. Lett. 79, 2795 共1997兲. X. Wang and G. Chen, Physica A 310, 521 共2002兲. 3 X. Li, X. Wang, and G. Chen, IEEE Trans. Circuits Syst., I: Regul. Pap. 51, 2074 共2004兲. 4 T. Chen, X. Liu, and W. Lu, IEEE Trans. Circuits Syst., I: Regul. Pap. 54, 1317 共2007兲. 5 W. Lu, Chaos 17, 023122 共2007兲. 6 J. Xiang and G. Chen, Automatica 43, 1049 共2007兲. 7 R. Li, Z. Duan, and G. Chen, Chin. Phys. B 18, 106 共2009兲. 8 F. Sorrentino, Chaos 17, 033101 共2007兲. 9 F. Sorrentino, M. Di Bernardo, F. Garofalo, and G. Chen, Phys. Rev. E 75, 046103 共2007兲. 10 M. Porfiri and M. Di Bernardo, Automatica 44, 3100 共2008兲. 11 P. De Lellis, M. Di Bernardo, and F. Garofalo, Chaos 18, 037110 共2008兲. 12 C. W. Wu, in Proceedings of IEEE International Symposium on Circuits and Systems, 2008, pp. 2530–2533. 13 L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 80, 2109 共1998兲. 14 J. M. Gonzalez-Miranda, Synchronization and Control of Chaos 共Imperial College Press, London, 2004兲. 15 S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. Hwang, Phys. Rep. 424, 175 共2006兲. 16 T. Nishikawa and A. E. Motter, Physica D 224, 77 共2006兲. 1 2
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D. J. Stilwell, E. M. Bollt, and D. G. Roberson, SIAM J. Appl. Dyn. Syst. 5, 140 共2006兲. 18 L. Fortuna, M. Frasca, and A. Rizzo, Chaos, Solitons Fractals 17, 355 共2003兲. 19 J. D. Skufca and E. M. Bollt, Math. Biosci. Eng. 1, 347 共2004兲. 20 R. Amritkar and C. Hu, Chaos 16, 015117 共2006兲. 21 M. Porfiri and D. J. Stilwell, IEEE Trans. Autom. Control 52, 1767 共2007兲. 22 M. Porfiri and F. Fiorilli, “Global pulse synchronization of chaotic oscillators through fast-switching: theory and experiments,” Chaos, Solitons Fractals 共in press兲. 23 M. Porfiri and R. Pigliacampo, SIAM J. Appl. Dyn. Syst. 7, 825 共2008兲. 24 M. Porfiri, D. J. Stilwell, and E. M. Bollt, IEEE Trans. Circuits Syst., I: Regul. Pap. 55, 3170 共2008兲. 25 C. Godsil and G. Royle, Algebraic Graph Theory 共Springer-Verlag, New York, 2001兲. 26 R. Grone and R. Merris, SIAM J. Discrete Math. 7, 221 共1994兲. 27 J. S. Li and X. D. Zhang, Linear Algebr. Appl. 285, 305 共1998兲. 28 R. Merris, Linear Algebr. Appl. 285, 33 共1998兲. 29 K. C. Das, Linear Algebr. Appl. 368, 269 共2003兲. 30 N. M. M. de Abreu, Linear Algebr. Appl. 423, 5 共2007兲. 31 L. Shi, Linear Algebr. Appl. 422, 755 共2007兲. 32 T. Biyikoglu, J. Leydold, and P. F. Stadler, Laplacian Eigenvectors of Graphs 共Springer-Verlag, New York, 2007兲. 33 R. W. Beard and W. Ren, Distributed Consensus in Multi-vehicle Cooperative Control 共Springer-Verlag, New York, 2007兲. 34 D. S. Bernstein, Matrix Mathematics 共Princeton University Press, Princeton, 2005兲. 35 V. Batagelj and A. Mrvar, Pajek. Program for Analysis and Visualization of Large Networks. Reference Manual. List of Commands with Short Explanation. Version 1.22 共University of Ljubljana, Lujbiana, 2008兲. 36 D. M. Pennock, G. W. Flake, S. Lawrence, E. J. Glover, and C. L. Giles, Proc. Natl. Acad. Sci. U.S.A. 99, 5207 共2002兲. 37 P. N. McGraw and M. Menzinger, Phys. Rev. E 77, 031102 共2008兲. 38 J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 共1985兲.
CHAOS 19, 013123 共2009兲
Influence of noise on the sample entropy algorithm Sofiane Ramdani,1,a兲 Frédéric Bouchara,2 and Julien Lagarde1 1
EA 2991 Efficience et Déficience Motrices, Université de Montpellier I, Montpellier 34090, France UMR CNRS 6168 LSIS, Université du Sud Toulon-Var, La Garde 83957, France
2
共Received 27 October 2008; accepted 26 January 2009; published online 3 March 2009兲 We study the effect of static additive noise on the sample entropy 共SampEn兲 algorithm 关J. S. Richman and J. R. Moorman, Am. J. Physiol. Heart Circ. Physiol. 278, 2039 共2000兲; R. B. Govindan et al., Physica A 376, 158 共2007兲兴 for analyzing time series. Using surrogate data tests, we empirically investigate the ability of the SampEn index to detect nonlinearity in simulated time series corrupted by increased amounts of noise. Discrete and continuous chaotic and nonchaotic systems are included in the numerical experiments. Both Gaussian and uniformly distributed noises are considered. The results indicate that the SampEn statistic is a robust index for detecting nonlinearity in time series corrupted by observational noise. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3081406兴 Since the seminal works of Packard et al.1 and Takens,2 the extraction of dynamical invariants from simulated or experimental time series became an important topic in chaos theory. Several algorithms for the quantification of the complexity of a time series underlying dynamics have been proposed. Here, we focus on a recent algorithm designed to compute a statistic, the so-called sample entropy, which provides a characterization of time series complexity in terms of their regularity. The sample entropy is also a good candidate for defining a criterion for the detection of determinism in relatively short time series. Since its original formulation to analyze specific biological data,3 this statistic has been used to study a wide range of other experimental time series. Real-world data are typically corrupted by a certain amount of observational noise; hence it is justified to investigate the limits of the algorithm in the case of noisy time series. We analyze several simulated sequences generated by lowdimensional nonlinear dynamical systems that are corrupted by static additive independent Gaussian or uniform noise. Experimental chaotic laser data are included in the simulations. Discussing the results leads us to propose recommendations for reliable detection of nonlinear determinism in noisy time series.
I. INTRODUCTION
Few years ago, Richman and Moorman3 proposed an algorithm to quantify the regularity of a time series. The output of this algorithm, namely, the sample entropy 共SampEn兲, is an unbiased estimation of the correlation entropy K2 共or Rényi entropy of order 2, see Ref. 4兲 which is an invariant quantity measuring the rate of generation of information in the context of nonlinear time series analysis 共see Ref. 5 for details兲. The sample entropy method is, in fact, derived from an algorithm proposed by Pincus6 who introduced a family of statistics called approximate entropy a兲
Electronic mail: sofiane
[email protected].
1054-1500/2009/19共1兲/013123/7/$25.00
共ApEn兲. These measures were introduced in order to analyze short and noisy time series and are theoretically related to the Kolmogoro–Sinai entropy. One of the main advantages of the ApEn and SampEn analyses is that they can be applied to both deterministic and stochastic systems. When associated with surrogate data tests, ApEn and SampEn can also be used as discriminant statistics to detect deterministic nonlinearities in time series.7,8 It has been shown3,9 that the SampEn approach produces more consistent results than ApEn and is less sensitive to the length of data. The SampEn method has been applied to various biological data such as cardiovascular time series,3,10–12 neural respiratory signals,9 and more recently to human postural sway data.8,13 In Ref. 14 Govindan et al. proposed to modify the original SampEn algorithm by incorporating a time delay in the construction of the delayed vectors. This modified method provided much better characterizations of the complexity of time series generated by deterministic chaotic and nonchaotic systems. The importance of including a time delay was recently confirmed15 for ApEn and SampEn estimations. Many studies have been devoted to the estimation of correlation entropy K2 for noisy data 共see, for instance, Refs. 16–19 and references therein兲. Generally, the proposed analyses were based on the classical algorithm proposed by Grassberger and Procaccia20 for the estimation of K2 which is based on the computation of the correlation sum 共see Ref. 5 or Ref. 16, for instance兲. In Ref. 4 a noise effect study was performed only for specific physiological time series using the multiscale entropy 共MSE兲 method. The MSE index21 is basically an extension of the SampEn computed by coarse graining the data at different time scales. In a recent comparative study between ApEn, SampEn, and Fuzzy entropy, Chen et al.22 studied the noise robustness for time series generated by the logistic map. However, these studies did not include the analysis of the sensitivity to noise of the entropy estimations in terms of their ability to detect nonlinearities. In addition, the SampEn estimations were performed by us-
19, 013123-1
© 2009 American Institute of Physics
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Ramdani, Bouchara, and Lagarde
ing the original algorithm3 which does not incorporate a time delay. In the present study, we analyze the behavior of the modified SampEn algorithm14 and its ability to detect nonlinearity when applied to time series corrupted by observational additive noise 共Gaussian and uniformly distributed兲. Section II is dedicated to the modified SampEn algorithm description. Section III contains the details of the performed numerical experiments including the surrogate data testing procedure. In Sec. IV, we present the noise influence results on the SampEn. Finally, in Sec. V, we discuss these results and compare the method to other approaches for detecting nonlinear determinism. II. THE MODIFIED SAMPLE ENTROPY ALGORITHM
Let x1 , x2 , . . . , xN be a standardized 共with zero mean and unit standard deviation兲 simulated or experimentally recorded time series. The algorithm3,14 is first based on the construction of subsequences, sometimes called template vectors, of length m 共similar to the time delay embedding procedure for phase space reconstruction1,2兲 given by y i共m兲 = 共xi,xi+, . . . ,xi+共m−1兲兲
共1兲
for i = 1 , 2 , . . . , N − 共m − 1兲, where m is the dimension of the vectors and is the time delay. The second step of the algorithm is to define, for each integer i, the following quantity: N−m
Bm i 共,兲 =
1 兺 ⌰共 − 储y j共m兲 − yi共m兲储兲, N − m j=1,兩j−i兩⬎
共2兲
where ⌰ is the Heaviside function, 储 . 储 is the maximum norm, and 共y j共m兲 , y i共m兲兲 are couples for which 兩j − i兩 ⬎ . In Eq. 共2兲, the sum simply represents the number of vectors y j共m兲 that are within a distance 共also called radius or tolerance兲 from y i共m兲 in the reconstructed phase space. In such a case, y j共m兲 and y i共m兲 are called neighbors. Note also that, as proposed in Ref. 14, the cases for which 兩j − i兩 ⱕ are excluded to avoid temporal correlation effects. This aspect of the algorithm is similar to the Theiler correction23 for the estimation of correlation dimension. Then the following quantity is calculated: Bm共,兲 =
1 N − m
兺 i=1
Bm i 共,兲.
共3兲
N−m
1 = 兺 ⌰共 − 储y j共m + 1兲 − yi共m + 1兲储兲, N − m j=1,兩j−i兩⬎ 共4兲
1 A 共,兲 = N − m m
冉
冊
1 Am共,兲 SampEn共m, ,兲 = − log m . B 共,兲
共6兲
III. NUMERICAL EXPERIMENTS
For all the numerical experiments reported here, we use 2000 sample time series. In order to compute the SampEn values of the simulated deterministic time series, we select the time delay using the classical average mutual information 共AMI兲 approach.24 For the radius, we set = 0.2 which is a recommended value14,15 and corresponds to 20% of the standard deviation of the analyzed signals. The dimension m is chosen such that we observe a “pseudoconvergence”14 of the SampEn value. To analyze the noise effect, we use independent and identically distributed 共iid兲 Gaussian and uniform noise defined in the range 关0,1兴. Both added noises are centered and have the specified standard deviations. A. The tested data
For the simulations, we first use data from two classical chaotic maps. The first one is the logistic map defined by xi+1 = rxi共1 − xi兲
共7兲
with r = 4. The second map is the Hénon map defined by the equations xi+1 = y i + 1 − ax2i ,
共8兲
y i+1 = bxi
共9兲
with a = 1.4 and b = 0.3. We also analyze a time series generated by a continuous nonlinear oscillator producing limit cycle dynamics, namely, the x-component data from the Van der Pol system defined by
N−m
Similar computations are performed in the 共m + 1兲-dimensional reconstructed state space leading to the definition of the following quantities: Am i 共,兲
number of available vectors in the 共m + 1兲-dimensional state space. Finally, the sample entropy is given by the following formula:14
N−m
兺 i=1
Am i 共,兲.
共5兲
Note from Eqs. 共2兲–共5兲 that Bm共 , 兲 and Am共 , 兲 are calculated for the same number of templates 共N − m兲 which is the
x˙ = y,
共10兲
y˙ = ␣共1 − x2兲y − x
共11兲
with ␣ = 1.5. Data obtained from the x-component of the Rössler system in a limit cycle regime are included in the study. The equations of this system are x˙ = − 共y + z兲,
共12兲
y˙ = x + ay,
共13兲
z˙ = b + 共x − c兲z
共14兲
with a = 0.2, b = 0.2, and c = 2.
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Effect of noise on sample entropy
The previous system is also used to generate a time series from the x-component in the chaotic regime with parameters a = 0.2, b = 0.2, and c = 5.7. Finally, we use the x-component time series obtained from the standard Lorenz system given by x˙ = 共y − x兲,
共15兲
y˙ = rx − y − xz,
共16兲
z˙ = xy − bz
共17兲
with = 10, r = 28, and b = 8 / 3. In the case of continuous systems, we use a fourth order Runge–Kutta numerical integration algorithm to simulate the original noise-free time series. The integration step is set to 0.01 for the Van der Pol and Rössler systems. For the Lorenz model, the integration step is 0.02. The sampling rates respectively used for the Van der Pol, Rössler limit cycle, chaotic Rössler, and Lorenz time series are 10, 15, 20, and 1. We have also included an experimental time series generated by a chaotic Santa Fe Institute competition25 NH3 laser. The details of the recording procedure of this data set can be found in Ref. 26. For all numerical experiments, we first select 10 000 point time series by removing all transients. Then, 2000 sample subsequences are extracted according to a criterion related to the surrogate data analysis procedure described in Sec. III C. B. Noise effect analysis
We add to each of these time series 共xi兲 iid samples 共i兲 of Gaussian or uniform noise such that the resulting time series 共si兲 is given by si = xi + i .
共18兲
Since the original time series 共xi兲 is standardized 共centered and normalized by its standard deviation兲, the amount of added noise is quantified by the standard deviation of the noise time series 共i兲. We use levels of noise ranging from 5% to 40% with 5% step for a first quantitative analysis. Other levels of noise are used for the surrogate data analysis and are given in Sec. III C. In order to define a statistic for the results of this noise influence analysis, we generate 100 realizations for each level and each type of added noise. C. Surrogate data analysis
To understand further the effect of noise on the SampEn algorithm, we investigate its ability to detect nonlinearity by means of phase randomized surrogate data tests introduced by Theiler et al.27 More specifically, we use iteratively refined amplitude adjusted Fourier transform 共iAAFT兲 surrogates.28,29 This surrogate data testing procedure is designed to test a null hypothesis associated with a 共rescaled兲 linear Gaussian stochastic underlying process. Practically, for each analyzed time series and each level of noise, we generate a number of surrogates and compare the SampEn of the original sequence to the distribution of the surrogates SampEn values. When a statistically signifi-
cant difference is observed, one can consider that a deterministic nonlinearity is detected. As recommended5,27,29 for such tests, we use here a rank based nonparametric statistical test. This is done by comparing the original SampEn value to the distribution of the SampEn estimates of a number of surrogates. Basically, if the original SampEn statistic lies inside the range defined by the surrogate SampEn values, we consider that the null hypothesis cannot be rejected. In the opposite case, the deterministic nature of the underlying process is detected. In order to reach a level of significance ␣ = 0.05, we generate 共2 / ␣兲 − 1 = 39 surrogates of each simulated time series.5,29 Another point that has to be considered when using iAAFT surrogates is related to the so-called end-to-end mismatch issue. It has been reported 共see Ref. 29 and references therein兲 that important mismatches between the beginning and the end of the time series produce altered surrogates with spectra containing high frequencies that are not present in the original signal. Such surrogates potentially lead to spurious rejections of the null hypothesis. In order to avoid these periodicity artifacts, we extract 2000 point time series from the original 10 000 point data according to the optimization of a criterion introduced in Ref. 30 and detailed in Ref. 29. This criterion is given by the average of two parameters quantifying the end-point mismatch and the first-derivative mismatch. For each system and noise distribution, we perform the surrogate data tests for the original time series and for data corrupted by noise with the following standard deviations: 10%, 20%, 40%, and 60%. In each case, 100 realizations of the noise sequences are used. Figure 1 shows the time series obtained from the chaotic Rössler system described in Sec. III A as well as one of its iAAFT surrogates. Figure 2 shows the same time series corrupted by a 40% level Gaussian noise with one of its iAAFT surrogates. IV. RESULTS
Figure 3 shows the evolution of the SampEn as function of the dimension m for the different deterministic models described in Sec. III A. We observe that m = 5 is a good choice for computing the SampEn estimations for all the time series since the pseudoconvergence is reached for this dimension value. For all the SampEn computations reported here we set m = 5. Figure 4 shows the results of the static Gaussian noise addition in the case of the logistic map. For the uniform distribution of the added noise, the results depicted in Fig. 4 are qualitatively equivalent. This remark holds for all the tested models. For each noise level, the statistics of the SampEn values 共mean⫾ standard deviation兲 are computed over 100 realizations of the noise sequences. Note that the SampEn value obtained for the pure 共non-noisy兲 2000 samples sequence is 0.6377. For the Hénon map, we obtain a value of 0.4041. The effect of the addition of static noise in this case produced very similar results to those of the logistic map. Figure 5 shows the results for the Van der Pol time series. The SampEn value for the original pure sequence is
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Ramdani, Bouchara, and Lagarde
(a)
(a)
(b)
FIG. 1. The 2000 point standardized chaotic Rössler time series used for the numerical experiments are shown in the upper panel. One of its iAAFT surrogates is shown in the lower frame. The SampEn values obtained for these two time series are 0.0330 and 0.1632, respectively.
0.0012. For the Rössler limit cycle, the SampEn obtained is 0.0043. These low values indicate the highly predictable dynamics of the analyzed systems. The evolution of the SampEn estimates for the Rössler limit cycle is similar to those of the Van der Pol oscillator. In Fig. 6 the results of the Lorenz time series are reported. The SampEn estimation of the pure sequence is 0.0357. For the chaotic Rössler system, we obtain the value of 0.0330. The noise effect for these two models is quite identical. Figure 7 shows the results for the chaotic laser time series. The SampEn value of the original noise-free sequence is 0.1046. As expected, the SampEn algorithm associated with the iAAFT surrogates is able to detect the deterministic nonlinearity for all the original non-noisy time series. In Table I, we present the results of the simulations performed with the noisy data. Four levels of noise are considered: 10%, 20%, 40%, and 60%. The tolerance distance is set to 0.2. Note that, when a deterministic nonlinearity is successfully detected, the SampEn values of all the surrogate data are higher than those of the original tested noisy time series. For the time series generated by the limit cycle dynamics, we observed that increasing the radius value to = 0.4 radically improved the detection of the nonlinearity when the noise intensity reached 20%. For the Van der Pol time series, we obtained the following rates of successful detections: 93% in the Gaussian noise case and 82% in the uniform noise case. For the Rössler limit cycle time series, the corre-
(b)
FIG. 2. The 2000 point chaotic Rössler time series of Fig. 1 corrupted by 40% level Gaussian noise and standardized are shown in the upper panel. One of its iAAFT surrogates is shown in the lower frame. The SampEn values computed for these two time series are 0.2080 and 0.2175, respectively.
sponding rates were 100% and 99%, respectively. We have applied the test procedure using = 0.4 to the chaotic Rössler and laser time series when corrupted by a 40% noise level as for both of these systems the SampEn produced low rates 共see Table I兲. We also observed an improvement of the capacity of the method to detect nonlinearity. For the chaotic Rössler data, we obtained the following rates of successful detections: 99% 共Gaussian noise兲 and 2.5
Logistic map Hénon map Van der Pol limit cycle Rössler limit cycle Rössler chaos Lorenz chaos Laser chaos
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97% 共uniform noise兲. For the chaotic laser time series, the corresponding rates were 100% and 98%, respectively. Finally, in order to investigate the role of the tolerance , we have computed the rates of successful detections for a wide range of values. This was done for two systems for which a noise level of 40% produced low detection rates, namely, the Van der Pol and chaotic Rössler time series 共see Table I兲. The results obtained with Gaussian noise are presented in Figs. 8 and 9. On these figures, one can observe that there is a wide range of values for which the algorithm successfully detected deterministic nonlinearity. V. DISCUSSION
The qualitative behavior of the SampEn algorithm when static Gaussian or uniform noise is added to the analyzed data is globally similar for most of the systems reported here 共see Figs. 4–7兲. The SampEn statistic is very sensitive to static noise in all cases. However, for both chaotic map data, we observe that a 5% noise level has a very weak effect on the entropy value of the non-noisy data. In the case of the logistic map data, this result is in agreement with a recent study.22 We observe a strong reliability of the relationship between the SampEn of the noisy time series and the added noise level across the different models. We also notice a
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FIG. 6. Same as Fig. 4 for the Lorenz system.
reduced relative variability of the SampEn values estimated for the noisy data across the different models 共see Figs. 4–7兲. Concerning the capacity of the SampEn index to detect deterministic nonlinearity in noisy data, the results are quite different 共see Table I兲. They are qualitatively similar whatever the nature of the added noise distribution. Generally, the SampEn test for detecting nonlinearity shows a good robustness to noise for all the chaotic data when the threshold radius is set to the classical value = 0.2. Indeed, the nonlinear dynamics are detected with 40% level of noise for at least 80% of the noise realizations except for the chaotic Rössler and laser time series. For both of these two systems, the efficiency of the SampEn algorithm for detecting nonlinearity is noticeably weakened with 40% noise level, and more specifically in the case of added uniform noise. However, as reported in Sec. IV, when the radius value is increased up to 0.4, the SampEn index is able to reveal the nonlinearity inherent to the underlying dynamics. This behavior of the algorithm is reminiscent of the well known false nearest neighbor issue that could arise when phase space reconstruction is performed for estimating dynamical invariants such as largest Lyapunov exponent 共see Ref. 5 for more details兲. Basically, when the threshold is smaller than the noise level, the neighbors involved in the definition of the SampEn could be false neighbors leading to spurious estimations and hence producing erroneous surrogate data tests. For both limit cycle data 共Van der Pol and Rössler兲, the results suggest that the SampEn is not efficient for detecting determinism with a noise level exceeding a 10% level. For 08
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FIG. 4. The SampEn values of the logistic map time series corrupted by static Gaussian additive noise. The value obtained for the pure time series is shown as well as the 共mean⫾ standard deviation兲 computed with eight levels of noise. The standard deviations are represented with error bars. One hundred realizations of the added noise sequences are used to compute these statistics.
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such systems exhibiting great predictability and very low SampEn values, the addition of noise gives rise to important perturbations and thus produces inconsistent surrogate data tests. Note that, as reported in Sec. IV and shown in Fig. 8, increasing the threshold can be a solution to improve the reliability of the method which is naturally very sensitive to the radius choice for very regular time series corrupted by noise. In order to compare the performance of the SampEn index to other criteria for detecting nonlinear determinism 共see Ref. 31 for a brief review of the field兲, we also performed the numerical experiments using two different statistics. The first one is based on the smoothness principle.32 This concept is based on the observation that the reconstructed trajectories generated by an underlying deterministic dynamical system are smooth and differentiable which is not the case for signals produced by stochastic processes. More specifically, we used a statistic called the central tendency measure 共CTM兲 proposed by Jeong et al.33 to quantify the smoothness of the time series. The CTM has been shown to be efficient for detecting determinism in relatively short time series 共2000 samples兲. The second statistic used for comparison is a root mean squared one-step nonlinear prediction error 共NPE兲 based on a locally constant predictor given by the nearest neighbor in the reconstructed phase space.5,34 The simulations were performed for three typical time series, namely, the data generated by the logistic, Van der Pol, and chaotic Rössler systems with the four levels of Gaussian noise reported in Table I. In the case of the logistic map, the results
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TABLE I. The results of the surrogate data tests for the different systems corrupted by increasing amounts of Gaussian 共G.n.兲 and uniform noise 共U.n.兲. One hundred realizations of each noise sequence are used. The percentage of cases for which the nonlinearity was successfully detected by the SampEn is reported in each case. These tests are based on SampEn values estimated with a radius of = 0.2.
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FIG. 8. The rates of successful detections of determinism obtained for the noisy Van der Pol time series as function of the tolerance . The original time series is corrupted by 40% level Gaussian noise.
of the CTM and NPE statistics were equivalent to those of the SampEn with 10%, 20%, and 40% noise levels. When the noise reached 60%, the CTM and NPE outperformed the SampEn 共computed with a radius of = 0.2兲 with at least 73% of successful detections. However, when we increased up to 0.6, the SampEn index produced a 100% rate of successful detections. In the case of noisy Van der Pol data, the SampEn outperformed the CTM for all the noise levels. The NPE outperformed the SampEn computed with = 0.2, but when the radius was increased, the SampEn showed equivalent results and performed even better than the NPE criterion with 60% of noise. Finally, in the case of noisy chaotic Rössler time series, the CTM produced the worst results and the NPE rates were equivalent to those of the SampEn estimated with = 0.2. Note that, as shown in Fig. 9, the performance of the SampEn index can also be improved in this case by increasing . VI. CONCLUSION
We have shown that the absolute entropy value estimated by the SampEn algorithm is sensitive to observational noise in the case of data produced by low-dimensional nonlinear dynamics. However, our results suggest that the SampEn index 共estimated with the typical radius value of = 0.2兲 provides a good criterion for detecting deterministic nonlinearity in noisy data when the underlying deterministic
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dynamics are low-dimensional chaotic and the noise level is about 40%. For some dynamical systems, we found that the radius must be increased to guarantee the detection of nonlinear determinism. The results 共see Figs. 8 and 9兲 indicate that the algorithm becomes reliable for a wide range of normalized tolerance provided the latter is large enough in comparison to the noise level. It is worth noting that the method 共applied with = 0.2兲 globally behaves better for noisy chaotic data than for noisy periodic time series. Our main recommendation is to compute the SampEn index and perform surrogate data tests with different increasing values of if a nonlinear structure is expected in noisy data especially when the noise level could exceed 40% of the standard deviation of the noise-free signal. 1
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N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, Phys. Rev. Lett. 45, 712 共1980兲. 2 F. Takens, in Lectures Notes in Mathematics, edited by D. A. Rand and L.-S. Young 共Springer, Berlin, 1981兲, Vol. 898, pp. 366–381. 3 J. S. Richman and J. R. Moorman, Am. J. Physiol. Heart Circ. Physiol. 278, 2039 共2000兲. 4 M. Costa, A. L. Goldberger, and C. K. Peng, Phys. Rev. E 71, 021906 共2005兲. 5 M. Kantz and T. Schreiber, Nonlinear Time Series Analysis, 2nd ed. 共Cambridge University Press, Cambridge, 2004兲. 6 S. M. Pincus, Proc. Natl. Acad. Sci. U.S.A. 88, 2297 共1991兲. 7 R. Nagarajan, Physica A 366, 530 共2006兲. 8 M. Roerdink, M. De Haart, A. Daffertshofer, S. F. Donker, A. C. H. Geurts, and P. J. Beek, Exp. Brain Res. 174, 256 共2006兲. 9 X. Chen, I. C. Solomon, and K. H. Chon, in Proceedings of the 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference, Shanghai, China, 2005 共IEEE, 2006兲, pp. 4212–4215. 10 D. E. Lake, J. S. Richman, P. M. Griffin, and J. R. Moorman, Am. J.
Physiol. Regulatory Integrative Comp. Physiol. 283, 789 共2002兲. M. Ferrario, M. G. Signorini, G. Magenes, and S. Cerutti, IEEE Trans. Biomed. Eng. 53, 119 共2006兲. 12 H. M. Al-Angari and A. V. Sahakian, IEEE Trans. Biomed. Eng. 54, 1900 共2007兲. 13 M. C. Costa, A. A. Priplata, L. A. Lipsitz, Z. Wu, N. E. Huang, A. L. Goldberger, and C.-K. Peng, Europhys. Lett. 77, 68008 共2007兲. 14 R. B. Govindan, J. D. Wilson, H. Eswaran, C. L. Lowery, and H. Preißl, Physica A 376, 158 共2007兲. 15 F. Kaffashi, R. Foglyano, C. G. Wilson, and K. A. Loparo, Physica D 237, 3069 共2008兲. 16 T. Schreiber and M. Kantz, Chaos 5, 133 共1995兲. 17 C. Diks, Phys. Rev. E 53, R4263 共1996兲. 18 K. Urbanowicz and J. A. Hołyst, Phys. Rev. E 67, 046218 共2003兲. 19 M. Strumik, W. M. Macek, and S. Redaelli, Phys. Rev. E 72, 036219 共2005兲. 20 P. Grassberger and I. Procaccia, Phys. Rev. A 28, 2591 共1983兲. 21 M. Costa, A. L. Goldberger, and C. K. Peng, Phys. Rev. Lett. 89, 068102 共2002兲. 22 W. Chen, J. Zhuang, W. Yu, and Z. Wang, Med. Eng. Phys. 31, 61 共2009兲. 23 J. Theiler, Phys. Rev. A 34, 2427 共1986兲. 24 A. M. Fraser and H. L. Swinney, Phys. Rev. A 33, 1134 共1986兲. 25 See http://www-psych.stanford.edu/~andreas/Time-Series/SantaFe.html. 26 U. Hübner, N. B. Abraham, and C. O. Weiss, Phys. Rev. A 40, 6354 共1989兲. 27 J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, Physica D 58, 77 共1992兲. 28 T. Schreiber and A. Schmitz, Phys. Rev. Lett. 77, 635 共1996兲. 29 T. Schreiber and A. Schmitz, Physica D 142, 346 共2000兲. 30 C. L. Ehlers, J. Havstad, D. Prichard, and J. Theiler, J. Neurosci. 18, 7474 共1998兲. 31 S. Ramdani, F. Bouchara, and J.-F. Casties, Phys. Rev. E 76, 036204 共2007兲. 32 D. T. Kaplan and L. Glass, Phys. Rev. Lett. 68, 427 共1992兲. 33 J. Jeong, J. C. Gore, and B. S. Peterson, IEEE Trans. Biomed. Eng. 49, 1374 共2002兲. 34 T. Schreiber and A. Schmitz, Phys. Rev. E 55, 5443 共1997兲. 11
CHAOS 19, 013124 共2009兲
Simple driven chaotic oscillators with complex variables Delmar Marshall1 and J. C. Sprott2 1
Department of Physics, Amrita Vishwa Vidyapeetham, Clappana P.O., Kollam, Kerala 690-525, India Department of Physics, University of Wisconsin, 1150 University Avenue, Madison, Wisconsin 53706, USA
2
共Received 12 November 2008; accepted 21 January 2009; published online 5 March 2009兲 Despite a search, no chaotic driven complex-variable oscillators of the form z˙ + f共z兲 = ei⍀t or z˙ ¯兲 = ei⍀t are found, where f is a polynomial with real coefficients. It is shown that, for analytic + f共z functions f共z兲, driven complex-variable oscillators of the form z˙ + f共z兲 = ei⍀t cannot have chaotic z solutions. Seven simple driven chaotic oscillators of the form z˙ + f共z ,¯兲 z = ei⍀t with polynomial f共z ,¯兲 are given. Their chaotic attractors are displayed, and Lyapunov spectra are calculated. Attractors for two of the cases have symmetry across the x = −y line. The systems’ behavior with ⍀ as a control parameter in the range of ⍀ = 0.1– 2.0 is examined, revealing cases of period doubling, intermittency, chaotic transients, and period adding as routes to chaos. Numerous cases of coexisting attractors are also observed. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3080193兴 It has become widely recognized that mathematically simple nonlinear systems can exhibit chaotic behavior. A logical question to ask is “How simple can a system be and still exhibit chaos?” Much of the work to date on this question has been on systems with real variables. This work asks the question for driven oscillators with complex variables. After a brief mathematical look at such systems, a search for simple examples is conducted. Seven systems found in the search are examined as to the degree of chaos present, i.e., the size of the largest Lyapunov exponent. The question of how these systems move into chaos, as a function of the driving frequency, is also investigated.
I. INTRODUCTION
Chaotic systems of equations with real variables have been and continue to be studied extensively,1–5 but the study of chaotic systems with complex variables is a more recent pursuit. There have been numerous theoretical studies, particularly involving nonlinear oscillators, often with periodic forcing.6–10 These systems have wide applications in physics, in areas as diverse as fluids, quantum mechanics, superconductivity, plasma physics, optical systems, astrophysics, and high-energy accelerators.3,11–15 Motivated by this more recent trend in research, and in line with previous work searching for simple systems of a given form that exhibit chaotic behavior,16–20 this work began with a search for simple driven chaotic oscillators with a complex variable z, of the form z˙ + f共z兲 = ei⍀t. This form is equivalent, with z = x + iy and f共z兲 = u共x , y兲 + iv共x , y兲, where u and v are real functions, to the driven two-dimensional system,
x˙ = − u共x,y兲 + cos ⍀t, y˙ = − v共x,y兲 + sin ⍀t, ˙t = 1. There are numerous examples of driven two-dimensional systems that exhibit chaotic behavior.5 In Sec. II, some preliminary theoretical and experimental observations indicate why the search was expanded to systems with functions of the form f共z ,¯兲. z In Secs. III and IV, seven simple quadratic and cubic chaotic polynomial systems with real coefficients are examined, along with their behavior when ⍀ is used as a control parameter, including routes to chaos. Section V is a brief summary. II. PRELIMINARY OBSERVATIONS
Excluding complex conjugates ¯z most simple complex functions of z are differentiable by z except at isolated points, and so are analytic. Analytic functions obey the Cauchy– Riemann equations, u / x = v / y and u / y = −v / x. The Jacobian of system 共2兲, assuming analytic f共z兲 and using the Cauchy–Riemann equations, is
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−
共1兲
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u u − ⍀ sin ⍀t − x y u u , ⍀ cos ⍀t − y x 0 0 0
−
=
which can be recast as the autonomous three-dimensional system,
冤 冤
u u − ⍀ sin ⍀t − x y v v J= ⍀ cos ⍀t − − x y 0 0 0
x˙ = − u共x,y兲 + cos ⍀t, y˙ = − v共x,y兲 + sin ⍀t,
共2兲
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which has eigenvalues 0 and −u / x ⫾ i u / y. The Lyapunov exponents 共which measure the exponential rates of
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separation of two nearby trajectories, and thus the degree of chaos兲 are the averages of the real parts of these eigenvalues along the trajectory: 0, 具−u / x典, and 具−u / x典. If the latter were both positive, nearby trajectories would separate exponentially in two directions, and so would have to be unbounded. Thus for bounded trajectories, any nonzero Lyapunov exponents must be negative: Nearby trajectories approach each other exponentially in two directions. The signature of chaos is sensitive dependence on initial conditions—nearby trajectories separating exponentially—so there can be no chaotic behavior for systems of the form z˙ + f共z兲 = ei⍀t, where f is analytic. A simple way to alter the situation is to introduce the ¯兲, where f共z兲 is analytic, variable ¯z = x − iy. For a function f共z the signs of the Cauchy–Riemann equations are reversed. The Jacobian becomes
冤 冤
u u − ⍀ sin ⍀t − x y v v J= ⍀ cos ⍀t − − x y 0 0 0 −
冥 冥
u u − ⍀ sin ⍀t − x y u u = . ⍀ cos ⍀t − y x 0 0 0 −
共4兲
¯兲 = ei⍀t, the trace of the Jacobian is Thus for systems z˙ + f共z zero, i.e., they are area preserving, like Hamiltonian 共roughly speaking, energy conserving兲 systems. A set of initial condition points 共x0, y 0兲 may change its shape, but the set will maintain its original area over time. However, for nonlinear ¯兲, these systems cannot be Hamiltonian in the sense of f共z having a mechanical analog because if x˙ = y, the reversedsign Cauchy–Riemann equations mandate a linear system, the harmonic oscillator. While we have not shown that systems of the form z˙ ¯兲 = ei⍀t cannot be chaotic, the observed behavior for + f共z ¯兲 with real coefficients is that most trajectories polynomial f共z diverge. In a few cases, there are stable limit cycles 共the trajectory repeatedly cycles around a fixed, closed path兲 or tori 共the trajectory remains on the surface of a torus兲. In two cases, a Poincaré section 共a plot of y versus x at a chosen phase of the drive cycle, over many cycles兲, shows the torus breaking up into island chains for some values of ⍀, as is commonly observed for a Hamiltonian system. With further changes in ⍀, reduction in these cases, the trajectory begins to form a chaotic sea, but soon diverges. The basins of attraction 共sets of initial conditions that end up on the attractor兲 are quite small, and it is the outermost of the nested tori that breaks up. With no stable, surrounding torus 共a so-called KAM torus兲, there is no containment for the trajectory—it soon wanders outside the basin and diverges. Given that analytic functions of z cannot produce chaos, and having found no polynomial functions of ¯z with real
coefficients that produce chaos, polynomial functions of the two variables z and ¯z with real coefficients were introduced. Two obvious choices, functions of 共z +¯兲 z / 2 and 共z −¯兲 z / 2i, i.e., of x and y separately, were avoided because much work has already been done on finding simple examples of such systems16–20 and because of the decision to restrict the search to polynomials with real coefficients. III. QUADRATIC SYSTEMS
A search was conducted for the simplest chaotic quadratic polynomial f共z ,¯兲 z with real coefficients that displays chaotic behavior. The most general such quadratic function is ¯兲 = a0 + a1z + a2z2 + a3zz ¯ + a4¯z + a5¯z2 . f共z,z
共5兲
The search assigned random coefficients ai and used random initial conditions. The random values were taken from the squared values of a Gaussian normal distribution with mean zero and variance of 1, with the original signs of the values restored after squaring. For ⍀, uniformly distributed random numbers between 0.1 and 2.0 were used. During the search, trajectories were followed using a fixed-step fourth-order Runge–Kutta integrator.21 When a chaotic system was identified, an effort was made to simplify its coefficients ai while retaining its chaotic behavior. The largest Lyapunov exponent was calculated using the method detailed in Ref. 5; there must also be a zero exponent, allowing the third to be obtained from the trace of the Jacobian.5 The three simplest chaotic quadratic systems 共Lyapunov exponents for ⍀ = 1 in braces兲 our search method discovered were z˙ + z2 − ¯z + 1 = ei⍀t
兵0.0473,0,− 0.9869其,
共6兲
z˙ + 共z − ¯兲z z + 1 = ei⍀t
兵0.0641,0,− 0.2031其,
共7兲
z˙ + 2z2 − ¯z2 + 2 = ei⍀t
兵0.0803,0,− 0.0805其.
共8兲
The behavior of these systems with ⍀ as a control parameter was investigated; initial conditions for each system were kept constant at 共z0 , t0兲 = 共−0.5i , 0兲. The investigation was conducted with a Cash–Karp adaptive-step fourth-order Runge–Kutta integrator,21 which avoided numerical trajectory divergences observed with the fixed-step integrator in association with occasional large excursions from the chaotic attractors. Some of the changes in the attractors occurred with very small changes in ⍀, so quite likely there are more to be discovered than those listed here. A. System „6…
The attractor for system 共6兲 is shown in Fig. 1. As ⍀ is reduced below 0.6, there is a limit cycle that adds a loop above the x-axis, then below, and so on, down to at least ⍀ = 0.05, where there are 26 loops, 13 above and 13 below. This phenomenon has been called period adding;22 the limit cycle’s overall period increases by one initial period at a time. If started from z0 = −0.5i, system 共6兲 diverges at
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FIG. 1. Attractor for system 共6兲.
⍀ = 0.54, where a loop grows to infinite size as it flips from positive to negative y values. From 0.6 to 0.85, there is a figure-8 shaped limit cycle similar to the figure-8 shaped strange attractor of Fig. 1. As ⍀ is increased from 0.85, the limit cycle period doubles 共the limit cycle’s period doubles repeatedly兲 three times to period 8 at 0.955, and then returns to period-4 before going chaotic near 0.959. The transition to chaos takes place by means of lengthening chaotic transients. Chaos continues until ⍀ = 1.023, where the system produces a period-5 figure-8 limit cycle. This cycle period doubles to a narrow chaotic window in ⍀ values at 1.034. It reforms at 1.036, as a period-6 cycle, so the overall transition has been period adding. The sequence of period doubling to chaos followed by period adding continues, up to period 14 at 1.059, which then period doubles to a window of weak, intermittent chaos. Increasing ⍀ further, intermittent periodic intervals lengthen, and then resolve into a limit cycle just above 1.064. This cycle, a single loop above the x-axis, grows ever larger with increasing ⍀ finally diverging at ⍀ = 1.88. Above 1.88, the loop returns, now below the x-axis, as observed at ⍀ = 0.54. The loop shrinks with further increase in ⍀ at least as far as ⍀ = 2.3. B. System „7…
The attractor for system 共7兲 is displayed in Fig. 2. For ⍀ = 0.1 upward, there is a limit cycle with varying-length chaotic transients. Competition between a torus and a limit cycle can be seen in some regions of ⍀-space. Near 0.25, the torus prevails; but by 0.252, the limit cycle wins. At 0.253, there is an intermittent period-2 cycle with a pair of subcycles that separate into chaos. However, by 0.255, the subcycles come together into a period-1.
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FIG. 2. Attractor for system 共7兲.
This sort of mix repeats in the intervals from 0.355 to 0.375, 0.45 to 0.49, 0.61 to 0.70, 0.82 to 0.85, and 0.89 to 0.92, with period-1 limit cycles in between. In some intervals, there are places where stable cycles either appear for certain ⍀ or form from the separating 2-cycle. With aslightly smaller ⍀, there may be a weakly chaotic torus modeled on the 2-cycle. In Poincaré sections, the tori often appear as groups or chains of islands. The interval between 0.89 and 0.92 differs from the others in two ways: A change in initial conditions suffices to recover the limit cycle; and coming out of the interval, a chaotic transient leads directly to the period-1 cycle, without intermittency. Above 0.92, the limit cycle becomes increasingly complex, finally period doubling between 0.954 and 0.955 to the chaotic attractor depicted in Fig. 2. The attractor persists until, between ⍀ = 1.22 and 1.27, it reverse period doubles 共i.e., its period is repeatedly halved兲 to a limit cycle. Limit cycles preceded by transients, at first chaotic, then almost periodic, continue as ⍀ is increased, although varying the initial conditions can result in either of two tori. Above 1.50, the chaotic transient is increasingly attracted to a weakly chaotic torus, which becomes established as an attractor by 1.56. The torus shrinks, then grows, stabilizes, and finally separates into islands above ⍀ = 1.69. Above 1.70 the islands shrink, grow, and come together. The previous limit cycle returns above 1.78, through a chaotic transient. The torus then tries to gain the upper hand, resulting in a lengthening transient to a narrow window of chaos below 1.81. Coming out of this window via reverse period doubling, with chaotic transients, the torus forces a compromise: a figure-8 limit cycle at ⍀ = 1.83. The chaotic transient shortens at 1.84, lengthens again, and leads into chaos by 1.88. By 1.95, there is a torus composed of a group of islands, which shrink, grow, and come together as a simple torus by ⍀ = 2.00.
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FIG. 3. Attractor for system 共8兲.
In addition to those mentioned above, system 共7兲 has other coexisting attractors. These exist for the same value of ⍀ but are reached from different initial conditions. For example, a trajectory that begins at z0 = −1.5i is attracted to the single-loop limit cycle for any ⍀ between 1.3 and 1.8; to the figure-8 limit cycle for 1.85; and at 1.90 to a figure-8 torus of small islands, with a shape similar to the figure-8 limit cycle. From 1.95 to above 2.10, −1.5i leads to chaos. The torus at ⍀ = 1.70 varies in shape as z0 varies from 0 to −i. C. System „8…
For system 共8兲, Fig. 3 shows the attractor, evident for ⍀ between 0.1 and 2.0, although the chaos is weaker with smaller ⍀, going through a minimum at 0.43. Above 1.0, the trajectory sometimes contracts onto a torus with weak chaos or no chaos; often these tori are composed of tiny islands when viewed in Poincaré sections. The contraction happens, for example, at approximately ⍀ = 0.42– 0.44, 1.28–1.33, at approximately 1.40, and again at 1.98. The transitions can be rather sudden as a function of ⍀. A small change in initial conditions is sufficient to recover the chaotic attractor common to most values of ⍀, so these tori are coexisting attractors. Other tori can be found by varying the initial conditions at values where the large attractor is seen for z0 = −0.5i. For example, z0 = + 0.5i produces a torus at ⍀ = 1.0. At ⍀ = 1.1, initial conditions from z0 = −0.1i to +5i produce a family of tori. System 共8兲 is almost area preserving; the trace of the Jacobian, the average of −8x over the trajectory, is nearly zero: −1.8⫻ 10−4. A Poincaré section of the trajectory starting from various initial conditions 共Fig. 4兲 has all the features of a Hamiltonian system—a chaotic sea, KAM tori, heteroclinic trajectories 共trajectories that connect saddle points兲, and island chains. Note, however, that the bulk of the chaotic sea resides outside the outermost KAM torus and is
FIG. 4. Poincaré section for system 共8兲, showing features typical of a Hamiltonian system.
thus not contained. This is the same as the situation encoun¯兲 = ei⍀t, which ditered earlier with the two cases of z˙ + f共z verged after becoming chaotic. The difference here is that the basin of attraction is quite large, providing containment of the trajectory. IV. CUBIC SYSTEMS
The search technique described in Sec. III was also used for cubic polynomial f共z ,¯兲 z with real coefficients. The resulting simplest four chaotic systems 共Lyapunov exponents for ⍀ = 1 in braces兲 were z˙ + 0.3z3 + ¯z + 0.3 = ei⍀t
兵0.1231,0,− 0.6053其, 兵0.1540,0,− 1.3517其,
z˙ + 共0.2z2 + 1兲z + ¯z = ei⍀t z˙ + 共z2 − ¯z2兲z + ¯z = ei⍀t
兵0.0892,0,− 0.1621其,
¯2 + zz ¯ + z2 + 1兲z ¯ = ei⍀t z˙ + 共z
共9兲 共10兲 共11兲
兵0.1157,0,− 6.1517其. 共12兲
Again the behavior of these systems with ⍀ as a control parameter was investigated, by the same method used for the quadratic cases, and using the same initial conditions for each system, 共z0 , t0兲 = 共−0.5i , 0兲. A. System „9…
The attractor for system 共9兲 is displayed in Fig. 5. A limit cycle for ⍀ = 0.1 becomes chaotic at 0.3457, reverse period doubles to period-3 at 0.347, and moves through a narrow window of chaos to period-2 at 0.348. With increasing ⍀, chaos returns, then gives way by shortening chaotic transients to period-3, back to period-2, and finally, by 0.351, back to period-1.
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FIG. 5. Attractor for system 共9兲.
The period-1 cycle breaks into chaos above 0.4095 and reverse period doubles back to period-1 by 0.422. The pattern repeats between 0.51 and 0.53, and again between 0.64 and 0.67. Just above 0.67, the system goes chaotic, until moving to period-6 and then to period-3 at 0.71. By 0.715, there is chaos again, which reverse period doubles to period-1 at 0.73. Transitions are gradual, with slow approaches to the final states. Except for a coexisting limit cycle that appears briefly at 0.77, the two-lobed limit cycle now bears a distinct resemblance to the two-lobed attractor in Fig. 5. The limit cycle continues until ⍀ = 0.89, then moves into chaos at 0.9, through periods 2, 4, and 6, with slow approaches rather than sudden transitions. Chaos continues until, at ⍀ = 1.07 and 1.0825, there are limit cycles, with chaos in between. Above 1.0825, chaos reverse period doubles to period-1 near 1.11; transitions are through chaotic transients. The limit cycle shrinks, but remains period-1, up to near 1.343, where it suddenly changes character and becomes substantially larger. The coexisting smaller cycle can be recovered from the initial condition z0 = −0.2– 0.5i. The larger cycle can also be reached at smaller ⍀. From the initial condition z0 = −0.4i, just below ⍀ = 1.09 there is a transition from the smaller limit cycle to the coexisting larger one as a period-4. It period doubles to chaos at around 1.12, then reverse period doubles to period-1 at 1.21. It remains period-1 up to near 1.343, where it first appeared when starting from z0 = −0.5i. Returning now to z0 = −0.5i: Above ⍀ = 1.35, the larger cycle develops a third and fourth lobes. From 1.42 to 1.45, there is a chaotic transient to the smaller limit cycle. At 1.46, the transient resolves into a cycle tracing just the third and fourth lobes of the four-lobed structure. This happens again
FIG. 6. Attractor for system 共10兲.
at 1.48, but above and below 1.48, the smaller cycle is the result. At 1.58, the chaotic transient becomes long-term chaos, persisting until above 1.90. There the transient resolves into a large single-loop limit cycle, which shrinks with increasing ⍀ through at least ⍀ = 2.00.
B. System „10…
Figure 6 shows the attractor for system 共10兲. For ⍀ = 0.1 to 0.888, there is a limit cycle, except for two narrow chaotic windows, from ⍀ = 0.457 to 0.463 and 0.467 to 0.479. Entry to the former window is by lengthening periodic transient and exit by reverse period doubling. Entry and exit to the latter are both by lengthening transient. From ⍀ = 0.1 to 0.48, the cycle has four lobes, although one lobe can be quite small. Above 0.48, it has two lobes, with the outline of the attractor of Fig. 6 beginning to form by 0.66. At 0.8879, the two-lobed limit cycle adds an inner loop, becoming chaotic. After briefly stabilizing again at 0.8880, chaos continues, with the attractor increasing in size, up to ⍀ = 1.31. Above 1.31, the two-lobed chaotic attractor develops two more lobes, but reverts to a two-lobed limit cycle at 1.3522. The cycle period doubles to period 4 at 1.355, reverse period doubles back to period 1 at 1.358, then period doubles to chaos again just above 1.36. The resulting chaotic attractor has just two lobes at first, but with increasing ⍀, soon recovers the other two. Chaotic transients of widely varying lengths lead into the limit cycles, where present. The chaotic attractor continues to exist well beyond ⍀ = 2. With increasing ⍀, the trajectory spends more time in a central helix formed from the inner loop that appeared at ⍀ = 0.8879, and less time in the outer loops that form the four lobes.
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FIG. 7. Attractor for system 共11兲.
C. System „11…
System 共11兲 produces the attractor shown in Fig. 7. Note the symmetry, where sometimes x and y 关i.e., Re共z兲 and Im共z兲兴 change signs, flipping the attractor across the x = −y line. This symmetry is also present in some of the limit cycles this system produces. There is hysteresis associated with the symmetry, since a chaotic trajectory can flip, but once established as a limit cycle, it cannot. For this system, there exists a variety of limit cycles from ⍀ = 0.1 to 0.95, some very complex, and some with long chaotic transients, interspersed with narrow windows of chaos. Worthy of note is a 7-cycle at 0.65, which doubles to 14 and again to 28 before becoming chaotic below 0.651. The chaotic window closest to 0.95 spans approximately 0.906–0.928. Entry is by intermittency; exit is by reverse period doubling. The resulting limit cycle at 0.9322 immediately period doubles back to chaos by 0.93 222. This is the low side of a wide window of mainly chaotic behavior, with some closely confined exceptions: At 1.09, the chaos is very weak, and at 1.19 and 1.23, there are limit cycles. The wide window closes at ⍀ = 1.243, where a limit cycle forms again. This cycle persists until above 1.45, where a torus appears, first as a gradually lengthening transient; but finally taking over as an attractor, just above ⍀ = 1.50. Meanwhile, a symmetric pair of the previous limit cycles continues, coexisting with the torus. The torus is a feature until 1.635, where it reclaims transient status, leading back to the limit cycle via a chaotic transient. At 1.64, the resulting limit cycle is quite distinct, but with further increase in ⍀, the original version of the cycle soon reappears. By ⍀ = 1.72, there is again a chaotic attractor. Chaotic behavior continues until above ⍀ = 2; however, through ⍀ = 1.90, the torus can still be reached from z0 = −0.7i.
FIG. 8. Attractor for system 共12兲.
At 1.66, the initial condition z0 = 0 leads to the distinct cycle noted at 1.64. Increasing ⍀ with this initial condition causes the cycle to become gradually more chaotic, until at 1.76 the resulting attractor is nearly indistinguishable from the one reached from z0 = −0.5i. Here is another case where the chaotic attractor seems to result from a contest between two different stable attractors. D. System „12…
The attractor for system 共12兲, shown in Fig. 8, displays symmetry across x = −y, like system 共11兲. The limit cycles and chaotic attractors described below can exist on either side of the line or on both at once, depending on ⍀ and on the initial condition. From ⍀ = 0.1 to 0.76, there is a limit cycle which, just above 0.76, period doubles to weak chaos. At 0.77 a narrow window for a period-5 cycle opens, while at 0.775 a narrow window begins for a period-3 cycle. Above the period-3 window, the system is chaotic, but at 0.84, a limit cycle reappears, period doubling back to chaos by 0.845. At 0.86, a limit cycle exists in a very narrow range. Another limit cycle appears and disappears between 0.89 and 0.895; several more occupy the range between 0.895 and 0.902. From 0.903 to 0.906, there is weak chaos. At 0.907, a narrow window opens for a period-4 cycle, above which chaos prevails, until reverse period doubling leads to a limit cycle at 1.02. A lengthening transient leads to chaos at 1.03, followed by reverse period doubling to a limit cycle just below 1.05. Intermittency then leads into chaos, out of it again by 1.12, and back in by 1.13. By 1.18, larger scale reverse period doubling begins, leading to a limit cycle just below ⍀ = 1.30. The transitions take place by means of a transient that gradually separates into a limit cycle. Then as ⍀ increases,
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the transient gradually comes together into the next lowest period limit cycle. The resulting period-1 cycle at 1.30 persists until well above ⍀ = 2. V. CONCLUSIONS
For complex-variable systems of the form z˙ + f共z兲 = ei⍀t, where f共z兲 is an analytic function, it was shown that there can be no chaotic solutions. Furthermore, no chaotic solu¯兲 = ei⍀t, with polynomial tions for systems of the form z˙ + f共z ¯兲, were observed, but in a brief search for systems of the f共z form z˙ + f共z ,¯兲 z = ei⍀t, several simple chaotic systems were found. With ⍀ as a control parameter, period doubling, intermittency, and lengthening chaotic transients as routes to chaos were observed in these systems, as well as period adding. In a number of cases, coexisting attractors occurred. Poincaré sections from some systems display characteristics of Hamiltonian systems, particularly in one case where the dissipation 共energy loss兲 was very slight. Further inquiries into chaotic complex-variable systems are planned. 1
Chaos 19, 013124 共2009兲
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S. H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering 共Perseus, Cambridge, 1994兲. 2 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 共Springer-Verlag, New York, 1983兲.
R. C. Hilborn, Chaos and Nonlinear Dynamics 共Oxford University Press, Oxford, 1994兲. 4 A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics 共Wiley, New York, 1995兲. 5 J. C. Sprott, Chaos and Time-Series Analysis 共Oxford University Press, Oxford, 2003兲. 6 G. Mahmoud and T. Bountis, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 3821 共2004兲. 7 G. Mahmoud, A. A. Mohamed, and S. A. Aly, Physica A 292, 193 共2001兲. 8 Z. Xu and Y. K. Cheung, J. Sound Vib. 174, 563 共1994兲. 9 R. Srzednicki and K. Wójcik, J. Differ. Equations 135, 66 共1997兲. 10 K. Wójcik and P. Zgliczynski, Nonlinear Anal. Theory, Methods Appl. 33, 575 共1998兲. 11 G. M. Mahmoud, H. A. Abdusalam, and A. A. M. Farghaly, Int. J. Mod. Phys. C 12, 889 共2001兲. 12 V. A. Rozhanski and L. D. Tsendin, Transport Phenomena in Partially Ionized Plasma 共Taylor and Francis, London, 2001兲. 13 A. C. Newell and J. V. Moloney, Nonlinear Optics, Reading 共Addison Wesley, Redwood City, 1992兲. 14 D. Benest and C. Froeschlé, Singularities in Gravitational Systems: Applications to Chaotic Transport in the Solar System 共Springer-Verlag, Berlin, 2002兲. 15 R. Dilão and R. Alves-Pires, Nonlinear Dynamics in Particle Accelerators 共World Scientific, Singapore, 1996兲. 16 J. C. Sprott, Phys. Rev. E 50, R647 共1994兲. 17 J. C. Sprott, Phys. Lett. A 228, 271 共1997兲. 18 J. C. Sprott, Am. J. Phys. 65, 537 共1997兲. 19 J. C. Sprott and S. Linz, Int. Journal Chaos Theory and Appl. 5, 3 共2000兲. 20 J.-M. Malasoma, Phys. Lett. A 264, 383 共2000兲. 21 W. H. Press, S. A. Teukolsky, W. P. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. 共Cambridge University Press, Cambridge, 1992兲. 22 M. Levi, SIAM J. Appl. Math. 50, 943 共1990兲. 3
CHAOS 19, 013125 共2009兲
Adaptive gain fuzzy sliding mode control for the synchronization of nonlinear chaotic gyros Mehdi Roopaei,1,a兲 Mansoor Zolghadri Jahromi,2,b兲 and Shahram Jafari1,c兲 1
School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran Centre for Computational Intelligence, De Montfort University, Leicester, LE1 9BH, United Kingdom
2
共Received 4 September 2008; accepted 29 December 2008; published online 13 March 2009兲 This paper proposes an adaptive gain fuzzy sliding mode control 共AGFSMC兲 scheme for the synchronization of two nonlinear chaotic gyros in the presence of model uncertainties and external disturbances. In the AGFSMC scheme, the hitting controller that drives the system to the sliding surface is constructed by a set of fuzzy rules. In the proposed method, the gain of the reaching controller is adaptively adjusted to provide robustness against bounded uncertainties and external disturbances. The AGFSMC scheme can provide robustness in the absence of any knowledge about the bounds of uncertainties and external disturbances. We show that the adaptive gain scheme used in AGFSMC, improves the performance in comparison with the same control methodology that uses a fixed gain. Theoretical analysis of the AGFSMC scheme based on Lyapunov stability theory is provided. Numerical simulation on the application of the proposed method for the synchronization of two chaotic gyros is provided to demonstrate the feasibility of the method. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3072786兴 Control and synchronization of chaos, important topics on the applications of nonlinear sciences, have been developed and studied extensively in the last few years. Fuzzy logic has been investigated as a mechanism to handle uncertainties present in most physical systems. In this paper, we propose an adaptive gain fuzzy scheme for the synchronization of uncertain chaotic systems, which is based on the sliding mode control (SMC). For this purpose, a fuzzy systems for the reaching part of the SMC method with adjustable gain is proposed. We show that the adaptive gain of our method improves the performance in comparison with the same control methodology that uses a fixed gain. The overall scheme is proven to be stable in the presence of bounded uncertainty and external disturbance. Simulation results on the application of the proposed method for the task of synchronizing two chaotic gyros is provided to demonstrate the feasibility of the method.
I. INTRODUCTION
Chaos is a very interesting nonlinear phenomenon and chaos synchronization has been intensively studied since 1990.1–4 Chaos synchronization5 is another basic feature in nonlinear science and is of fundamental importance in a variety of complex physical, chemical, and biological systems.6 Applications of chaos synchronization are in the fields of secure communications, performance optimization of nonlinear systems, modeling the brain activity, and pattern a兲
Author to whom correspondence should be addressed. Telephone: ⫹989177137538. Electronic mail: mehdi
[email protected]. b兲 On sabbatical leave from Shiraz University. Electronic mail:
[email protected]. c兲 Electronic mail:
[email protected]. 1054-1500/2009/19共1兲/013125/9/$25.00
recognition.2,4–8 It must be mentioned that in a wider sense, nonlinear dynamics can play an important role in resolving outstanding problems in theoretical physics.9 Recently, chaos synchronization has become a hot spot in the field of nonlinear dynamics and many researchers have studied a variety of problems on chaos synchronization.2–5,10–14 These include the stability conditions for chaos synchronization, the realization for a successful synchronization, and applications of chaos synchronization. Pecora and Carroll2 developed the first synchronization technique. In their seminal paper, they addressed the synchronization of chaotic systems in a drive-response coupling scheme. A chaotic system, called the driver 共or master兲, generates a signal which is sent over a channel to a responder 共or slaver兲, using this signal to synchronize itself with the master. Many techniques for chaos control and synchronization have been developed in recent years. These include the periodic parametric perturbation method,15 adaptive control,16–24 sliding mode control 共SMC兲,25,26 backstepping control,27–30 H⬁ control,31 and many others.5 Since Zadeh32 initiated the fuzzy set theory, fuzzy logic control 共FLC兲 schemes have been widely developed and successfully applied to many real world applications.33 Besides, adaptive FLC schemes have been used to control and synchronize the chaotic systems.34,35 Using FLC in combination with SMC for ensuring stability and improved performance is an active research topic in the field of fuzzy control.36–38 Using the framework of the sliding mode control, in this paper, we use fuzzy inference model to propose an adaptive scheme for the synchronization of a class of chaotic systems in the presence of random perturbations and external disturbances. Although, external disturbances and perturbations
19, 013125-1
© 2009 American Institute of Physics
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Roopaei, Zolghadri Jahromi, and Jafari
are assumed to be bounded, no knowledge of the bounds of these uncertainties are required in our proposed method. Using Lyapunov stability analysis, we provide the proof of stability of the proposed method. Numerical simulation on the application of the proposed method for the synchronization of two chaotic gyros is provided to demonstrate the feasibility of the method. We show that the adaptive gain scheme improves the performance in comparison with the same control methodology that uses a fixed gain. The rest of this paper is organized as follows: Sec. II presents the system description. Design of the proposed controller is introduced in Sec. III. Simulation results are given in Sec. IV. Finally, conclusions are given in Sec. V.
The control problem considered in this paper is that for different initial conditions of systems 共1兲 and 共2兲, the two coupled systems 关i.e., the master system 共1兲 and the slave system 共2兲兴, would become synchronized by designing an appropriate control u共t兲 in system 共2兲 such that lim 储x共t兲 − y共t兲储 → 0,
t→⬁
共3兲
where 储·储 is the Euclidian norm of a vector. Let us define the state errors between the master and slave system as e 1 = y 1 − x 1,
e2 = y 2 − x2, . . . ,en = y n − xn .
共4兲
e˙1 = e2 , II. SYSTEM DESCRIPTION AND PROBLEM FORMULATION
e˙2 = e3 ,
Chaos synchronization is a problem that may arise when two 共or more兲 chaotic oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator. Because of the butterfly effect, which causes the exponential divergence of two chaotic systems started with different initial conditions, having two chaotic systems evolving in synchrony might appear quite surprising. The goal is to synchronize the slave 共response兲 system behavior to the master 共drive兲 system. In this paper, we consider a class of the following two n-dimensional chaotic systems: x˙i = xi+1,
1 艋 i 艋 n − 1, 共1兲
x˙n = f共x,t兲, x = 关x1,x2, . . . ,xn兴 苸 Rn , y˙ i = y i+1,
1 艋 i 艋 n − 1,
y˙ n = f共y,t兲 + ⌬f共y兲 + d共t兲 + u,
共2兲
y = 关y 1,y 2, . . . ,y n兴 苸 Rn , where u 苸 R is the control input, f is a given nonlinear function of x, and t, ⌬f共y兲 is an uncertain term representing the unmodeled dynamics or structural variation of system 共2兲 and d共t兲 is the disturbance of system 共2兲. In general, the uncertainty and the disturbance are assumed to be bounded as follows: 兩⌬f共y兲兩 艋 ␣
and 兩d共t兲兩 艋  ,
where ␣ and  are two unknown positive constants. System 共1兲 is mostly applied in physical systems, such as, the Duffing–Holmes damped spring system, Van der Pol equation, Genesio system,39 robot systems, flexible-joint mechanisms,40 and nonlinear chaotic gyros.41 It is also assumed that F共·兲 satisfies all the necessary conditions, such as, systems 共1兲 and 共2兲 having a unique solution in the time interval 关t0 , + ⬁兲 , t0 ⬎ 0, for any given initial condition x0 = x共t0兲 and y 0 = y共t0兲. The dynamics of system 共1兲 displays chaotic motion without control input 共u = 0兲.
]
共5兲
e˙n−1 = en , e˙n = g共x1,x2, . . . ,xn,e1,e2, . . . ,en兲 + ⌬f共e + x兲 + d共t兲 + u共t兲, where g = f共y , t兲 − f共x , t兲 is a known nonlinear function. The synchronization problem can be viewed as the problem of choosing an appropriate control law u共t兲 in such a way that the error states ei共i = 1 , 2 , . . . , n兲 in Eq. 共5兲 generally converge to zero. Here, robust adaptive gain fuzzy sliding mode control design is used to achieve the objective. III. CONTROLLER DESIGN METHODOLOGY
SMC is an efficient tool to control complex high-order dynamic plants operating under uncertainty conditions due to its order reduction property and low sensitivity to disturbances and plant parameter variations. In SMC, the states of the controlled system are first guided to reside on a designed surface 共i.e., the sliding surface兲 in state space and then keeping them there with a shifting law 共based on the system states兲. In traditional SMC, a switching surface representing the following desired system dynamics is constructed, n−1
S = e n + 兺 c ie i .
共6兲
i=1
The switching surface parameters 兵ci , i = 1 , . . . , n − 1其 are chosen based on the following two criteria. First, the values are chosen to stabilize the system during the sliding mode. Routh–Hurwitz criterion42 is used to determine the range of coefficients ci that produce stable dynamics. That is, all the roots of the following characteristic polynomial describing the sliding surface have negative real parts with desirable pole placement: P共兲 = n + cn−1n−1 + ¯ + c2 + c1 .
共7兲
Second, the values are chosen such that the system during sliding mode has fast and smooth response. When the closed loop system is in the sliding mode, it satisfies s˙ = 0. The equivalent control law is obtained by
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Adaptive gain fuzzy sliding mode control
FIG. 1. Block diagram of the AGFSMC scheme.
n−1
ueq = − g共x,e兲 − ⌬f共e + x兲 − d共t兲 − 兺 ciei+1 .
共8兲
to determine an appropriate reaching law. In the following sections, each of the above components is described in detail.
i=1
According to the Lyapunov stability theory,42 a Lyapunov function is defined as 1 V = 2 s2 .
共9兲
The derivative of V becomes
冉
n−1
冊
V˙ = ss˙ = s e˙n + 兺 cie˙i . i=1
共10兲
In the above equation, if V˙ is negative for all s ⫽ 0, then the so-called reaching condition42 is satisfied. That is, the control u is designed to guarantee that the states are hitting on the sliding surface S = 0. In the traditional SMC, the reaching control law is selected as ur = kwuw and the overall control u is determined by u = ueq + ur = ueq + kwuw ,
共11兲
where kw is the switching gain, which is positive and uw is obtained by uw = sgn共s兲.
共12兲
Based on Lyapunov theory, the system states approach the hyperplane if V˙ 艋 −kw兩s兩. The error vector asymptotically reduces to zero once the system states are on s = 0. The finite time delays and limitations of practical control systems render the implementation of such control signals problematic in real-world systems. In other words, the sign function in overall control u will cause the control input to produce the chattering phenomenon. In the current study, this problem is resolved through the application of a FLC scheme
A. Designing an AGFSMC
Since its introduction in 1965, fuzzy systems have been used in many real world applications. These systems have mainly been used in the field of control to handle uncertainties present in most real-life applications. Since the introduction of fuzzy sets, many researchers have shown interest to apply this theory to system identification. Fuzzy identification has become a very important area in fuzzy system theory. Many approaches have been proposed to obtain fuzzy models from input-output measurement data. A mathematical tool, which in some way uses fuzzy sets, is called a fuzzy model. In fact, this model consists of if-then rules with fuzzy antecedents and mathematical functions in the consequent part. The relationships between variables are represented by means of if-then rules with imprecise predictions, such as: If heating is high then temperature increase is fast. This rule defines in a rather qualitative way the relationship between the heating and the temperature in a room. To make such a model operational, the meaning of the terms “high” and “fast” is defined by using fuzzy sets, i.e., sets where the membership is changing gradually rather than in an abrupt way. Fuzzy sets are defined through their membership functions, which map the elements of the considered universe to the interval 关0, 1兴. The extreme values 0 and 1 denote complete membership and nonmembership, respectively, while a degree between 0 and 1 means partial membership in the fuzzy set.43 The dynamic behavior of a fuzzy controller is characterized by a set of linguistic rules based on expert’s knowledge.
FIG. 2. Membership functions of the fuzzy sets assigned to 共a兲 output variable 共u fs兲, 共b兲 input variables 共s and s˙兲.
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TABLE I. Rule base of FSMC. S u fs S˙
PB PM PS ZE NS NM NB
PB
PM
PS
ZE
NS
NM
NB
NB NB NB NB NM NS ZE
NB NB NB NM NS ZE PS
NB NB NM NS ZE PS PM
NB NM NS ZE PS PM PB
NM NS ZE PS PM PB PB
NS ZE PS PM PB PB PB
ZE PS PM PB PB PB PB
From this set of rules, the inference methodology of FLC provides the appropriate control action. When the rules of fuzzy controller are based on SMC, It is usually denoted as fuzzy sliding mode controller 共FSMC兲.44 Our proposed AGFSMC scheme is shown in Fig. 1. It contains an equivalent control part, a two-input single-output FSMC and an adaptive filter. The filter generates the adaptive gain using the relation 共15兲. The equivalent control part is the same as that in Eq. 共7兲 and the reaching law is selected as 共13兲
ur = kafsu fs ,
where kafs is the reaching gain achieved by adaptation law and u fs is the output of the FSMC, determined by the normalized s and s˙. The overall control u is chosen as u = ueq + ur = ueq + kafsu fs ,
共14兲
k˙afs = ␥兩s兩,
共15兲
where ␥ is a positive constant. The fuzzy control rules of FSMC provide the mapping of input linguistic variables s and s˙ to the output linguistic variable uFSMC as follows: u fs = FSMC共s,s˙兲.
共16兲
The membership function of input variables s and s˙, and output variable u fs are shown in Fig. 2. They are decomposed into seven fuzzy partitions expressed as negative big 共NB兲, negative medium 共NM兲, negative small 共NS兲, zero 共ZE兲, positive small 共PS兲, positive medium 共PB兲, and positive big 共PB兲. The fuzzy rule table of FSMC is designed as in Table I.44 In practical systems, the system uncertainty ⌬f共e + x兲 and external disturbance d共t兲 are unknown and the implemented equivalent control input is modified as
1. Theorem 1
Assume that the uncertain nonlinear system 共5兲 is controlled by u共t兲 in Eq. 共14兲, where ueq is Eq. 共17兲, u fs is Eq. 共16兲, and kafs is Eq. 共15兲. Then, the error state trajectory converges to the sliding surface s共t兲 = 0. Consider the following Lyapunov function: 1 1 V = s2 + 共kˆ − kafs兲2 . 2 2␥
共17兲
i=1
共18a兲
Then we have 1 V˙ = ss˙ − 共kˆ − kafs兲k˙afs ␥
冋 冋
n−1
册
1 = s · e˙n + 兺 cie˙i − 共kˆ − kafs兲k˙afs ␥ i=1 n−1
= s · − g共x,e兲 + ⌬f共x + e兲 + d共t兲 + ueq + kafsu fs + 兺 cie˙i i=1
册
1 − 共kˆ − kafs兲k˙afs ␥ 1 = s · 关⌬f共x + e兲 + d共t兲 + kafsu fs兴 − 共kˆ − kafs兲k˙afs ␥ 1 艋 ␣兩s兩 + 兩s兩 − kafs兩s兩 + kˆ兩s兩 − kˆ兩s兩 − 共kˆ − kafs兲k˙afs ␥ 1 艋 共␣ +  − kˆ兲兩s兩 + 兩s兩共kˆ − kafs兲 − 共kˆ − kafs兲k˙afs ␥
冉
冊
1 艋 共␣ +  − kˆ兲兩s兩 + 共kˆ − kafs兲 − k˙afs + 兩s兩 . ␥ It is clear that the scalar kˆ can be chosen in such a way that the value of ␣ +  − kˆ remains negative 共i.e., ␣ +  − kˆ = −, where ⬎ 0兲. Considering that −1 / ␥k˙afs + 兩s兩 = 0, the adaptive gain of fuzzy controller is k˙afs = ␥兩s兩, and the derivative of the mentioned Lyapunov function will satisfy the following condition: V˙ 艋 − 兩s兩.
n−1
ueq = − g共x,e兲 − 兺 ciei+1 .
Note that, we can always choose an arbitrary scalar kˆ such a way that ␣ +  − kˆ would be negative 共i.e., ␣ +  − kˆ = − where ⬎ 0兲.45
共18b兲
Using Barbalat’s lemma,42 it can be concluded that s, s˙ 苸 L⬁, and s approaches zero as t → ⬁. It means the system is stable and the error asymptotically converges to zero.
B. The stability analysis
In the following theorem, we provide the proof of stability of the scheme presented in Eq. 共14兲. With the aid of this, it is possible to derive the nonlinear system 共1兲 onto the sliding mode S = 0. That is, the reaching condition s共t兲s˙共t兲 ⬍ 0 is guaranteed.
2. Remark 1
It must be noticed that in our proposed method, no knowledge of the bounds of uncertainty and disturbance terms is required in advance while in the scheme presented in Ref. 46, this knowledge is required.
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Chaos 19, 013125 共2009兲
Adaptive gain fuzzy sliding mode control
IV. SIMULATION RESULTS
In this section, we evaluate the performance of our control scheme by applying it to synchronize two coupled chaotic gyro systems.
terms, respectively. The term 2共共1 − cos 兲2 / sin3 兲 − sin is a nonlinear resilience force. Given the states x1 = , x2 = ˙ , and g共兲 = −2共共1 − cos 兲2 / sin3 兲, this system can be transformed into the following nominal form: x˙1 = x2 ,
A. Description of nonlinear chaotic gyros
The dynamics of a symmetric gyro mounted on a vibrating base is given by41 共1 − cos 兲 ¨ + 2 − sin + c1˙ + c2˙ 3 = f sin t sin , sin3 2
共19兲 where is the angle that the spin axis of the gyro makes with the vertical axis, the term f sin t represents a parametric excitation, c1˙ and c2˙ 3 are linear and nonlinear damping
x˙2 = g共x1兲 − c1x2 − c2x32 + 共 + f sin t兲sin共x1兲.
共20兲
This gyro system exhibits complex dynamics and has been studied by Chen41 for values of f in the range 32⬍ f ⬍ 36 and constant values of 2 = 100, = 1, c1 = 0.5, c2 = 0.05, and = 2. Consider the following coupled gyro systems: x˙1 = x2 , x˙2 = g共x1兲 − c1x2 − c2x32 + 共 + f sin t兲sin共x1兲
FIG. 3. Our proposed AGFSMC. 共a兲 and 共b兲 States of master and slave systems; 共c兲 controller signal; 共d兲 error dynamics.
共21兲
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Chaos 19, 013125 共2009兲
Roopaei, Zolghadri Jahromi, and Jafari
e˙1 = e2 ,
y˙ 1 = y 2 , y˙ 2 = g共y 1兲 − c1y 2 − c2y 32 + 共 + f sin t兲sin共y 1兲
共22兲
+ ⌬f共y 1,y 2兲 + d共t兲 + u, where ⌬f共y 1 , y 2兲 expresses model uncertainty or structural variation and d共t兲 is used to denote the time-varying disturbances. In general, the uncertainty and the disturbance are assumed to be bounded as follows:
兩⌬f共y兲兩 艋 ␣ˆ and 兩d共t兲兩 艋 ˆ . where ␣ˆ and ˆ are two unknown positive constants. Denoting the error of the coupled systems as e1 = y 1 − x1 and e2 = y 2 − x2, we have
e˙2 = g共e1 + x1兲 − g共x1兲 − c1e2 − c2共e2 + x2兲3 + c2x32
共23兲
+ 关sin共e1 + x1兲 − sin共x1兲兴 + ⌬f共e1 + x1,e2 + x2兲 + d共t兲 + u, where 共t兲 =  + f sin t. In the conventional SMC method, the sliding surface s is usually selected as
s = e2 + e1 ⇒ s˙ = e˙2 + e˙1 = e˙2 + e2 .
共24兲
Using the above surface, the equivalent control law can be written as
FIG. 4. FSMC 共Ref. 44兲. 共a兲 and 共b兲 States of master and slave systems; 共c兲 controller signal; 共d兲 error dynamics.
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Chaos 19, 013125 共2009兲
Adaptive gain fuzzy sliding mode control
ueq = − e2 + c1e2 − g共e1 + x1兲 + g共x1兲 + c2共e2 + x2兲3
V˙ = ss˙ − 1/␥共kˆ − kafs兲k˙afs
− c2x32 − 关sin共e1 + x1兲 − sin共x1兲兴
= s关− c1e2 + g共e1 + x1兲 − g共x1兲 − c2共e2 + x2兲3 + c2x32
− ⌬f共e1 + x1,e2 + x2兲 − d共t兲.
共25兲
In real-world applications, the system uncertainties 关i.e., ⌬f共e1 + x1 , e2 + x2兲兴 and external disturbance d共t兲 are unknown. Therefore, the equivalent control law is modified to ueq = − e2 + c1e2 − g共e1 + x1兲 + g共x1兲 + c2共e2 + x2兲 −
c2x32
− 关sin共e1 + x1兲 − sin共x1兲兴.
3
+ 关sin共e1 + x1兲 − sin共x1兲兴 + ⌬f共e1 + x1,e2 + x2兲 + d共t兲 + ueq + kafsu fs − e2兴 − 1/␥共kˆ − kafs兲k˙afs = s关⌬f共e1 + x1,e2 + x2兲 + d共t兲 + kafsu fs兴 − 1/␥共kˆ − kafs兲k˙afs 艋 兩s兩共ˆ + ␣ˆ − kˆ兲 − kafs兩s兩 + kˆ兩s兩 − 1/␥共kˆ − kafs兲k˙afs
共26兲
艋 兩s兩共ˆ + ␣ˆ − kˆ兲 + 共kˆ − kafs兲共兩s兩 − 1/␥k˙afs兲兩,
共28兲
Using this, the overall control input is defined as u = ueq + kafsu fs = − e2 + c1e2 − g共e1 + x1兲 + g共x1兲 + c2共e2 + x2兲3 − c2x32 − 关sin共e1 + x1兲 − sin共x1兲兴 + kafsu fs .
兩s兩 − 1/␥k˙afs = 0 ⇒ k˙afs = ␥兩s兩.
共27兲
Defining the system Lyapunov function as V = 共1 / 2兲s + 共1 / 2␥兲共kˆ − kafs兲2, we have
2
共29兲
As we use Eq. 共15兲 to specify k˙afs and select the parameter kˆ in such a way that the condition ˆ + ␣ˆ − kˆ ⬍ 0 is satisfied, the error asymptotically converges to zero 共Theorem 1兲.
FIG. 5. Traditional SMC. 共a兲 and 共b兲 States of master and slave systems; 共c兲 controller signal; 共d兲 error dynamics.
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Chaos 19, 013125 共2009兲
Roopaei, Zolghadri Jahromi, and Jafari
FIG. 6. Traditional SMC with boundary layer. 共a兲 and 共b兲 States of master and slave systems; 共c兲 controller signal; 共d兲 error dynamics.
In simulations, the parameters of the gyro systems were specified as ␣2 = 100,  = 1, c1 = 0.5, c2 = 0.05, = 2, and f = 35.5. Using these values, the gyro systems show chaotic behavior. The initial conditions were specified as x1共0兲 = 1, x2共0兲 = −1, y 1共0兲 = 1.6, y 2共0兲 = 0.8. It was assumed that the uncertainty term, i.e., ⌬f共y 1 , y 2兲 = 0.1 sin共y 1兲, and the disturbance term, i.e., d共t兲 = 0.2 cos共t兲, are bounded by 兩⌬f共y 1 , y 2兲兩 艋 ␣ˆ = 0.1 and 兩d共t兲兩 艋 ˆ = 0.2, respectively. For the sliding surface design, a value of = 6 was selected and the overall control input is expressed as
TABLE II. Integral absolute error 共IAE兲 values for various methods.
u = ueq + kafsu fs = − 6e2 + c1e2 − g共e1 + x1兲 + g共x1兲 + c2共e2 + x2兲3 − − 共t兲关sin共e1 + x1兲 − sin共x1兲兴 + kafsu fs .
method presented in Ref. 46 and the traditional SMC design, respectively. To remove the chattering phenomena, boundary layer with the thickness of 0.5 was used. In Figs. 3–6, simulation results of our proposed AGFSMC scheme, the FSMC method,46 traditional SMC with/ without boundary layer is given. As seen, the error trajectory converges to S = 0, and the synchronization error also converges to zero, which confirms our theoretical analysis. Overall, our scheme performs well and the two chaotic non-
c2x32 共30兲
In our simulations, k fs = 0.5 共normalization factor of the output variable兲 and kr = 0.6 共reaching gain兲 were used for the
Method
The method of Ref. 46
Our proposed method
Traditional SMC
Traditional SMC with boundary layer technique
IAE
787.1
442.1
484.8
186.1
013125-9
linear gyros started with different initial values did achieve synchronization in the presence of uncertainty and external disturbance. In comparison with the FSMC method, our proposed scheme performs better, which in turn proves that our adaptive gain law is effective. We can also see that the chattering phenomenon is avoided in our scheme. In order to have a quantitative comparison of different methods, we used the integral absolute error 共IAE兲 as the criterion which is defined mathematically as follows:47 IAE =
冕
⬁
兩e共t兲兩dt.
0
Table II gives the IAE values for different methods. The results of Table II indicate that the performance of our method is better than the method proposed in Ref. 46 and conventional SMC. However, the conventional SMC method with boundary layer technique has better performance than ours. V. CONCLUSION
In this paper, an AGFSMC was introduced to handle the problem of synchronizing a class of uncertain chaotic systems with bounded uncertainty. In this method, no knowledge of the bound of uncertainty is required in advance. To verify the effectiveness of our method, we implemented our methodology to synchronize two uncertain chaotic gyros and compared our results with some recent method proposed for the same task. Simulation results proved the effectiveness of our adaptive gain strategy in comparison with the same control methodology that uses a fixed gain. For general applicability of the method presented in this paper, the work should be extended to the case of autonomous systems, and systems as the Lorenz having nonlinearity in more than one equation. This is more representative of the type of problems that are abundant in nature. A. W. Hubler, Helv. Phys. Acta 62, 343 共1989兲. L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 共1990兲. 3 T. L. Carroll and L. M. Pecora, IEEE Trans. Circuits Syst. 38, 453 共1991兲. 4 S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, Phys. Rep. 366, 1 共2002兲. 5 G. Chen and X. Dong, From Chaos to Order Methodologies, Perspectives and Applications 共World Scientific, Singapore, 1998兲. 1 2
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H. G. Schuster, Handbook of Chaos Control 共Wiley-VCH, Weinheim, 1999兲. 7 J. C. Sprott, Chaos and Time-Series Analysis 共Oxford University Press, Oxford, 2003兲. 8 E. Ott, C. Grebogi, and JA. Yorke, Phys. Rev. Lett. 64, 999 共1990兲. 9 H. H. Diebner and F. Grond, Chaos, Solitons Fractals 25, 765 共2005兲. 10 A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Science 共Cambridge University Press, Cambridge, 2001兲. 11 J. S. González Salas, E. Campos Cantón, F. C. Ordaz Salazar, and I. Campos Cantón, Chaos 18, 023136 共2008兲. 12 C. Li, L. Chen, and K. Aihara, Chaos 18, 023132 共2008兲. 13 T. Huang, C. Li, and X. Liao, Chaos 17, 033121 共2007兲. 14 X-Y. Wang and X-J. Wu, Chaos 16, 033121 共2006兲. 15 V. V. Astakhov, V. S. Anishchenko, T. Kapitaniak, and A. V. Shabunin, Physica D 109, 11 共1997兲. 16 M. Roopaei and M. Zolghadri Jahromi, Chaos 18, 043112 共2008兲. 17 Z. Li, G. Chen, S. Shi, and C. Han, Phys. Lett. A 310, 40 共2003兲. 18 K. Y. Lian, P. Liu, T. S. Chiang, and C. S. Chiu, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 49, 17 共2002兲. 19 T. L. Liao, Chaos, Solitons Fractals 9, 1555 共1998兲. 20 Y. Wang, Z. H. Guan, and X. Wen, Chaos, Solitons Fractals 19, 899 共2004兲. 21 C. W. Wu, T. Yang, and L. O. Chua, Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 455 共1996兲. 22 H. Salarieh and M. Shahrokhi, Chaos, Solitons Fractals 37, 125 共2008兲. 23 X. Li, W. Xu, and Y. Xiao, J. Sound Vib. 314, 526 共2008兲. 24 J-J. Yan, M-L. Hung, and T-L. Liao, J. Sound Vib. 298, 298 共2006兲. 25 J. Q. Fang, Y. Hong, and G. Chen, Phys. Rev. E 59, R2523 共1999兲. 26 X. Yin, Y. Ren, and X. Shan, Chaos, Solitons Fractals 14, 1077 共2002兲. 27 M. T. Yassen, Chaos, Solitons Fractals 27, 537 共2006兲. 28 M. T. Yassen, Phys. Lett. A 360, 582 共2007兲. 29 J. Lu and S. Zhang, Phys. Lett. A 286, 145 共2001兲. 30 C. Wang and S. S. Ge, Chaos, Solitons Fractals 12, 199 共2001兲. 31 J. A. K. Suykens, P. F. Curran, J. Vandewalle, and L. O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 44, 891 共1997兲. 32 L. A. Zadeh, Fuzzy Sets, Inf. Control 8, 338 共1965兲. 33 L. A. Zadeh, Inf. Sci. 共N.Y.兲 178, 2751 共2008兲. 34 G. Feng and G. Chen, Chaos, Solitons Fractals 23, 459 共2005兲. 35 Y. J. Xue and S. Y. Yang, Chaos, Solitons Fractals 17, 717 共2003兲. 36 M. Roopaei and M. Zolghadri Jahromi, Chaos 18, 033133 共2008兲. 37 B. Yoo and W. Ham, IEEE Trans. Fuzzy Syst. 6, 315 共1998兲. 38 J. Wang, A. B. Rad, and P. T. Chan, Fuzzy Sets Syst. 122, 21 共2001兲. 39 J. H. Park, Chaos, Solitons Fractals 27, 369 共2006兲. 40 J. E. Slotine, Int. J. Robot. Res. 4, 49 共1985兲. 41 H. K. Chen, J. Sound Vib. 255, 719 共2002兲. 42 J. E. Slotine and W. Li, Applied Nonlinear Control 共Prentice-Hall, New Jersey, 1991兲. 43 S. H. Zak, Systems and Control 共Oxford University Press, Oxford, 2003兲. 44 L. A. Zadeh, IEEE Comput. Graphics Appl. 21, 83 共1988兲. 45 M. Roopaei, P. Karimaghaee, and M. Soleimanifar, Nonlinear Stud. 13, 167 共2006兲. 46 H-T. Yau and C-l. Chen, Mech. Syst. Signal Process. 22, 408 共2008兲. 47 J. A. Romagnoli and A. Palazoğlu, Introduction to Control Process 共CRC, Boca Raton, 2006兲. 6
CHAOS 19, 013126 共2009兲
Self-organization of a neural network with heterogeneous neurons enhances coherence and stochastic resonance Xiumin Li,a兲 Jie Zhang, and Michael Small Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
共Received 23 July 2008; accepted 7 January 2009; published online 13 March 2009兲 Most network models for neural behavior assume a predefined network topology and consist of almost identical elements exhibiting little heterogeneity. In this paper, we propose a self-organized network consisting of heterogeneous neurons with different behaviors or degrees of excitability. The synaptic connections evolve according to the spike-timing dependent plasticity mechanism and finally a sparse and active-neuron-dominant structure is observed. That is, strong connections are mainly distributed to the synapses from active neurons to inactive ones. We argue that this selfemergent topology essentially reflects the competition of different neurons and encodes the heterogeneity. This structure is shown to significantly enhance the coherence resonance and stochastic resonance of the entire network, indicating its high efficiency in information processing. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3076394兴 The topology presented in a neural network exerts significant impacts on its function. Traditionally, a predefined topological structure is adopted in neural network modeling, which may not reflect the true situation in real-world networks such as the brain network. In this paper we propose a self-organized network (SON) whose synaptic connections evolve according to the spike-timing dependent plasticity (STDP) mechanism. Specifically, we study how the heterogeneity of neurons will influence the dynamical evolution and the emergent topology of the network. We find that our network obtained from STDP learning can significantly enhance the coherence resonance (CR) and stochastic resonance (SR) of the entire network. This result may have important implications on how the brain network is able to achieve a high efficiency in information processing by encoding the inherent heterogeneity in its topology. I. INTRODUCTION
Complex neural systems from either living biological entities or biophysical models have attracted great attention in recent years. Neural networks of various topologies have been investigated, such as globally coupled networks,1 smallworld networks,2,3 and scale-free networks.4 Specifically, instead of a prior imposition of a specific topology, it is more reasonable to consider self-organized neural networks, which have been broadly studied in Refs. 5–10. The selforganization is usually managed through STDP, which is a form of long-term synaptic plasticity both experimentally observed11 and theoretically studied.12,13 We note, however, that most network models in previous work did not take into account the heterogeneity of neurons, a feature ubiquitous for real neural networks. For example, neurons located near the canard region exhibit complex behaviors in the presence a兲
Electronic mail:
[email protected].
1054-1500/2009/19共1兲/013126/6/$25.00
of noise,14–16 where they are more sensitive to external signals and thus enhance information transfer in biological systems. Neurons having different dynamical activities will lead to the network heterogeneity, which can trigger competitions between individuals and play an important role in the CR 共Ref. 17兲 and phase synchronization.18 Moreover, in fact, the evolution of the synaptic connectivity or the network structure is closely related to the intrinsic heterogeneous dynamics of neurons. In this paper the network connection is evolved according to the STDP rule over a set of heterogeneous neurons. The heterogeneity is introduced into the network by choosing the key parameter from a uniform distribution covering a wide variety of neuronal behavior. We start from a network with global constant connections among neurons subject to a common input signal in a noisy background. At this time the neurons are in different states and fire at various frequencies. We find that with the STDP rule, the initial global connection among neurons is self-organized into a particular topology that eventually gives rise to synchronous spiking behavior, during which the competitions are mainly caused by the heterogeneous dynamics of each neuron rather than the initial conditions or different external inputs, as studied in Refs. 5–7. After the reorganization, the active cells tend to have high out-degree synapses and low in-degree synapses, while the inactive ones are just the opposite. This self-emergent topology essentially reflects the relationships of influence and dependence among the heterogeneous neurons and thus achieves energy consumption. In order to test the efficiency of this SON in signal processing, we have made comparisons to three other networks of different topologies in terms of CR and SR, which have been analyzed in various neural networks recently.16,17,19,20 We show that the network obtained from the STDP learning achieves a higher efficiency in information transfer.
19, 013126-1
© 2009 American Institute of Physics
Chaos 19, 013126 共2009兲
Li, Zhang, and Small 1
The network used in this paper is composed of N FitzHugh–Nagumo neuron models21 described by
0.25
Percentage of gij
II. NEURON MODEL AND STDP DESCRIPTION
0.8
0.2
0.6
0.15
gij<=0.1*gmax g >=0.9*g
0.4
V˙i = Vi − V3i /3 − Wi + Iex + Isyn i ,
F
013126-2
ij
0.1
max
0.05
0.2
others
共1兲
(a)
0 0
2000
t
4000
0.1
gijs j共Vi − Vsyn兲,
b
0.5
i
0.6
0.7
0.2 0.06
s˙ j = ␣共V j兲共1 − s j兲 − s j ,
0.15
0.04 0.02
(b)
0 0
b=0.4530 2000
Here the synaptic recovery function ␣共V j兲 can be taken as the Heaviside function. When the presynaptic cell is in the silent state V j ⬍ 0, s j can be reduced to s˙ j = −s j; otherwise s j jumps quickly to 1 and acts on the postsynaptic cells. The synaptic conductance gij from the jth neuron to the ith neuron will be updated through STDP that will be shown later. Note that in this paper both the excitatory and inhibitory synapses are considered. The type of synapse is determined by the synaptic reversal potential Vsyn, which we set to be 0 and ⫺2 for excitatory and inhibitory synapses, respectively. In this model, b is a critical parameter that can significantly influence the dynamics of the system. For a single neuron free from noise, the Andronov–Hopf bifurcation occurs at b0 = 0.45. For b ⬎ b0, the neuron is in the rest state and is excitable; for b ⬍ b0, the system has a stable periodic solution generating periodic spikes. Between these two states, there exists an intermediate behavior, known as canard explosion.22 In a small vicinity of b = b0, there are small oscillations near the fixed point before the sudden elevation of the oscillatory amplitude. In our system, bi is uniformly distributed in 关0.45, 0.75兴. Hence each neuron when uncoupled has a different activity when subject to external input and noisy background, and neurons with b located near the bifurcation point are prone to fire in a much higher frequency than the others 关see Fig. 1共d兲兴. According to the experimental report on STDP,11 there are no obvious modifications of excitatory synapses onto inhibitory postsynaptic cells after their repetitive and relative activities. Hence, we set inhibitory synaptic conductance and excitatory-to-inhibitory synaptic conductance to be constants. The remaining excitatory synapses are updated by the STDP modification function F, which selectively strengthens the pre- to postsynapse with relatively shorter latencies or stronger mutual correlations while weakening the remaining synapses.6 The synaptic conductance is updated by
t
4000
0.1 6000
(e) 0
0.1
4000
6000
0.3
b=0.6008
0.04
0 0
t
0.4
b=0.4530
0.06
(c)
2000
0.5
0.08
0.2
b=0.5 b=0.6
0.1
0.02
共2兲
Gin
where i = 1 , 2 , . . . , N. a, bi, and are dimensionless parameters with small enough 共 Ⰶ 1兲 to make the membrane potential Vi a fast variable compared to the slow recovery variable Wi. i is the independent Gaussian noise with zero mean and intensity D that represents the noisy background, is the and Iex stands for the externally applied current. Isyn i total synaptic current through neuron i, where the dynamics of the synaptic variable s j is governed by
b=0.6008
out
1共j⫽i兲
␣共V j兲 = ␣0/共1 + e−V j/Vshp兲.
0
0.08
G
兺
(d)
0.25
b=0.7350
N
Isyn i =−
6000
F
˙ = V + a − b W + D , W i i i i i
b=0.7
b=0.7350 2000
t
4000
6000
(f)
0 0
0.05
0.1
0.15
0.2
0.25
Noise Intensity (D)
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FIG. 1. 共Color online兲 Evolution of the network structure. 共a兲 Percentage of synapses at three value levels: gij ⱕ 0.1gmax 共red line兲, gij ⱖ 0.9gmax 共blue line兲, and the others 共black line兲. 关共b兲 and 共c兲兴 The average in-degree and out-degree synapses of three neurons with a different excitability, which is controlled by b. The red line represents the more excitable one with b = 0.4530, the blue line shows the less excitable one with b = 0.7350, and the black line is the one with medial excitability b = 0.6008. 共d兲 The initial firing rate 共F兲 distribution of individual cells with different b for the first 200 s. 共e兲 The average firing rate 共具F典兲 of all cells during the learning process. 共f兲 Influence of noise intensity 共D兲 on the firing rate of single neuron with different values of b.
⌬gij = gijF共⌬t兲, F共⌬t兲 =
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冎
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where ⌬t = ti − t j and F共⌬t兲 = 0 if ⌬t = 0. + and − determine the temporal window for synaptic modification. A+ and A− determine the maximum amounts of synaptic modification. Experimental results suggest that A−− ⬎ A++, which ensures the overall weakening of synapses. Here, we set − = + = 2, A+ = 0.05, and A− / A+ = 1.05, as used in Ref. 6. Only the excitatory-to-excitatory synapses are modified by this learning rule and are restricted to the range 关0 , gmax兴, where gmax is the limiting value. Other parameters used in this paper are a = 0.7, = 0.08, ␣0 = 2,  = 1, Vshp = 0.05, and gmax = 0.1. The other parameters are given in each case. Numerical integration of the system is done by the explicit Euler–Maruyama algorithm,23 with a time step of 0.005. III. SELF-ORGANIZATION OF NEURAL NETWORK
We consider a network of N = 60, which consists of 50 excitatory and 10 inhibitory neurons. All the neurons are bidirectionally and globally coupled at the beginning, and we assign gmax / 2 and 3gmax / 2 to excitatory and inhibitory syn-
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apses, respectively. The whole network is subject to an external current 共Iex = 0.1兲 and noisy background 共D = 0.06兲 as a learning environment. The influence of noise intensity 共D兲 on the firing rate of single neuron with different values of b is shown in Fig. 1共f兲. We now check how the network structure evolves during the learning process. As shown in Fig. 1, after competition, most of the synaptic connections converge to either 0 or the maximum gmax from the initial value gmax / 2 关see Fig. 1共a兲兴. This structure becomes stable after about 6000 s. From Figs. 1共b兲 and 1共c兲 we can see how the average in-degree synapses Gin and out-degree synapses Gout of different cells evolve in this competition. For the active cell, such as the one with bi = 0.4530, it fires so frequently that it is more likely to activate the others and thus strengthen its out-degree synapses Gout to gmax while weakening its in-degree synapses Gin to 0. This exactly reflects that such neurons are highly dominant and therefore less dependent on the others, while for the inactive cells 共e.g., bi = 0.6008 and 0.7350兲, they typically need large Gin to be excited and have small Gout due to their low influence. This contributes to the sparse connection of the network and benefits energy consumption. Figure 1共d兲 shows the initial firing rate distribution of each neuron with different b. The firing rate of the whole network plateaus after about 1500 s when the number of synapses with gij ⱖ 0.9gmax equals to that of the synapses with gij ⱕ 0.1gmax 关Fig. 1共e兲兴. So the following update of the synapses is in fact a refining procedure that further weakens those unnecessary connections. Network structures at learning times of 200 and 6000 s are shown in Fig. 2. The synaptic connection finally becomes sparse with about 50% being 0 and 20% being gmax 关Fig. 2共c兲兴. Figure 2共d兲 gives a clear picture of the active-neurondominant synaptic connections in this network, where strong connections are mainly distributed to the synapses from active neurons 共those with small values of bi兲 to inactive ones 共those with large values of bi兲. The reason for generating such a special structure is that, under the same learning environment, active neurons can fire with a high frequency and thus are more likely to act as the precells whose out-degree and in-degree synapses are then strengthened and weakened by STDP, respectively. Such synapse distribution renders the active cells a powerful drive to the inactive ones. Hence, through the STDP learning process, the high level of excitability of those active neurons is fully exploited to trigger the whole network to fire synchronously, which becomes more excitable than the original network 关Fig. 1共e兲兴. It should be noted that when the driving of external applied signal is removed, the sustained synchronous firing after learning will terminate and the whole network returns the normal rest state. Instead of the synchronous activity, the main point of this paper is the reorganized network topology. Its enhancement on coherence and SR will be discussed in the Sec. IV. As the inhibitory synapses are not involved in the update procedure, the size and distribution of the number of excitatory and inhibitory neurons will not influence the formation of final network topology, but just the speed of the convergence process. Also, if the initial excitatory synapses are set to be gmax or randomly distributed in 关0 , gmax兴, similar re-
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FIG. 2. 共Color online兲 Histogram and distribution of the synaptic matrix G at learning times of 200 and 6000 s. Synapses gij from cell j to cell i with b j and bi, respectively, are plotted. The black dots are the strong synapses satisfying gij ⱖ 0.9gmax, the blue circles are the weak synapses satisfying gij ⱕ 0.1gmax, and the red plus signs are intermediate values of synapses.
sults can be obtained but need longer convergence time. Moreover, to ensure that our results do not depend on the specific realization of the uniform distribution of parameter bi among neurons, we have performed the learning process over several different realizations and find no significant changes of the final network topology.
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FIG. 3. 共Color online兲 Comparisons of four types of neural networks on 关共a兲 and 共b兲兴 CR and 关共c兲 and 共d兲兴 SR. SON is the self-organized network obtained via STDP. RNS is the network with the same synaptic distribution as SON but shuffled. RNG is the random network with synapses uniformly distributed in 关0 , gmax兴. CN is the globally coupled network with constant synapses gmax / 2. 关共a兲 and 共b兲兴 S and Tmean vs noise intensity D, respectively. 共c兲 Q vs noise intensity D, where B1 = 0.75. 共d兲 The influence of inactive cells on SR. Qmax is the maximum of Q. Only cells with parameter bi 苸 关0.45, B1兴 are subject to external signal. This figure is the average result of ten trials.
IV. CR AND SR
In this section, we investigate the efficiency of the SON obtained via STDP in signal processing by comparing its performance on CR and SR with three other networks, i.e., the network with the same synaptic distribution as SON but shuffled 共RNS兲, the random network with synapses uniformly distributed in 关0 , gmax兴 共RNG兲, and the globally coupled network with constant synapses gmax / 2 共CN兲. All these four types of network are composed of heterogeneous cells that are bidirectionally coupled and have the same mean value of synapses being about gmax / 2. Ten trials are conducted for each network. CR is a noise-induced effect, which describes the occurrence and optimization of periodic oscillatory behavior due to noise perturbations.14 With an intermediate noise intensity, the system can behave the most regular periodic oscillations. We take S and Tmean as the coherence factors of the firing events. They are defined as N
1 S = 兺 S i, N i=1
Si =
具Tik典t/
冑Var共Tik兲,
N
1 Tmean = 兺 具Tik典t . N i=1
共5兲
Tik = ik+1 − ik is the pulse internal, where ik is the time of the kth firing of the ith cell. 具·典t denotes average over time. S describes the degree of spiking regularity in neural systems.
Tmean is the average interspike interval 共ISI兲. Here, Iex = 0 and all the cells are in the subthreshold region in the absence of noise. Figure 3共a兲 shows that the optimal regularity occurs when the noise intensity D equals about 0.06. The corresponding S in SON is much larger than the other networks, indicating the high coherent output of SON. The best performance of SON on CR with intermediate noise intensity D = 0.06 is shown in Fig. 4共a兲. The flat curve of Tmean near the optimal case 关see Fig. 3共b兲兴 reflects that the regular ISIs in SON can exist in a relatively wide range of noise intensity, while due to the inefficient connectivity, the other networks display unsynchronized and inactive activities, causing the small S and large Tmean 共ISI兲. This is because, under the driving of the same noise intensity, neurons with different levels of excitability show diverse firing patterns. Only the SON that has a reasonably selected synapse distribution can couple the neurons efficiently and generate regular spiking. SR describes the cooperative effect between a weak signal and noise in a nonlinear system, leading to an enhanced response to the periodic force. The neuron model is an excitable system, which can potentially exhibit SR.24 To evaluate SR, we set the periodic input to be Iex = B sin共t兲, with B = 0.1 and = 0.3. The amplitude of the input signal is small enough to ensure that there is no spiking for all the neurons in the absence of noise. Also, the frequency is much
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cell-dominant connection in SON regulates well the network activity and eventually achieves a balanced energy distribution among neurons. The best performance of SON on SR with intermediate noise intensity D = 0.02 is shown in Fig. 4共b兲. In order to investigate the importance of active cells, only cells with bi 苸 关0.45, B1兴, where 0.47ⱕ B1 ⱕ 0.75, are subject to the periodic input. Figure 3共d兲 shows that whether the inactive cells are subject to external signal or not has little effect on SR. This indicates that the contributions of inactive cells to SR are negligible, while the active cells are critical and play a vital role to trigger the whole network response with external signal.
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FIG. 4. 共Color online兲 Performance of SON network on CR and SR with intermediate noise intensity D = 0.06 in 共a兲 and D = 0.02 in 共b兲. The top figures in 共a兲 and 共b兲 show the spike trains among the first 50 excitatory neurons and the last 10 inhibitory neurons. V1, V2, and V3 represent the membrane potentials of three neurons with different values of b: V1, b = 0.4530; V2, b = 0.6008; V3, b = 0.7306. The black line in 共b兲 共bottom兲 is the input signal.
slower than that of the neuron’s inherent periodic spiking. Fourier coefficient Q is used to evaluate the response of output frequency to input frequency. It is defined as25 Q = 冑Q2sin + Q2cos,
Qcos = 2n
冕
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Qsin =
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2Vi共t兲sin共t兲dt,
0
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0
Here n is the number of periods 2 / covered by the integration time. Vi is the average membrane potential among the network. The quantity Q measures the component from the Fourier spectrum at the signal frequency . The maximum of Q shows the best phase synchronization between input signal and output firing. Again, SON exhibits greater SR than the other cases 关Fig. 3共c兲兴. In the three other networks that have inefficient connections, active cells fire much more frequently than the periodic driven signal while the inactive ones may be even at the rest state. The active-
V. CONCLUSION
In this paper, a new type of self-organized neural network with heterogeneous neurons is obtained via STDP learning. The internal dynamics of different neurons is shown to be clearly encoded in the topology of the emergent network after learning. During the STDP learning process, the synaptic strengths of the network are renewed by increasing the influence of active cells over the others and the dependence of inactive cells on the active cells. This process mediates the internal dynamical properties of different neurons and renders the whole network more synchronous and therefore more sensitive to weak input. This effect is clearly reflected from its improved performance on CR and SR. Therefore, we believe that this self-organized heterogeneous neural network is much efficient in signal processing tasks. The network model we proposed may be biologically relevant, considering the highly diversified behaviors of different neurons and the time-varying synaptic connectivity. Our result may be further extended to the study of functional or hierarchical connections in complex brain networks,26 where heterogeneity is essential for certain brain activities. Recently STDPs of inhibitory synapses are also observed and investigated.27,28 This kind of synaptic plasticity has been shown to play an important role in the neuronal function, although the cooperation between these two types of STDP is still unclear. It could be considered by using more physiological neuron and synapse model in the future. For simplicity, we use the excitatory postsynaptic current 共EPSC兲 of AMPA 共␣-amino-3-hydroxyl-5-methyl-4isoxazole-propionate兲 type as is used in Refs. 5–7, which is a kind of fast synaptic current mediated by AMPA receptors.29 Further advancements on NMDA 共N-methyl-D-aspartic acid兲 receptors, which activate EPSC much slower than AMPA, need to be studied in detail in terms of long-term synaptic plasticity.30 ACKNOWLEDGMENTS
This research was funded by a Hong Kong University Grants Council Competitive Earmarked Research Grant 共CERG兲 No. PolyU 5269/06E. D. H. Zanette and A. S. Mikhailov, Phys. Rev. E 58, 872 共1998兲. L. G. Morelli, G. Abramson, and M. N. Kuperman, Eur. Phys. J. B 38, 495–500 共2004兲. 3 J. W. Bohland and A. A. Minai, Neurocomputing 38-40, 489–496 共2001兲. 1 2
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D. Stauffer, A. Aharony, L. da Fontoura Costa, and J. Adler, Eur. Phys. J. B 32, 395–399 共2003兲. 5 Z. Palotai, G. Szirtes, and A. Lorincz, Proceedings of the 2004 IEEE International Joint Conference on Neural Networks, 2004 共unpublished兲. 6 S. Song, K. D. Miller, and L. F. Abbott, Nat. Neurosci. 3, 919–926 共2000兲. 7 C. W. Shin and S. Kim, Phys. Rev. E 74, 045101共R兲 共2006兲. 8 E. M. Izhikevich, J. A. Gally, and G. M. Edelman, Cerebral Cortex 共Oxford University Press, New York, 2004兲. 9 S. Kang, K. Kitano, and T. Fukai, Neural Networks 17, 307–312 共2004兲. 10 N. Levy, D. Horn, I. Meilijson, and E. Ruppin, Neural Networks 14, 815–824 共2001兲. 11 B. Guo-qiang and P. Mu-ming, J. Neurosci. 18, 10464–10472 共1998兲. 12 Y. Dan and M. Poo, Physiol. Rev. 86, 1033–1048 共2006兲. 13 P. Roberts and C. Bell, Biol. Cybern. 87, 392–403 共2002兲. 14 E. Ullner, “Noise-induced phenomena of signal transmission in excitable neural models,” Ph.D. thesis, University Potsdam, 2004. 15 V. A. Makarov, V. I. Nekorkin, and M. G. Velarde, Phys. Rev. Lett. 86, 3431 共2001兲. 16 X. Li, J. Wang, and W. Hu, Phys. Rev. E 76, 041902 共2007兲. 17 C. Zhou, J. Kurths, and B. Hu, Phys. Rev. Lett. 87, 098101 共2001兲.
18
Y. Tsubo, J. N. Teramae, and T. Fukai, Phys. Rev. Lett. 99, 228101 共2007兲. 19 Z. Gao, B. Hu, and G. Hu, Phys. Rev. E 65, 016209 共2001兲. 20 M. Perc, Phys. Rev. E 76, 066203 共2007兲. 21 R. FitzHugh, Biophys. J. 1, 445 共1961兲. 22 M. Wechselberger, SIAM J. Appl. Dyn. Syst. 4, 101 共2005兲. 23 D. J. Higham, SIAM Rev. 43, 525 共2001兲. 24 T. Wellens, V. Shatokhin, and A. Buchleitner, Rep. Prog. Phys. 67, 45 共2004兲. 25 L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223–287 共1998兲. 26 L. Zemanová, C. Zhou, and J. Kurths, Physica D 224, 202–212 共2006兲. 27 J. S. Haas, T. Nowotny, and H. D. I. Abarbanel, J. Neurophysiol. 96, 3305–3313 共2006兲. 28 S. S. Talathi, D. U. Hwang, and W. L. Ditto, J. Comput. Neurosci. 25, 262–281 共2008兲. 29 A. Destexhe, Z. F. Mainen, and T. J. Sejnowski, Methods in Neural Modeling 共MIT Press, Cambridge, 1998兲. 30 M. I. Rabinovich, P. Varona, A. I.Selverston, and H. D. I. Abarbanel, Rev. Mod. Phys. 78, 1213–12165 共2006兲.
CHAOS 19, 013127 共2009兲
Dynamical mechanism of intrinsic localized modes in microelectromechanical oscillator arrays Qingfei Chen,1 Liang Huang,1 Ying-Cheng Lai,1,2 and David Dietz3 1
Department of Electrical Engineering, Arizona State University, Tempe, Arizona 85287, USA Department of Physics, Arizona State University, Tempe, Arizona 85287, USA 3 Air Force Research Laboratory, Directed Energy Directorate, 3550 Aberdeen Ave. SE, Kirtland Air Force Base, New Mexico 87117, USA 2
共Received 11 November 2008; accepted 15 January 2009; published online 13 March 2009兲 Experimental evidence of intrinsic localized modes 共ILMs兲 in microelectromechanical oscillator arrays has been reported recently. In this paper, we carry out a detailed analysis of a new mechanism for ILMs in typical experimental settings; that is, spatiotemporal chaos is ubiquitous and it provides a natural platform for actual realization of various ILMs through frequency control. We find that unstable periodic orbits associated with ILMs are pivotal for spatiotemporal chaos to arise and these orbits are the keys to stabilizing ILMs by frequency modulation. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3078706兴 Small-sized systems such as microelectromechanical (MEM) resonators have become common in many fields of science and engineering. These systems have a relatively simple structure but they show surprisingly rich nonlinear-dynamical behaviors such as bistability, chaos, and energy-localized oscillations. This paper focuses on intrinsic localized modes (ILMs) in MEM oscillator arrays. The phenomenon is characterized by the oscillations of a few oscillators with significantly larger amplitudes than the average amplitude. While ILMs have been identified in a wide variety of physical systems such as Josephson junctions, optical waveguide arrays, photonic crystals, and antiferromagnets, their discovery in MEM systems has been relatively recent. We shall report results from numerical computations and dynamical analysis of a generic class of MEM oscillator arrays, which suggest the fundamental role played by chaotic dynamics in generating ILMs. In particular, we find that spatiotemporal chaos provides a natural platform for ILMs. As MEM systems are employed extensively in device research and development, we expect our finding to be potentially useful. I. INTRODUCTION
ILMs, also known as “discrete breathers” or “lattice solitons,” can occur in a defect-free nonlinear lattice, extending over only a few lattice sites.1–4 In basic physics, ILMs represent an interesting phenomenon as they are the result of purely nonlinear interactions. Theoretically, for conservative systems, the localized modes are exact solutions.5,6 In device applications, ILMs can be of significant concern as localized high-energy states can have undesirable effects on the operation of the device. ILMs have in fact been observed in many physical systems, such as Josephson junctions,7 optical waveguide arrays,8 photonic crystals,9 and antiferromagnets.10 Recently, ILMs have been discovered experimentally in MEM oscillator arrays.11,12 As such systems are the key com1054-1500/2009/19共1兲/013127/9/$25.00
ponents in state-of-the-art technologies that are having ever increasing impacts on various areas of science and engineering,13 it is of considerable interest to explore the dynamical mechanism of ILMs in MEM oscillator arrays. The purpose of this paper is to offer a detailed analysis for the mechanism of ILMs in experimental systems. In particular, a route for exciting ILMs in MEM oscillator arrays through spatiotemporal chaos is presented. While the dynamical mechanism of ILMs in conservative systems has been understood reasonably well,5,6 systems of MEM oscillator arrays are typically dissipative. Prior to our work, the status of understanding of ILMs in MEM oscillator arrays is as follows. It has been suggested that artificial impurities in MEM cantilever arrays can induce ILMs,14 and ILMs induced by a forced nonlinear vibration mode have also been realized.15 However, there is experimental evidence indicating that ILMs can be generated without any impurities.12 In Ref. 16, besides ILMs, phenomena such as hopping and repulsion are reported, but the underlying mechanisms of these phenomena are not given. Computational study of MEM cantilever arrays of identical beam length has revealed that it is not possible to have modulational instability in the system, but ILMs can be generated by noise in combination with frequency chirping modulation.17 The conclusion appears to be that, in order for a MEM oscillator-array system to exhibit ILMs, the following two conditions are needed: 共1兲 random heterogeneous initial conditions or noise and 共2兲 modulation of driving frequency in time 共frequency chirping兲. The starting point of our analysis is a prototype system described by a set of coupled differential equations derived from experiments.11 Since typical experimental settings to excite ILMs include the case where periodic forcing is applied to MEM arrays of alternating lengths, i.e., a long 共short兲 beam has two short 共long兲 beams as its nearest neighbors, one on each side,11,12 we will consider both the uniform-length and the alternating-length cases. The method of averaging can then be employed to reduce the system to a
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FIG. 1. 共Color online兲 Schematic of a single cantilever unit in a typical experimental MEM array system, where L1, L2, W1, W2, T, P, and H denote the beams’ lengths, widths, thickness, length of the pitch, and length of the overhang, respectively.
form for which, in the zero-coupling limit, all possible equilibrium solutions can be obtained and their stabilities can be determined. Analytic continuation of the solutions into the finite-coupling regime reveals the coexistence of both locally high- and low-energy states 共LESs兲, which is a necessary condition for ILMs in MEM oscillator arrays. We find that, for the alternating-length case, spatiotemporal chaos can arise when the LES loses its stability, and ILMs can be excited naturally from chaos by frequency modulation, which can be abrupt with even uniform initial conditions for all oscillators in the absence of any external noise. A brief account of this phenomenon has been reported recently.18 In this paper, we carry out a detailed, systematic analysis of the dynamical mechanism for ILMs. Issues such as the existence of ILM states, their stabilities and bifurcations, the occurrence of spatiotemporal chaos, and the stabilization of ILMs from chaos by frequency modulation will be addressed. In Sec. II, we describe the physics that leads to a generic model for MEM oscillator arrays, suitable for nonlineardynamics-based analysis. In Sec. III, the dynamics of ILMs are studied in detail. The mechanism for generating spatiotemporal chaos is investigated in Sec. IV, and the route to ILMs via chaos is demonstrated in Sec. V. Conclusions and discussion are presented in Sec. VI.
II. MODEL
The geometry of a single cantilever unit in a coupled MEM oscillator array employed in experimental studies12 is shown in Fig. 1, where two cantilevers with alternating length are coupled by an overhang. The corresponding MEM array system can be fabricated by low-stress silicon nitride. Driving of the array is realized by a piezoelectric transducer. The material properties of the system are11,16 L1 = 50 m, L2 = 55 m, W1 = W2 = 15 m, T = 0.3 m, H = 23 m, and P = 40 m. The density and Young’s modulus of the material are = 2300 kg/ m3 and E = 110 GPa, respectively. The dynamics of a MEM cantilever beam is in general described by a nonlinear partial differential equation19 that involves complicated mechanical and electrical interactions between the beam and its surroundings. For a single cantilever beam, the continuum equation of motion under driving force is11
共1兲
where A = W ⫻ T is the cross-sectional area of the beam and I = 共W ⫻ T3兲 / 12 is the moment of inertia. The displacement variable u共s , t兲 can be expanded based on a set of orthonormal shape functions, denoted by n共s兲, as follows: u共s,t兲 = 兺 n共s兲n共t兲,
共2兲
n
where n共t兲 is the beam tip’s displacement associated with n. The shape functions satisfy the boundary conditions n共0兲 = 0 and n共L兲 = 1. Here, only the lowest frequency mode is kept. By substituting Eq. 共2兲 into Eq. 共1兲, multiplying 1共s兲 on both sides, and then integrating the equation over the length of the beam, one can obtain m
d2x共t兲 + k2x共t兲 + k4x3共t兲 = m␣ cos共⍀t兲, dt2
共3兲
where x共t兲 is the displacement of the beam’s tip and k2 = 共12.36EI兲 / L3 and k4 = 共24.79EI兲 / L5 are the harmonic and the quadratic spring constants,11 respectively. For coupled microcantilever arrays, the full dynamical equations are more complicated. The cantilevers are affected by their environment, generating various damping forces. They also interact with each other with coupling forces. Without losing the essential dynamics of the system, we focus on the motions of the free ends of the beams, taking into account damping and coupling effects. The system equations can then be described by the following set of nonlinear ordinary differential equations:12 mix¨i + bix˙i + k2ixi + k4ix3i + kI共2xi − xi+1 − xi−1兲 = mi␣ cos共2 ft兲,
共4兲
where xi 共i = 1 , . . . , N兲 is the displacement of the end point of the ith cantilever beam of effective mass mi, bi is the damping coefficient, k2i and k4i are the on-site harmonic and quadratic spring constants of the ith beam, respectively, and kI is the harmonic coupling spring constant which we choose as a bifurcation parameter of the system. Each beam is subject to a common sinusoidal driving characterized by acceleration ␣ and frequency f. Let ⍀ = 2 f and xi共t兲 = Ui共t兲cos共⍀t兲 + Vi共t兲sin共⍀t兲 and define Qi = 冑mik2i / bi and ⍀0i = 冑k2i / mi as the quality factor and the resonant frequency of beam i, respectively. When the driven frequency ⍀ is close to the resonant frequency and when the driving is strong, the beam dynamics can be strongly nonlinear, exhibiting a bistable response with either large- or small-amplitude oscillations. Moreover, in the bistable region, there exists an unstable solution between the two stable solutions. We first consider the situation where all beams have identical length. The MEM array system equation 共4兲 is in fact a system of coupled driven Duffing oscillators. The averaging technique20,25 can then be employed for moderate driven force. Let the averaged functions of Ui共t兲 and Vi共t兲 be ui共t兲 and vi共t兲, respectively. A straightforward averaging procedure leads to
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kI 共2ui − ui+1 − ui−1兲 , mi
i = 1, . . . ,N,
1
1
kI (N/m) −5
In this section, we systematically investigate the dynamics of ILMs in MEM oscillator arrays. We first study the dynamics of two coupled MEM oscillators and then extend the analysis to systems of larger numbers of oscillators.
S3
k
B
0.6
U2
0.4
U1
k
B
2
1
2
III. DYNAMICS OF ILMS IN MEM OSCILLATOR ARRAYS
0.015 S1
U4 U3
0.2 S2
0 (b) 0
1
k (N/m) I
0.01
0.015
−5
x 10
A6
(m)
0.8
k
1,2,...,11
for i = 1 , . . . , N, where i = ui + jvi. For uniform beam length, Eq. 共6兲 has a continuum limit. However, for the case of alternating beam length, Eq. 共6兲 does not have the continuum limit and can only be analyzed as a coupled lattice system. Another difference between Eq. 共6兲 and the standard coupled NLSE is that, for the latter, the coefficient of the first-order term on the right-hand side is complex.
0.01
x 10
0.8
共6兲
2
S2
0 0 1
kB
B
B
A
kI + 共2i − i+1 − i−1兲 2⍀mi
k
A (m)
册
U4
0.2
which is in a form of a discrete nonlinear Schrodinger equation 共NLSE兲20 driven by a constant force ␣ / 共2⍀兲,
冋
U3
U2
0.4
(a)
3 k4i 1 ⍀0i di ␣ =− 共⍀2 − ⍀20i兲 + j i + 兩 i兩 2 i − j 8 ⍀mi dt 2Qi 2⍀ 2⍀
S3
U1
0.6
共5兲
1 3 k4i dvi ⍀0i = 共⍀2 − ⍀20i兲ui − ui共u2i + v2i 兲 − ⍀vi + ␣ 4 mi dt 2⍀ Qi −
S1
0.8
kI 共2vi − vi+1 − vi−1兲 , mi
冋
−5
x 10
A (m)
−
1
0.2
(c)
0 0
k (N/m) I
0.008 0.01
A. Two coupled MEM oscillators
A key feature of ILMs is the coexistence of two groups of oscillators with high and low energies, respectively. To gain insight, we first study the case of two coupled oscillators. Although the case of coupled undamped NLSEs has been discussed in Ref. 21, to our knowledge, the driven damped NLSEs with complex coefficient which describe coupled MEM oscillator systems have not been studied. Here, we discuss the case of coupled beams of identical lengths. Since Eq. 共5兲 governs the evolution of the averaged motion, its equilibrium solutions correspond to oscillatory solutions of the actual system. A viable approach is to set kI = 0 to find all possible equilibrium solutions for the decoupled system and then analytically continue the solutions as kI is increased from zero. Figures 2共a兲 and 2共b兲 show projections of the amplitudes of the averaged oscillations A1 = 冑u21 + v21 and A2 = 冑u22 + v22 versus kI, respectively, where A1 ⬎ A2 共the case A2 ⬎ A1 is symmetrical兲. Parameters are chosen according to their corresponding experimental values:12
(d) FIG. 2. Projections of solution locus: 共a兲 A1 = 冑u21 + v21 vs kI and 共b兲 A2 = 冑u22 + v22 vs kI for N = 2. As kI is decreased through the value denoted by the vertical line kB1, a S-N bifurcation occurs. The vertical line kB2 denotes an unstable-unstable pair bifurcation. Oscillations of distinct amplitudes occur for 0 ⱕ kI ⬍ kB1. 共c兲 For a system of N = 11 beams, equilibrium solutions Ai vs the coupling parameter kI. ILMs are possible for 0 ⬍ kI ⬍ kB ⬇ 0.008 N / m. 共d兲 A representative ILM state.
共mi,bi,k2i,k4i, f, ␣兲 = 共5.46 ⫻ 10−13 kg, 6.24 ⫻ 10−11 kg/s, 0.303 N/m, 5 ⫻ 108 N/m3, 1.25 ⫻ 105 Hz, 1.56 ⫻ 104 m/s2兲.
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Chen et al.
there are three homogeneous solutions 共denoted as HOSs兲 where the dynamical states of the two oscillators are identical, that is, HOS1:兵共A1, 1,A2, 2兲兩A1 = M 1, 1 = T1,A2 = M 1, 2 = T1其, HOS2:兵共A1, 1,A2, 2兲兩A1 = M 2, 1 = T2,A2 = M 2, 2 = T2其, and HOS3:兵共A1, 1,A2, 2兲兩A1 = M 3, 1 = T3,A2 = M 3, 2 = T3其, and three heterogeneous solutions 共HESs兲, HES1:兵共A1, 1,A2, 2兲兩A1 = M 1, 1 = T1,A2 = M 2, 2 = T2其, HES2:兵共A1, 1,A2, 2兲兩A1 = M 1, 1 = T1,A2 = M 3, 2 = T3其, and FIG. 3. 共Color online兲 Time series x1共t兲 and x2共t兲 共thin solid and dashed traces兲 of unsynchronized solution U1 and their amplitudes obtained by the averaged model 共thick horizontal solid and dashed lines兲 for KI = 0.005 N / m.
There are three synchronized solutions denoted by S1, S2, and S3 and four unsynchronized solutions labeled U1, U2, U3, and U4. The stable 共unstable兲 solutions are represented by solid 共dashed兲 lines. The vertical line denoted by kB1 indicates a saddle-node 共S-N兲 bifurcation point. We see that in the parameter region kI ⬍ kB1, there is a stable equilibrium solution 共U1兲 for which the oscillation amplitudes of the two beams are quite different, besides the existence of stable synchronized motions, which has the feature of ILM. To justify the use of the averaged system for approximating the driven system equation 共4兲, we plot the time series x1共t兲 and x2共t兲 obtained directly from Eq. 共4兲 共thin traces兲 and the amplitudes from the averaged system, as shown in Fig. 3 for KI = 0.005 N / m. We observe that the amplitudes of the time series agree with those from the averaged system very well. We can now analyze the distinct dynamical states and their origins by focusing on the solutions in the decoupled limit 共or anticontinuous limit兲 of Eq. 共5兲 where kI = 0. A decoupled system can be transformed into polar coordinate as
冋
册
1 dAi ⍀⍀0imiAi = − − mi␣ sin i , dt 2⍀ Qi
冋
册
3k4iA3i 1 di − mi␣ cos i , = 共⍀20i − ⍀2兲Ai + 4 dt 2⍀Ai
共7兲
where Ai = 冑u2i + v2i and i are the radial and angular coordinates of 共ui , vi兲 共Ai and i also denote the amplitude and the phase angle of xi, respectively兲. Consider the static solutions at the decoupled limit of system 共5兲. There are in total three hyperbolic equilibria for each subsystem 共7兲: S1i ª 兵共Ai , i兲 兩 Ai = M 1 , i = T1其, 2 Si ª 兵共Ai , i兲 兩 Ai = M 2 , i = T2其, and S3i ª 兵共Ai , i兲 兩 Ai = M 3 , i = T3其, where i = 1 , 2, M 1 = 2.5218⫻ 10−7 m, M 2 = 9.6173 ⫻ 10−6 m, M 3 = 9.3655⫻ 10−6 m, T1 = 1.0005, T2 = −0.0176, and T3 = −0.9828. For the case of A1 ⱖ A2, there are six solutions at the decoupled limit. In particular,
HES3:兵共A1, 1,A2, 2兲兩A1 = M 2, 1 = T2,A2 = M 3, 2 = T3其 共the other three HESs for A2 ⬎ A1 are considered to be the same as HES1, HES2, and HES3 due to symmetry兲. Among these solutions, HOS3, HES2, and HES3 are unstable since they have the unstable equilibrium S3i in at least one of the two subsystems. Since all the solutions at the decoupled limit are hyperbolic, when the value of kI is continued from 0, the stabilities of the solutions can be maintained, as stipulated by the local stability theorem.26 In Figs. 2共a兲 and 2共b兲, the solutions S2, S1, and S3 originate from HOS1, HOS2, and HOS3, respectively, and the solutions U1, U3, and U2 are extensions of HES1, HES2, and HES3, respectively. Since ILMs are spatially heterogeneous and are physically observable, they can only be continued from stable HESs. Therefore, among the solutions in Figs. 2共a兲 and 2共b兲, only U1 satisfies the condition and represents an ILM. B. N „N > 2… coupled MEM oscillators
We now treat the general case of N ⬎ 2. Following the same approach as for the N = 2 case, we note that, in the bistable regime, a single, decoupled beam has three hyperbolic equilibria: two stable and one unstable. For kI = 0, there are 3N hyperbolic equilibria. As these solutions are analytically continued from kI = 0, their number and stabilities remain the same, as guaranteed by the local stability theorem. There thus exists a finite parameter regime kI ⲏ 0 in which each beam has two possible stable solutions: one of small and the other of large amplitude. An ILM corresponds to the situation where a few of the beams are in the large-amplitude state, while the remaining beams oscillate with small amplitudes. An examination of the basin structure of the singlebeam dynamics reveals that the basin of the large-amplitude stable equilibrium is typically much smaller than that of the small-amplitude stable equilibrium. Thus, from random heterogeneous initial conditions, majority of the beams oscillate with small amplitude in the steady state while only a few may oscillate with relatively large amplitude. This indicates that, from the viewpoint of dynamics, ILMs are generic in the sense that the opposite situation where many more beams oscillate with large amplitude is typically unlikely. Physi-
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Dynamical mechanism of ILMs
cally, ILMs also represent collective motions of the system that are energetically favorable. To illustrate how our analysis works, we present an example of finding an ILM for N = 11, where the sixth beam oscillates with a high amplitude. In this case, we study the beam arrays with identical beam lengths and the parameters are set to be the same as the ones employed in Figs. 2共a兲 and 2共b兲. A solution for kI = 0 thus represents the state where this beam is initiated in the high-amplitude basin while the remaining ten beams are initially set to oscillate with low amplitude. Numerically obtained continuation of these solutions as kI is increased from zero is shown in Fig. 2共c兲, where the vertical line at kB denotes a S-N bifurcation point, beyond which only synchronized beam dynamics can occur. ILMs can be found for 0 ⬍ kI ⬍ kB ⬇ 0.008 N / m. The spatiotemporal evolution of the beam energies is shown in Fig. 2共d兲 for kI = 0.005 N / m, where a spatially localized behavior in the energy can be seen. While a stability analysis indicates that ILMs can arise typically in MEM oscillator arrays, it does not guarantee that ILMs can actually be observed in, for instance, a specific experiment. In previous works concerning MEM arrays with alternating beam lengths,12,16,17 it has been argued that, in order for ILMs to occur, 共1兲 the initial state of the MEM beam system should be random to allow for spatial heterogeneity and 共2兲 the frequency of the external driving should be increased gradually with time 共frequency chirping兲 to enhance the heterogeneity. Our computations reveal, however, that both conditions can be relaxed. In particular, the degree of heterogeneity in the initial condition distribution can be made arbitrarily small. For instance, we can actually use null initial conditions for all beams. A new finding is that spatiotemporal chaos can occur typically in the parameter regime of low driving frequency due to the uneven distribution of the beam length, which can serve as the source of spatial heterogeneity for beam dynamics. Because of chaos, the required frequency chirping scheme can be replaced by a more abrupt frequency-changing scheme. Qualitatively, this can be seen by noting that chaos contains an infinite number of possibilities for dynamical state. Once the system is in a chaotic regime, a change in the driving frequency can stabilize the system in one of the ILM states. Depending on the amount of frequency change, different ILMs can be realized. Figure 4 presents a space-time plot of the amplitudes of oscillating beams in the same system that has been used in previous experimental and numerical studies.12 There are N = 256 beams with alternating lengths. The parameters are12 共mi,bi,k2i,k4i兲 = 共5.46 ⫻ 10−13 kg, 6.24 ⫻ 10−11 kg/s, 0.303 N/m, 5 ⫻ 108 N/m3兲
Chaos 19, 013127 共2009兲
FIG. 4. Spatiotemporal chaos and ILM in a MEM oscillator array of N = 256 beams for kI = 0.0241 N / m and ␣ = 1.56⫻ 104 m / s2. The driving frequency is increased abruptly from 1.47⫻ 105 to 1.50⫻ 105 Hz at tstep = 0.013 s.
the driving frequency is increased abruptly, generating an ILM about site 14. This result shows that with the heterogeneity associated with spatiotemporal chaos, ILMs can be generated in a noise-free system with uniform initial conditions. IV. ORIGIN OF SPATIOTEMPORAL CHAOS
We consider systems of alternating lengths to demonstrate that spatiotemporal chaos is typical. Chaos plays the role of “internal noise” so that additional amplification as offered by frequency chirping is not necessary for the occurrence of ILMs. In this regard, we note that in the coupled NLSE, chaos has been found to be ubiquitous.22,23 In both the original MEM array system and the averaged system as described by the driven damped NLSE, we find that chaos is common. In fact, when the system size is not small 共say, larger than a few coupled units兲, spatiotemporal chaos can arise. One example is shown in Fig. 4. Another example is shown in Fig. 5共a兲 for a system of N = 16 coupled oscillators. The physical parameters and initial conditions are set to be the same as the ones used in Fig. 4 except for the smaller number of beams. The occurrence of chaos in the averaged system is exemplified in Fig. 5共b兲 in which the parameters and initial conditions of Fig. 5共a兲 are employed. Qualitatively, the spatiotemporal patterns in Figs. 5共a兲 and 5共b兲 are similar to each other, indicating that the averaged model equation 共5兲 captures not only the static but also the dynamical behaviors of the original system equation 共4兲. Since the averaged system is amenable to bifurcation analysis, we shall use it to explore the origin of spatiotemporal chaos.
for odd i, the long beams, and 共mi,bi,k2i,k4i兲 = 共4.96 ⫻ 10−13 kg, 5.67 ⫻ 10−11 kg/s, 0.353 N/m, 5 ⫻ 108 N/m3兲 for even i, the short beams. The initial displacements and velocities are all set to be zero. Figure 4 reveals a highly irregular behavior before tstep = 0.013 s both in space and in time, which is characteristic of spatiotemporal chaos. At tstep,
A. Key bifurcations in the averaged model
In Figs. 2共a兲–2共c兲, it is shown that the stable HESs disappear at certain bifurcation points in kI. In the averaged model for the alternated-beam-length array system, there are two types of bifurcations leading to the destruction of some ILM state and the rise of LESs where all the beams oscillate with low amplitudes. These are the S-N and the stability-
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Chaos 19, 013127 共2009兲
Chen et al. −5
1.4
x 10
S−N bifurcation
1.2
A1,2,...,16 (m)
1 0.8 0.6 0.4 0.2
(a) 0 0
(a)
0.005
0.01
0.015
0.02
k (N/m)
0.025
I
−5
1.4
x 10
S−T bifurcation
1.2
A1,2,...,16 (m)
1
0.2
FIG. 5. 共a兲 Spatiotemporal chaos in an original MEM array system of size N = 16. 共b兲 Spatiotemporal plot of Ai 共i = 1 , 2 , . . . , 16兲 in the corresponding averaged system 共5兲.
B. Occurrence of spatiotemporal chaos
Figure 7 shows a magnified bifurcation diagram combining Figs. 6共a兲 and 6共b兲. The two types of bifurcation divide the relevant parameter interval into three distinct regions. A LES exists in regions I and II, and it is destroyed by the S-N bifurcation that occurs at the value of kI indicated by the right vertical line. In region I, an ILM fixed point also exists, and it becomes an unstable saddle at the value of KI denoted by the left vertical line via an S-T bifurcation. Thus, in region I, there are two stable fixed points, one corresponding to
0.6 0.4
(b)
(b)
0 0
0.005
0.01
0.015
0.02
kI (N/m)
0.025
FIG. 6. 共Color online兲 For the averaged system equation 共5兲 of N = 16 oscillators: 共a兲 A S-N bifurcation for an LES and 共b兲 an S-T bifurcation for an ILM.
x 10
1.2
S−N bifurcation
S−T bifurcation
−5
1.4
II
I
III
1
A8(m)
transition 共S-T兲 bifurcations, where the latter occurs when a stable fixed point loses its stability and becomes an unstable saddle. In a MEM oscillator array, the destruction of ILMs and LESs can be attributed to the two types of bifurcations. Two examples for N = 16 are presented in Fig. 6 where the stabilities of the particular solutions are determined by the sign of the largest eigenvalue of the Jacobian matrix in system 共5兲. Figure 6共a兲 is a bifurcation diagram of an LES. It can be seen that the LES is destroyed when it collides with another unstable fixed point at the S-N bifurcation that occurs at kI ⬇ 0.0238 N / m. The bifurcation diagram of an ILM is shown in Fig. 6共b兲, where the ILM becomes unstable for kI ⲏ 0.0234 N / m. These bifurcations thus provide a base for understanding the occurrence of spatiotemporal chaos.
0.8
Stable ILM 0.8
ILM saddle
0.6 0.4 LES
0.2 0 0.02
0.021
0.022
0.023
0.024
kI(N/m) FIG. 7. 共Color online兲 Typical bifurcation diagram of LES and ILM for the case of N = 16.
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Chaos 19, 013127 共2009兲
Dynamical mechanism of ILMs −6
4
x 10
3 2
x30 (m)
1 0 −1 −2
FIG. 8. Schematics of the dynamics in different regions of kI.
−3
(a)
FIG. 9. Space-time plot of chaotic motion for N = 16. Signatures of various ILM saddles are indicated by circles.
0.005
0.01
0.015
0.02
0.025
150
200
250
t (s)
−6
2
x 10
0
Displacement (m)
the LES and another to ILM. In region II, only the LES is stable, and there are no stable fixed points in region III. Shown in Fig. 7 is a particular pair of ILM and LES solutions. Note that, in the full system, there are many pairs of such solutions. A schematic illustration of the dynamics about these stable and unstable fixed points is shown in Fig. 8. In region I, the ILMs are denoted by solid circles and the LESs are denoted by solid squares. Because of the existence of multiple stable attractors 共multistability兲, the phase space is divided according to the basins of attraction of these attractors. In region II, the ILMs are unstable but the LESs are still stable attractors. In region III, all orbits are unstable saddles, and their stable and unstable manifolds typically form a network of homoclinic and heteroclinic crossings, which generates horseshoe dynamics and henceforth chaos. The chaotic attractor thus contains all the unstable saddles as its skeleton, and a typical trajectory will visit the neighborhoods of the saddles alternately in time. Signatures of such saddles can be found in the space-time plot of the chaotic attractor, as shown in Fig. 9, where the ILM saddles are circled. When proper external perturbations are applied, some of the ILM saddles can be stabilized, generating stable, physically observable ILMs. As we will demonstrate next, frequency modulation is a natural means of such perturbation.
−4 0
−2
−4
−6
−8
(b)
−10 0
50
100
Site
FIG. 10. 共Color online兲 共a兲 Time series from spatial site 30 and 共b兲 spatial profile at t = 0.0021 s for a MEM oscillator-array system of size N = 256.
V. ROUTE TO ILMS FROM SPATIOTEMPORAL CHAOS
Previous works have revealed that modulational instability is necessary to induce energy-localized motions in a nonlinear lattice.24 This principle can also be applied to MEM oscillator-array system. In particular, we can set the system in a spatiotemporally chaotic state, a kind of modulational instability. Applying proper adjustments to the driving frequency can then stabilize the system about one of the ILMs. To obtain insight into the working of this mechanism, we plot in Figs. 10共a兲 and 10共b兲 a typical time series obtained at an arbitrary spatial site and a spatial plot at a fixed instant of time, respectively. It can be seen from Fig. 10共a兲 that there are bursts of the trajectory in various short time intervals, signifying modulational instability. A distinct feature of Fig. 10共b兲 is, however, a large spike at a certain spatial site. The intermittent bursts in the time series in Fig. 10共a兲 and the spatial spike in Fig. 10共b兲 suggest that the trajectory visits the neighborhoods of various ILM saddles. When the system trajectory moves near a particular ILM saddle, a perturbation in the form of sudden frequency change can stabilize the saddle. When this occurs, the system is likely to be in the basin of attraction of the corresponding stable ILM attractor
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1.485
x 10
β
2
β1
f (Hz)
1.48
1.475
1.47
LES 1−ILM 2−ILMs 3−ILMs 0.02
β0
0.022
Spatiotemporal chaos 0.024
kI (N/m)
0.026
FIG. 11. 共Color online兲 Boundaries between various asymptotic states including spatiotemporal chaos and various ILMs for a MEM system of N = 16 beams in a two-dimensional, experimentally accessible parameter region.
since it is the continuation of the corresponding saddle. It is in this sense which we say that chaos provides a platform for generating ILMs through frequency modulation. To further explore the chaos route to ILMs, it is useful to focus on the two-dimensional parameter space 共kI , f兲, which is experimentally accessible. To facilitate testing of our predictions, we shall choose the ranges of the parameters as in typical experiments.11,12,16 Figure 11 shows, for a system of N = 16 beams, boundaries between distinct asymptotic states including spatiotemporal chaos and various ILMs. In the figure, the region to the right of the “square” boundary is for spatiotemporal chaos and the region to the left denotes the state where the low-amplitude oscillation mode is stable but the high-energy mode is unstable 共LES region兲; one ILM can be expected in the region to the left of the “diamond” boundary, two ILMs can occur in the region to the left of the “triangle” boundary, and so on. Figure 11 indicates that, for example, LESs and 1-ILM states are both possible in the parameter region in between the diamond and the triangle boundaries. For a given set of parameter values, distinct ILM states have different basins of attraction in the phase space. Now imagine an experimental situation where kI is fixed and the driving frequency is increased, as indicated by the vertical dashed line in Fig. 11 where 0, 1, and 2 are the intersections between the dashed line and the boundaries of the LES, 1-ILM, and 2-ILM regions, respectively. For f ⬍ 0, the system has no stable equilibrium and it exhibits spatiotemporal chaos. For 0 ⬍ f ⬍ 1, there is one stable equilibrium in the averaged system, corresponding to stable oscillations of low oscillation energy state in the actual system. In this case, all initial conditions generate trajectories that approach the LES. For 1 ⬍ f ⬍ 2, both LESs and 1-ILM states are possible. That is, depending on the choice of initial conditions, all beams in the system can oscillate with low energy or the system can exhibit one ILM. For 2 ⬍ f ⬍ 3, the sys-
FIG. 12. 共Color online兲 A schematic illustration of allowed and forbidden transitions between various states in a MEM oscillator array by frequency control.
tem can be in one of the three possible asymptotic states: LES, 1-ILM, and 2-ILM. How does one physically realize ILMs? Focus, for instance, on the situation where the system exhibits one ILM. Figure 11 suggests one approach: initially set the system in spatiotemporal chaos by choosing a relatively small value of f and then increase the frequency to some value in between 1 and 2. There is then a finite probability to drive the system into a 1-ILM state. Suppose now the frequency is reduced to some value in between 0 and 1; the system will then be in an LES. Equivalently, we say that there is a transition from a 1-ILM state to LES. A key point is that, it is practically impossible to change the system back into the 1-ILM state from the LES by increasing the frequency from some value in between 0 and 1 to some value in between 1 and 2. The reason is that, once the system settles down in an LES, it will always be in its basin regardless of any frequency increase. In this sense, the transition from an LES directly to a 1-ILM state is forbidden. This line of reasoning suggests that, in the absence of any random factors, the only way to excite the system to a 1-ILM state is through spatiotemporal chaos. The same consideration applies to higherorder ILM states. For instance, a sufficient amount of frequency increase can bring the system from spatiotemporal chaos to a 2-ILM state. There can be transitions from a 2-ILM state to an LES or to a 1-ILM state, but the opposite transitions are not allowed. The possible transitions between various dynamical states are shown schematically in Fig. 12. The message is that, for a noiseless system, spatiotemporal chaos provides a natural platform for exciting ILMs in MEM oscillator arrays. VI. CONCLUSION
We have carried out a nonlinear-dynamical-system analysis of the phenomenon of ILMs in MEM oscillator arrays. Utilizing the averaging method allows the origin of various oscillations, including ILMs, to be identified and their stabilities to be analyzed. In particular, the dynamical origin of spatiotemporal chaos is clarified in the averaged model. A bifurcation analysis then enables the scenarios for physically realizing various ILMs to be predicted. While previous works have emphasized that ILMs can occur without any defects in the system, some sort of random perturbation
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Dynamical mechanism of ILMs
is thought to be essential for experimentally observing ILMs in MEM oscillator arrays. Our study suggests that even this requirement can be relaxed. Insofar as the system is nonlinear, spatiotemporal chaos can arise, which contains all possible unstable modes; some of them are reminiscent of various ILMs as they become unstable 共ILM saddle兲. These saddles can connect together as a heteroclinic network, inducing spatiotemporal chaos. Parameter modulations such as frequency control are therefore capable of “restoring” the ILMs from the corresponding ILM saddles. This dynamicsbased view represents an alternative but more comprehensive approach to ILMs in coupled MEM oscillators, whose occurrence in physical systems has proven to be ubiquitous. An implication of our results to experimental study of ILMs in MEM array systems is that the requirement of external noise can be relaxed. In particular, the experimental system in Ref. 12 employed bi-element beam arrays with perturbations 共noise兲 to induce modulational instability and spatial heterogeneity, which are essential for creating ILMs. In MEM systems, it is generally nontrivial to generate spatially heterogeneous thermal noise and to use it to excite ILMs. It is thus desirable to search for an alternative mechanism to generate spatial heterogeneity required for ILMs. Our results have shown that, even in the absence of noise, modulational instability is possible with appropriate design of the system, such as using beams of alternating lengths. In fact, spatiotemporal chaos can serve effectively as a robust type of modulational instability, from which ILMs can be generated as a natural consequence of the collective dynamics of the system. External noise and amplification are not necessary for ILMs. We expect this finding to be useful for experimental study of ILMs in MEM array systems. ACKNOWLEDGMENTS
This work was supported by AFOSR under Grant No. FA9550-06-1-0024.
1
S. A. Kiselev, S. R. Bickham, and A. J. Sievers, Comments Condens. Matter Phys. 17, 135 共1995兲. 2 P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, Int. J. Mod. Phys. B 15, 2833 共2001兲. 3 S. Flach and C. R. Willis, Phys. Rep. 295, 181 共1998兲. 4 R. Lai and A. J. Sievers, Phys. Rep. 314, 147 共1999兲. 5 S. Aubry, Physica D 103, 201 共1997兲. 6 R. S. MacKay and S. Aubry, Nonlinearity 7, 1623 共1994兲. 7 P. Binder, D. Abraimov, A. V. Ustinov, S. Flach, and Y. Zolotaryuk, Phys. Rev. Lett. 84, 745 共2000兲. 8 R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 2726 共1999兲. 9 S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474 共2001兲. 10 M. Sato and A. J. Sievers, Nature 共London兲 432, 486 共2004兲. 11 M. Sato, B. E. Hubbard, and A. J. Sievers, Rev. Mod. Phys. 78, 137 共2006兲. 12 M. Sato, B. E. Hubbard, A. J. Sievers, B. Ilic, D. A. Czaplewski, and H. G. Craighead, Phys. Rev. Lett. 90, 044102 共2003兲. 13 Classical and Seminar Papers to 1990, edited by W. S. Trimmer 共IEEE, New York, NY, 1997兲. 14 M. Sato, B. E. Hubbard, A. J. Sievers, B. Ilic, and H. G. Craighead, Europhys. Lett. 66, 318 共2004兲. 15 A. J. Dick, B. Balachandran, and C. D. Mote, Jr., Proc. SPIE 6166, 61660N 共2006兲. 16 M. Sato, B. E. Hubbard, L. Q. English, A. J. Sievers, B. Ilic, D. A. Czaplewski, and H. G. Craighead, Chaos 13, 702 共2003兲. 17 P. Maniadis and S. Flach, Europhys. Lett. 74, 452 共2006兲. 18 Q. Chen, L. Huang, and Y.-C. Lai, Appl. Phys. Lett. 92, 241914 共2008兲. 19 S. K. De and N. R. Aluru, Phys. Rev. Lett. 94, 204101 共2005兲; J. Microelectromech. Syst. 15, 355 共2006兲. 20 Y. S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 共1992兲. 21 J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott, Physica D 16, 318 共1985兲. 22 M. J. Ablowitz, B. M. Herbst, and C. M. Schober, Physica A 228, 189 共1996兲. 23 E. Shlizerman and V. Rom-Kedar, Phys. Rev. Lett. 96, 024104 共2006兲. 24 I. Daumont, T. Dauxoisz, and M. Peyrard, Nonlinearity 10, 617 共1997兲. 25 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 共Springer, New York, 1983兲. 26 E. Groesen and E. Jager, Studies in Mathematical Physics 共Elsevier Science, Amsterdam, The Netherlands, 1991兲, Vol. 2.
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Four twins for a paradox: On “sensitive” twins and the biological counterpart of the “twin paradox” Fortunato A. Asciotia兲 Faculty of Engineering, University of Reggio Calabria, Feo di Vito, I-89100 Reggio Calabria, Italy
共Received 24 November 2008; accepted 22 January 2009; published online 13 March 2009兲 Monozygotic twin 共MZT兲 epigenetic development, i.e., aging, diverges largely in time despite the initially very small genetic differences between MZTs. This fact is interpreted as a “sensitivity to initial conditions” phenomenon, a common property of either deterministic or stochastic chaotic systems. Some of the biotheoretical implications stemming from this empirical observation are briefly discussed here, while an actual measure of MZT epigenetic time divergence is given through an estimate of the 共Stochastic兲 Lyapunov exponents 共LEs兲 共i.e., the rate of exponential time divergence兲. These results suggest a reconsideration of the Langevin–Einstein thought experiment known as the “twin paradox.” At least four twins are necessary in order to take into account the inertially independent divergent aging described here. Alternatively, LE estimates, like those given here, should be used. Finally suggested in the actual special-relativity experiments is the replacement of clocks with some nonlinear 共chaotic兲 forced oscillator. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3081043兴 The recent findings showing a sharp divergence in the epigenetic development, and thus aging, of monozygotic twins (MZTs), despite their initial (i.e., at birth) very small genetic difference „È0.1%…, were interpreted as a phenomenon of “sensitivity to initial conditions,” a property typical of those nonlinear systems that are capable of exhibiting various types of (noisy) chaotic dynamics (latu sensu). Some intriguing theoretical aspects were briefly discussed and an actual measure of the MZT epigeneticaging divergence was then made through an estimate of the (stochastic) Lyapunov exponents (LEs) (i.e., the rate of the exponential divergence in time). Because of this exclusively biological divergent aging, a “biological correction” (normalization) was suggested for the Langevin– Einstein “twin paradox,” MZT aging in a (sometimes very) different way although moving at the same speed in space-time (i.e., on the same inertial reference frame). The estimated stochastic LEs (SLEs) were proposed as a possible “normalization factor” capable of measuring the biological contribution to the differential aging caused by the relativistic time contraction that would affect, in the above thought experiment, the “speedy-traveler” twin. Finally, the suggestion was made of replacing, in the actual experiments aimed at testing the twin paradox, clocks with some appropriate nonlinear mechanical devices capable of exhibiting a somehow recordable chaotic behavior.
I. INTRODUCTION
In a recent paper, Fraga et al.1,2 showed that the epigenetic expression in monozygotic 共MZ兲 human twins diverges largely in time despite the very small differences 共⬃0.1%兲 in the initial 共i.e., at birth兲 genetic pattern between them 共by definition of MZTs兲. a兲
Electronic mail: [email protected].
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This fact represents clearly a sensitivity to initial conditions phenomenon3,4 which is a typical property of either deterministic or stochastic nonlinear dynamical systems that may exhibit various types of chaotic dynamical behavior and possess either a “strange attractor” 共purely deterministic case兲3,4 or various types of noisy attractors5 and “chaotic repellors”6 共stochastic-deterministic case兲. It is well known that even a tiny difference between two initial conditions at the start of a process whose nature is either deterministically 共noisy contaminated or not兲 or stochastically nonlinear will grow exponentially with time and that the trajectories representing the system’s states will diverge accordingly.3,6 This seems to be exactly the case for the MZTs of Fraga et al.1 In effect, the occurrence of nonlinearity in the MZT genetic-epigenetic system is evident from the variation in the number of differential bands of sibling-specific changes of DNA methylation 关Fig. 2共A兲 of Fraga et al.兴 that increases from about 5 in 3 year old twins to 75 in 50 year old ones, i.e., a rate of ⬃1.5 differential bands/year. As for the number of overexpressed and repressed genes 关Fig. 4共D兲 of Fraga et al.兴, they both increase in 47 years from 920 to 3757 共i.e., a rate of ⬃60 genes/ year兲 and from 200 to 786 共i.e., a rate of ⬃12 genes/ year兲, respectively. It should be stressed, however, that the findings of Fraga et al.1 indicate also that the larger the difference in the environmental and lifestyle conditions of twins, the stronger the epigenetic-aging divergence will be. A great deal of external “environmental stochasticity” and not only an “internal” exclusively genetic nonlinear “machinery” is therefore implied in the origin of the observed sensitivity to initial conditions 共see also the discussion in Refs. 1 and 2兲. On the other hand, evidence reported in Refs. 7 and 8, as outlined in Ref. 2, clearly shows that even under rigorously controlled environmental conditions, divergence in mortality, and thus aging, is still the case in the siblings of clonal animals. Therefore, internal genetic-epigenetic nonlinear
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mechanisms seem capable of generating, even without the intervention of any sort of external environmental noise, a sensitive to initial condition process. Whether these mechanisms are deterministically chaotic 共as hypothesized in Refs. 9 and 10兲, or not remains to be ascertained. However, from the above results it is evident that a noisy-contaminated or even noisy-driven nonlinear dynamics is occurring in the epigenetic “unfolding” of MZTs. The precise nature of this dynamics needs to be understood and further appropriate investigations should be pursued in order to do so 共also through a further analysis of the same data of Fraga et al.,1 see the Appendix兲. In effect, understanding if an evidently noisy-contaminated or even noisy-driven deterministic chaotic dynamics is actually governing the epigenetic unfolding of genetic information may have important implications in the development of an alternative paradigm in life sciences, as stressed, since 1997, by Strohman.11 II. LYAPUNOV EXPONENT ESTIMATES OF MZ TWIN EPIGENETIC DIVERGENCE
Apart from the above more speculative arguments 共which will be broadly explored in a forthcoming paper, see the Appendix兲, there is, though, a rigorous way of quantifying the epigenetic exponential divergence in twins: i.e., the so-called LEs3,5 or, better, their generalized stochastic version, the SLEs.12–14 These latter can be defined and measured regardless of the dynamics governing the process,13,14 even though the interpretations of the results would vary according to the underlying dynamics.12 SLEs 共as well as their deterministic version兲 are the coefficients of the exponential divergence of initially nearby points and they thus measure the rate of divergence 共convergence, when negative兲 of a sensitive to initial conditions phenomenon.3,13–15 LEs are defined in the following usual way:3–6,12,13 Given ␦共0兲, the difference 共in absolute value兲 between two initial states of a 共D-dimensional兲 dynamical system, and t, the time elapsed from the initial time t0 = 0, then the increase in ␦共t兲 will be described by the exponential equation
␦共t兲 ⬇ ␦共0兲eDt ,
共1兲
where the Ds are the LEs of the D-dimensional dynamical system we are considering, and 兵1 , 2 , 3 , . . . , D其 is the socalled LE spectrum. Positive indicate a divergent behavior and negative means a decrease in divergence with a possible “folding” of the trajectories and a consequent “reinjection” of them near the original initial states from which a new divergent dynamics restarts, although this is not always the case for stochastic or “mixed” stochastic-deterministic systems.13 The stochastic version of LEs 共i.e., the SLEs estimated here兲 does not differ in the computational procedure 共in the simple case discussed here兲; SLEs include though a “noisy part” that cannot be separated from the deterministic one. Their significance is therefore different from that of 共purely deterministic兲 LEs.12–14 Here this mathematical-physical approach is applied to the MZT aging data shown in Ref. 1, namely, 共i兲 global 5 mC
TABLE I. SLE estimates of twin divergent aging 关from the data of Fraga et al. 共Ref. 1兴: MZT aging variables 共first column兲; time of measurements in years 共second column兲; ␦共t兲 differences 共in absolute value兲 between aging variables at 3 and 50 years 共gathered from the paper of Fraga et al. according to the procedure given in Sec. II兲 共third column兲; SLE, rate of exponential divergence ⌬ ln ␦共t兲 / ⌬t, where ⌬ ln ␦共t兲 = ln ␦共50兲 − ln ␦共3兲, and ⌬t = 47 共i.e., 50–3 years兲 共fourth column兲. t 共years兲
␦共t兲
SLE 共year−1兲
Global 5 mC DNA content
3 50
0.002 0.010
0.034
Histone H4 acetylation
3 50
0.003 0.125
0.079
3 50 3 50
0.002 0.054 0.19 0.79
MZT aging variables
Histone H3 acetylation 共No. of overexpressed genes兲 − 共No. of repressed genes兲a
0.070 0.030
a
Note that here it is the difference between overexpressed and repressed genes in 3 and 50 year old twins that has been measured. It should not be confused with the exponential increase in overexpressed and repressed genes taken separately, as previously given in the text 共although they are evidently related兲.
DNA content, 共ii兲 histone H4 and histone H3 acetylation 关Fig. 1共C兲 of Fraga et al.兴, and 共iii兲 the number of overexpressed and repressed genes 关DNA array analysis, Fig. 4共D兲 of Fraga et al.兴. From Figs. 1共C兲 and 4共D兲 of Fraga et al. one can easily estimate 共with a reasonably good approximation兲 the divergence values for the above variables. Thus, Figs. 1共C兲 and 4共D兲 of Fraga et al. were appropriately magnified from the PDF version of their paper via ADOBEPHOTOSHOP® and the values of each variable of interest were then acquired through the appropriate ruler. Since the original graph of Fraga et al., where the twin epigenetic divergences were plotted, did show only a subportion of each Cartesian plane, data required to be normalized in order to get homogeneous and comparable values. Therefore, 5 mC DNA and H3 and H4 acetylation histone percentages were first rescaled to 0.0 共i.e., to the origin of each plot兲 simply by subtracting the x-axis shift actually observed in the original graphs of Fraga et al. These same values were then divided by 100 in order to get 关0 1兴 rescaled values from the percentage data 共see Table II in the Appendix兲. Overexpressed-repressed gene numbers required instead simply to be divided by their absolute maximum 共since their values were already given from the origin in the graphs of Fraga et al.兲 共see Table III in the Appendix兲. The time differences 共in absolute values兲, ␦共t兲, between such normalized values were finally used to estimate the SLEs accordingly to Eq. 共1兲 共see Table I and Sec. III below兲. III. RESULTS
Table I shows the difference data for each MZT aging variable along with the estimates 关according to the Eq. 共1兲兴 of the 共first and single, i.e., one-dimensional兲 SLE 共i.e., the rate of exponential divergence measured over 47 years兲. From this table, it is evident that the rate of divergence in aging is higher for H3 and H4 histones and lower for 5
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mC DNA and the number of overexpressed versus repressed genes. The former are, thus, better tracers of the MZT aging divergence than the latter. IV. THE NONLINEARLY REVISITED “TWIN PARADOX”
The reason for having pursued the quantification of the twin epigenetic-aging divergence through the SLEs 共as given in the above table兲 lies in the main point this paper seeks to address. It is as follows: In the famous thought experiment of special relativity known as the twin paradox,15–17 one of two twins is hypothetically capable of traveling at a speed close to the limit one 共the velocity of light兲 while the other would stay “still” on our 共or his/her兲 planet 共actually he/she would move “slowly,” i.e., at the same velocity as the planet兲. At the end of his/her journey the “traveler” twin landing on Earth 共or his/her planet兲 would find his/her “sedentary” twin older than himself/herself 共within the special-relativity frame兲. This would be due to the purely physical effect of the difference in the velocity at which the twin moved 共i.e., the relativistic contraction of time for the traveler twin兲. The fact that 共MZ兲 twins do exponentially diverge in their epigenetic expression, and thus aging, because of purely biological reasons 共although they move at the same speed on the same reference frame, e.g., the same planet兲 would make it necessary now for a biological correction or normalization which is needed in order to take into account this “inertially independent” divergent aging. Note also that the “paradoxical twin” would live separate lives 共on the planet the sedentary and on the spacecraft the traveler兲 and this, as before stated, would accentuate their biological aging divergence according to the results of Fraga et al.1 This inertially independent divergence can be considered the “biological counterpart” of the twin paradox, with the above SLE estimates just giving now the magnitude of this exclusively biological effect 共be it deterministically chaotic or not兲. Another possible way of removing this biological inertially independent part from the physical “speeding up” effect could be that of using at least four twins: two travelers and two sedentary ones. The difference in the diverging of aging, i.e., the differences in the SLEs between these two pairs, gives the correct measure of the 共physical兲 relativistic time-contraction effect. The above considerations suggest also that it would be interesting in the actual experiments aimed at testing the twin paradox17 to replace the Newtonian periodical systems 共i.e., clocks兲 with some nonlinear devices capable of exhibiting, despite their simplicity, a somehow recordable chaotic behavior 共i.e., a forced periodical system兲. For example, it could be possible to modify the classical special-relativity time-dilation/contraction experiments conducted by Hafele and Keating18 where cesium beam atomic clocks were sent flying around the world in opposite West/East directions and the relativistic time contraction/dilation was measured through a comparison with reference “still clocks.” A nonlinear version of these same experiments, would see the replacement of those 共by definition兲 harmonic oscillators 共i.e., clocks兲 with a forced version of them 共e.g., an atomic version of Chua’s electronic circuit19兲. Forcing should
be chosen such that both devices would reach and keep an equally chaotic state while flying in opposite West/East directions, and the same should hold for the “still” reference devices. At this point, the independent 共and divergent兲 chaotic time evolution of these forced devices could be easily 共electronically兲 recorded and the relative 共this time nonstochastic兲 LEs estimated as well. Finally, the differences between these LEs would give a measure of the “right” 共relativistic兲 different-velocity effect that would affect the 共this time兲 nonlinear chaotic oscillators 共i.e., time dilation/contraction minus the “autonomous” chaotic time divergence due to the nonlinear nature of the systems used for the experiment兲. Indeed, future investigations 共both theoretical and experimental兲 on the dynamical behavior nonlinear systems may exhibit within the special-relativity framework seem worth to be pursued. ACKNOWLEDGMENTS
It is a pleasure to thank C. Franceschi who first directed my attention to the divergent aging in twins during his very interesting speech delivered at 15–19 July 2008 SPAISSummer School held in Isnello, Palermo, Sicily. I thank also D. Lucchesi and the other organizers of the same Summer School. I thank F. Oliveri, F. T. Arecchi, and the anonymous referees for their critical reading of the manuscript and for their helpful suggestions. I wish to dedicate this work to my wife Silvia who is presently just at the beginning of her lovely “work in progress” of generating our baby 共…twins?!兲. APPENDIX: DATA AND COMPUTATIONAL DETAILS
For actual and normalized MZT epigenetic-aging data gathered from Fraga et al.1 see Tables II and III. For further studies, note that a further analysis of the data of Fraga et al.1 TABLE II. Actual and normalized MZT epigenetic-aging data gathered from Fig. 1共C兲 of Fraga et al. 共Ref. 1兲 through ADOBEPHOTOSHOP® ruler: 5 mC, AcH4, and AcH3 twin aging variables 共first column兲 and their actual values at 3 and 50 years 共second column兲; rescaling to 0.0 共third column兲; ⫻100−1 共fourth column兲; differences at 3 and 50 years 共absolute values兲 共fifth column兲.
5 mC DNA
AcH4
AcH3
Xi共t兲 共t in years兲
Rescaling to 0.0
⫻100−1
X1共3兲 = 2.37 X2共3兲 = 2.57 X1共50兲 = 3.62 X2共50兲 = 4.65
−1.5= 0.87 −1.5= 1.07 −1.5= 2.12 −1.5= 3.15
0.009 0.011 0.021 0.031
X1共3兲 = 46.9 X2共3兲 = 47.2 X1共50兲 = 49.1 X2共50兲 = 61.6
−30= 16.9 −30= 17.2 −30= 19.1 −30= 31.6
0.169 0.172 0.191 0.316
X1共3兲 = 38.4 X2共3兲 = 38.6 X1共50兲 = 24.1 X2共50兲 = 18.7
−15= 23.4 −15= 23.6 −15= 9.10 −15= 3.70
0.234 0.236 0.091 0.037
␦共t兲 0.002 0.010
0.003 0.125
0.002 0.054
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TABLE III. As above 共Table II兲 but for the number of overexpressed versus repressed genes 关Fig. 4共D兲 of Fraga et al.兴: actual values at 3 and 50 years 共second column兲; normalized values 共to the maximum value, i.e., 3757兲 共third column兲; differences at 3 and 50 years 共absolute values兲 共fourth column兲.
No. of overexpressed genes vs No. repressed genes
Xi共t兲 共t in years兲
Xi共t兲 / max Xi共t兲
X1共3兲 = 200 X2共3兲 = 920 X1共50兲 = 786 X2共50兲 = 3757
0.053 0.245 0.209 1.000
␦共t兲 0.192 0.791
should consider the aging divergence 共measured through the SLEs兲 of those twins that lived nonseparate lives and showed the smallest differences in lifestyle and life habits. This would allow a better evaluation of the influence of the “environmentally independent” factors on twin aging divergence 共still exponential?兲. For speculative arguments, a broader account of some of the speculative aspects stemming from this note will be given in a further paper 共now in preparation兲 the title of which 共slightly “autobiographic”兲 is as follows: “To be chaotic or not to be, this is the question… Are complex
non-linear dynamics keeping alive 共and healthy兲 living systems?” M. F. Fraga et al., Proc. Natl. Acad. Sci. U.S.A. 102, 10604 共2005兲. G. M. Martin, Proc. Natl. Acad. Sci. U.S.A. 102, 10413 共2005兲. 3 L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life 共Princeton University Press, Princeton, NJ, 1988兲. 4 R. M. May, Proc. R. Soc. London, Ser. B 228, 241 共1986兲. 5 J. P. Crutchfield, J. D. Farmer, and P. A. Hubermann, Phys. Rep. 92, 45 共1982兲. 6 D. A. Rand and H. B. Wilson, Proc. R. Soc. London, Ser. B 246, 179 共1991兲. 7 J. R. Vanfleteren, V. A. De, and B. P. Braeckman, J. Gerontol., Ser. A 53, B393 共1998兲. 8 J. W. Vaupel et al., Science 280, 855 共1998兲. 9 P. C. M. Molenaar, D. I. Boomsma, and C. V. Dolan, Behav. Genet. 23, 519 共1993兲. 10 L. J. Eaves, K. M. Kirk, N. G. Martin, and R. J. Russell, Twin Res. 2, 43 共1999兲. 11 R. C. Strohman, Nat. Biotechnol. 15, 194 共1997兲. 12 B. Dennis, R. A. Desharnais, J. M. Cushing, S. M. Henson, and R. F. Costantino, Oikos 102, 329 共2003兲. 13 J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 共1985兲. 14 K. S. Chan and H. Tong, J. R. Stat. Soc. Ser. B 共Methodol.兲 56, 301 共1994兲. 15 P. Langevin, Scientia 10, 31 共1911兲. 16 A. Einstein, Die Naturwissen. 6, 697 共1918兲. 17 J. Preston and L. F. Wanex, Found. Phys. Lett. 19, 75 共2006兲. 18 J. C. Hafele and R. E. Keating, Science 177, 166 共1972兲. 19 T. Matsumoto, IEEE Trans. Circuits Syst. 31, 1055 共1984兲. 1 2
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Low dimensional description of pedestrian-induced oscillation of the Millennium Bridge Mahmoud M. Abdulrehem and Edward Ott University of Maryland, College Park, Maryland 20742, USA
共Received 21 October 2008; accepted 4 February 2009; published online 13 March 2009兲 When it opened to pedestrian traffic in the year 2000, London’s Millennium Bridge exhibited an unwanted, large side-to-side oscillation, which was apparently due to a resonance between the stepping frequency of walkers and one of the bridge modes. Models for this event, and similar events on other bridges, have been proposed. The model most directly addressing the synchronization mechanism of individual walkers and the resulting global response of the bridge-pedestrian system is the one developed by Eckhardt et al. 关Phys. Rev. E 75, 021110 共2007兲兴. This model treats individual walkers with a phase oscillator description and is inherently high dimensional with system dimensionality 共N + 2兲, where N is the number of walkers. In the present work we use a method proposed by Ott and Antonsen 关Chaos 18, 037113 共2008兲兴 to reduce the model of Eckhardt et al. to a low dimensional dynamical system, and we employ this reduced description to study the global dynamics of the bridge/pedestrian interaction. More generally, this treatment serves as an interesting example of the possibility of low dimensional macroscopic behavior in large systems of coupled oscillators. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3087434兴 In order to celebrate the new Millennium, a footbridge over the River Thames was designed and built. After an opening ceremony, an eager crowd of people streamed onto the bridge. As the number of people on the bridge increased, a relatively large lateral wobble of the bridge set in. Subsequently, the bridge was close, and a fix was designed and implemented (at considerable cost). This phenomenon has also been observed in other footbridges but is also of more general scientific interest as a clear example of emergent behavior in a large system of many coupled oscillators (where we view the individual walkers as oscillators with oscillator frequencies equal to their stepping frequencies). In this paper, we will show that an inherently high dimensional microscopic description of many individual walkers self-consistently coupled to each other through the bridge response can be reduced to a low dimensional system of just a few coupled ordinary differential equations, and we will use this dimensionreduced description to study the wobbling that was observed when the bridge opened.
I. INTRODUCTION
On opening day the Millennium Bridge, a pedestrian footbridge crossing the Thames River in London, was observed to exhibit a pronounced lateral wobbling as more and more people streamed onto the bridge.1 This phenomenon apparently was due to a resonance between a low order bridge oscillation mode and the natural average stepping frequency of human walkers: a small initial oscillation of the bridge induces some of the walkers to synchronize the timing of their steps to that of the bridge oscillations, thus exerting a positive feedback force on the bridge that derives the bridge oscillation to higher amplitude, eventually resulting in 1054-1500/2009/19共1兲/013129/5/$25.00
a large steady-state oscillation. Subsequent studies by the bridge builder showed that oscillations did not develop unless the number of pedestrians on the bridge was greater than a critical value.1,2 After this event became widely publicized, it began to emerge that this phenomenon had also been observed on other footbridges.1,2 More generally, the phenomenon on these footbridges may be viewed as a particularly dramatic example of the emergence of global collective behavior in systems of many coupled heterogeneous oscillators3 关examples of this general type of emergent behavior span biology 共e.g., synchronization of pacemaker cells in the heart4 and of neurons governing day-night rhythms in mammals5兲, chemistry 共e.g., Ref. 6兲, physics 共e.g., Ref. 7兲, etc.兴. Several mathematical models have been proposed to describe the phenomenon observed on the Millennium Bridge.1,2,8–11 The models of Refs. 1, 2, and 8–10 are low dimensional but do not address the issue of the mechanism by which individual walkers synchronize with the bridge, nor do they address the very important issue that different walkers have different natural stepping frequencies. Eckhardt et al.10 addressed these issues, and they give a detailed argument for their method of modeling the response of walkers to bridge motion. The greater degree of realism of walker modeling achieved in Ref. 10, however, results in a high dimensional model, of dimensionality 共N + 2兲. Here N is the number of pedestrians and the bridge is described by a second order ordinary differential equation 共hence +2兲. Furthermore, to facilitate analysis, it proves convenient to take N → ⬁. The main goal of our current paper is to show how to reduce the N → ⬁ version of the model of Eckhardt et al.10 to low dimensionality, thus producing a formulation combining the advantages of both classes of Millennium Bridge models 共Refs. 1, 2, and 8–10, on the one hand, and Ref. 10, on the
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other兲. Furthermore, we will utilize our reduced description to study the dynamics of the bridge/pedestrian system. We note that our dimension reduction makes use of the recently developed technique of Ref. 11. This technique may be applicable to a wide variety of problems involving large systems of coupled phase oscillators, the simplest example of which is the well-known Kuramoto12 model. The organization of this paper is as follows. In Sec. II we develop our reduced model for the case where the number of walkers on the bridge N is constant in time. In Sec. III we offer analytical and numerical treatments of the reduced model developed in Sec. II. Motivated by an experimental study by the bridge builder1,2 in which the number of walkers on the bridge was increased in with time, we consider this situation in Sec. IV; in particular, we generalize our model of Sec. II to this case, and we numerically examine the resulting behavior. II. REDUCED MODEL A. Review of the model of Eckhardt et al.
The model proposed in Ref. 10 describes the lateral modal bridge displacement y共t兲 as satisfying a damped harmonic oscillator equation forced by the walkers, N
My¨ + 2My˙ + M⍀2y = 兺 f i共t兲,
共1兲
i=1
where M is the modal mass, is the modal damping rate, ⍀ is the mode resonant frequency, f i共t兲 is the lateral modal force exerted by pedestrian i, and there are N pedestrians on the bridge. As argued in Ref. 10, at least for low bridge oscillation amplitude, the pedestrians can be modeled as phase oscillators.12 That is, the pedestrian force is ˜F 共t兲 = F cos关 共t兲兴, i i i
共2兲
where Fi is the peak modal force applied by walker i and i共t兲 is the phase of the walker i. In the absence of bridge motion, ˙ i = i, where i is the natural stepping frequency of walker i. In the presence of bridge motion, since the walkers move in the frame of the bridge, they are influenced by the inertial force due to the bridge acceleration y¨ 共t兲. As a consequence, and as argued in Ref. 10, a reasonable model of this modification is
˙ i = i − iy¨ cos关i共t兲兴,
共3兲
where i is a coupling constant reflecting the strength of the response of walker i to a lateral force. Data on the natural stepping frequency of humans show that the frequency of a typical group of humans is, on average, peaked at about 1 Hz, with a spread of the order of 0.1 Hz.13 This average stepping frequency is slightly below the resonant frequency of the lowest order lateral mode of the north span of the Millennium Bridge. The bridge damping was fairly low / ⍀ Ⰶ 0.1. The closeness of the average walker frequency, ¯ , to the bridge resonance ⍀, as well as the small denoted value of / ⍀, was used in the arguments of Ref. 10 to justify the phase oscillator walker model 关Eqs. 共2兲 and 共3兲兴.
While, as we have noted, there is available data on the distribution of i for a typical group of humans, we do not know of any such data for Fi and i. Ideally, for input to this model 关Eqs. 共1兲–共3兲兴, we would like to have the joint probability distribution function for all these walker parameters G共 , Fi , 兲. In the absence of such data, we will set all Fi and all i equal, Fi = ¯F,
i = ¯ ,
共4兲
but we will use a distribution function for i, and we denote this distribution function g共兲. We do not view the assumption 关Eq. 共4兲兴 as capable of making a crucial qualitative change in the phenomenon we investigate. The inclusion of a distribution of the walker frequencies is, however, essential, and, as we will later see, it plays the key role in determining the critical number of pedestrians necessary for the onset of bridge oscillation. ¯ and ¯, referring to Ref. 10, we use Concerning F as nominal values for these quantities ¯F = 25 N and ¯ = 1.75 s / m. We note, however, that the value estimated for ¯ is only an order of magnitude estimate and that no relevant direct experimental measurements of this quantity have, to our knowledge, ever been made. The model 关Eqs. 共1兲–共3兲兴 is a dynamical system of dimensionality 共N + 2兲. We now assume that the number of walkers is large and approximate the ensemble of walkers using a distribution function f共 , , t兲 such that f共 , , t兲dd is the fraction of walkers at time t whose phases are in the interval 共 , + d兲 and whose frequencies are in the range 共 , + d兲. Formally, this description corre¯ held fixed. Note that in sponds to the limit N → ⬁ with NF terms of f共 , , t兲, the distribution function of natural frequencies g共兲 is g共兲 = 兰20 f共 , , t兲d. In this continuum limit the model 关Eqs. 共1兲–共4兲兴 becomes10 ¯ Re关R共t兲兴, Myត共t兲 + 2My˙ 共t兲 + M⍀2y共t兲 = NF
冕 冕 ⬁
R共t兲 =
d
−⬁
2
f共, ,t兲eid ,
共5兲 共6兲
0
f f + − ¯y¨ 共f cos i兲 = 0. t
共7兲
The quantity R共t兲, introduced above, represents the averaged normalized walker forcing of the bridge. When stepping of the walkers is uncorrelated, the distribution function f is uniform in , f = g共兲/2 ,
共8兲
for which Eq. 共6兲 yields R = 0, which is consistent with the at-rest solution of Eqs. 共5兲 and 共7兲, y共t兲 = 0. As shown by Ref. 10, this solution becomes unstable to the onset of bridge wobbling 关y共t兲 ⫽ 0兴 and synchronization of the walkers 共兩R兩 ⬎ 0兲, if N exceeds a critical number of walker Nc, and the value found in Ref. 10 for Nc was in reasonable agreement 共to within inherent uncertainties兲 with the experimental value1,2 for the Millennium Bridge.
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Millennium Bridge: Low dimensional description
B. Reduction in the model to low dimensionality
Following the technique of Ott and Antonsen, pand f共 , , t兲 in a Fourier series in , f共, ,t兲 =
再冋
g共兲 1+ 2
⬁
f n共,t兲ein + c.c. 兺 n=1
册冎
11
,
we ex-
we obtain
共9兲
再
共10兲
冎
¯ ␣共,t兲 + i ␣ − ˜y¨ 关␣2 + 1兴 = 0. 2 ˜t
Rⴱ共t兲 =
冕
g共兲␣共,t兲d ,
共12兲
−⬁
ⴱ
where R is the complex conjugate of R. The special class of distribution functions given by Eq. 共10兲 constitutes an invariant manifold M in the space of distribution functions; i.e., an initial condition satisfying Eq. 共10兲 evolves to another state satisfying Eq. 共10兲, and it does so according to the evolution equations 关Eqs. 共11兲 and 共12兲兴, for ␣共 , t兲. We are interested in evolution from near the incoherent state 关Eq. 共8兲兴, which is on M 关Eqs. 共9兲 and 共10兲 with ␣ = 0兴. Thus our solution on M will be the relevant one, if perturbations from M do not grow with time. That this is indeed the case is supported by the observation that our reduced solutions agree with those of Eqs. 共1兲 and 共3兲 numerically obtained in Ref. 10. To proceed, we now adopt a convenient form of the distribution function of natural walker frequencies. In particular, as in Ref. 11, we take g共兲 to be Lorentzian, g共兲 =
1 ⌬ , ¯ 兲2 + ⌬2 共 −
共13兲
¯ is the mean walker frequency and ⌬ is the spread in where the frequencies of the walkers. The integral in Eq. 共12兲 can now be done by analytically continuing into the complex -plane, taking ␣共 , t兲 to be analytic and bounded in Im共兲 ⬍ 0 共see Ref. 11兲, and closing the integration path in the lower-half -plane with a large semicircle of radius approaching infinity. The integral is then given by the residue of the integrand at the pole of g共兲 that ¯ − i⌬, is at = ¯ − i⌬,t兲. Rⴱ共t兲 = ␣共
冎
¯ ˜y¨ 关R2 + 1兴 = 0. 2
共15兲
III. ANALYSIS OF THE REDUCED MODEL A. Linear analysis
Linearizing Eqs. 共5兲 and 共15兲 and assuming that y ⬃ exp共s⍀t兲 and R ⬃ exp共s⍀t兲, we obtain the following equation for s: ˜ ˜ 兲2 + ˜ 2兴共s2 + 2 ˜ s + 1兲 = s2 ˜, 关共s + ⌬ 2
共16兲
where we have introduced the normalizations, 共11兲
Now using Eqs. 共9兲 and 共10兲 in Eq. 共6兲, we obtain ⬁
再
dR ¯ + i⌬兲Rⴱ − − i 共 dt˜
Our exact reduction in the model of Ref. 10 to low dimension thus consists of Eqs. 共5兲 and 共15兲.
where c.c. stands for complex conjugate. Substituting Eq. 共9兲 into Eq. 共7兲 and considering a restricted class of distribution functions such that f n共,t兲 = 关␣共,t兲兴n ,
Chaos 19, 013129 共2009兲
共14兲
¯ − i⌬ in Eq. 共11兲, we obtain a simple first order Inserting = ordinary differential equation for the evolution of complex quantity, R共t兲,
˜ = ⌬/⍀, ⌬
˜ = ¯ /⍀,
˜ = /⍀,
¯ ˜ = ¯ NF . M⍀
共17兲
˜ and ˜ are small 共⬍0.1兲 and that We note that both ⌬ ˜ − 1兩 is also small. Expanding Eq. 共16兲 for 兩
冑
˜ − 1兩 ⬃ ˜ ⬃ ˜ ⬃ ˜ , 1 Ⰷ 兩
共18兲
˜ = ˜ − 1, Eq. 共16兲 becomes where ˜ − i ˜ 兲共 + ˜兲 = ˜/8, 共 + ⌬
共19兲
where = s − i. We define a critical value of ˜, denoted as ˜c, as the value of ˜ at which the real part of transitions from negative to positive as ˜ increases. Once ˜c is known, the critical number of walkers for wobble onset is predicted to be Nc =
M⍀˜c . ¯F¯
共20兲
In the sense of giving the lowest ˜c 共equivalently the lowest number of walkers兲, the worst situation is the case ¯ matches the bridge when the walkers’ mean frequency ˜ ˜ mode frequency ⍀, i.e., = 0. With = 0, Eq. 共19兲 gives a critical value of ˜ of ˜ = 8⌬ ˜ ˜ , c
共21兲
and a solution for the instability growth rate of
=
1 2
冋冑
册
˜ + ˜兲2 − 1 共˜ − ˜ 兲 − 共⌬ ˜ + ˜兲 . 共⌬ c 2
共22兲
If ⫽ 0, then ˜c is given by ˜ ˜2 c =1+ . ˜ ˜兲 ˜ + ˜兲2 共8⌬ 共⌬
共23兲
Thus the critical number of walkers increases quadratically ¯ with the difference between the mean walker frequency and the bridge resonant frequency ⍀. Also note that inclu˜ ⫽ 0兲 is sion of the spread in natural walker frequencies 共⌬ crucial, in which, without it, ˜c from Eq. 共23兲 would be zero
013129-4
Chaos 19, 013129 共2009兲
M. M. Abdulrehem and E. Ott
FIG. 2. A time trace of lateral acceleration of the bridge deck and the number of pedestrians 关taken from Arup’s measurements 共Ref. 2兲兴. 共The numbers on the vertical axis are in units of 103 of the acceleration of gravity.兲
FIG. 1. Plot of the normalized oscillation amplitude ¯ 兲 as obtained from a numerical solution of Eqs. 共5兲 ˜y max = 2M⍀y max / 共NF ¯ / ⍀ = 1, and ⌬ / ⍀ = 0.072 共circles兲. Also plotted and 共15兲 with / ⍀ = 0.0075, is the theoretical result from Eq. 共26兲 共solid curve兲.
very good. The slight downward displacement of the circles from the theoretical curve is, we believe, due to our neglect of the 2⍀ harmonic in our derivation of Eq. 共26兲. IV. TIME VARIATION IN THE NUMBER OF WALKERS
and there would thus be no threshold for instability 关i.e., Nc = 0 from Eq. 共20兲兴. Equations 共20兲 and 共21兲 with our nominal values of parameter for the Millennium Bridge ¯ = 25 N, ¯ = 1.75 s / m, ⍀ / 2 = 1.03 Hz, and walkers 共F 5 ¯ / 2 = 1.03, and ⌬ / ¯ = 0.072兲 give M = 1.13⫻ 10 kg, Nc = 73 in the range of the observed value.1,2 B. Nonlinear saturation
¯ =⍀ We now examine the saturated state for the case ˜ 共i.e., = 0兲. Using Eq. 共18兲 we assume that y共t兲 = A cos ⍀t and suppose that the amplitude of the component of y共t兲 oscillating at the frequency 2⍀ is negligible. With this assumption, we substitute y共t兲 = A cos ⍀t into Eq. 共15兲 and solve for the component of R共t兲 that oscillates as ei⍀t 关we are interested in this component since it is the only one that drives y共t兲 resonantly兴. Denoting this component of R共t兲 by irei⍀t and substituting into Eqs. 共15兲 and 共5兲, respectively, give A ⌬r =  共1 − r2兲, 4
共24兲
¯ r, 2MA⍀A = NF
共25兲
Following the discovery of the walker-induced wobble of the Millennium Bridge, the bridge builder conducted a controlled test. Using company employees, they introduced successively more walkers onto the bridge, increasing the number of walkers by the addition of groups at discrete times. The steplike curve in Fig. 2 shows the number of walkers on the bridge as a function of time. Also plotted is the sideways lateral acceleration as measured by an accelerometer attached to the bridge. It is seen that the oscillation onset occurs rather abruptly as the number of walkers increases. When the bridge began to oscillate strongly, the people were rapidly removed from the bridge. Thus a steady state 关as in Eq. 共25兲兴 was apparently not attained in this experiment. We now wish to adapt our formulation in Sec. II to simulate this situation. Since we regard the newly introduced walkers to initially be randomly distributed in phase at the
which yield the steady saturated state r = 冑1 − c/,
A=
¯ NF r. 2M⍀
共26兲
Figure 1 shows the peak value ˜y max of the steady-state ¯ / 共2M⍀兲 as obtained from oscillation y共t兲 normalized to NF the numerical solution of Eqs. 共5兲 and 共15兲 plotted versus  / c 共circles兲. The parameters used for this plot are ¯ / ⍀ = 1, and ⌬ / ⍀ = 0.072, which are realistic / ⍀ = 0.0075, representative values appropriate for the Millennium Bridge. Also plotted in Fig. 1 as a solid line is the theoretical result from Eq. 共26兲. We see that the agreement between them is
FIG. 3. Effect of adding more walkers to the bridge as a function of time. The dashed line is the number of walkers vs time and the solid line is the lateral acceleration vs time.
013129-5
Chaos 19, 013129 共2009兲
Millennium Bridge: Low dimensional description
time of introduction, we will allow for a different Lorentzian distribution function for each group of walkers. Adapting our previous formulation to this situation, we have in place of Eq. 共5兲, J共t兲
Myត + 2My˙ + M⍀2y = ¯F 兺 N j Re关R j共t兲兴,
共27兲
j=1
where N j walkers are introduced onto the bridge at the time t j, the number J共t兲 of groups of walkers on the bridge at time t is defined by t j+1 ⬎ t ⱖ t j ,
共28兲
and the complex quantity R j共t兲 characterizes the distribution of walkers in group j. For t ⱖ t j, R j共t兲 satisfies Eq. 共15兲 with the initial condition R j共t j兲 = 0,
共29兲
corresponding to the walker phases being randomly distributed at the time when they first enter the bridge. Thus, by virtue of their different times of entry, Eq. 共29兲 implies a different distribution of oscillator phases of each group. Proceeding in this way, we introduce 20 groups of 10 walkers each, at a rate of 1 group/min. The result in Fig. 3 is in rough agreement with that seen in Fig. 2. In conclusion, we have obtained a low dimensional formulation of a previous high dimensional model for walker synchronization on the Millennium Bridge, and we have demonstrated the usefulness of our formulation for studying various aspects of the dynamics of the bridge/pedestrian system. More generally, our work serves as an interesting ex-
ample of the potential for low dimensional macroscopic behavior in systems consisting of many coupled oscillators. ACKNOWLEDGMENTS
This work was supported by NSF and by ONR 共Award No. N00014-07-0734兲. 1
P. Dallard, A. J. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. M. R. Smith, and M. Willford, Struct. Eng. 79, 17 共2001兲. 2 P. Dallard, A. J. Fitzpatrick, A. Flint, A. Low, R. M. R. Smith, M. Willford, and M. Roche, J. Bridge Eng. 6, 412 共2001兲. 3 S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order 共Penguin Science, London, 2004兲. 4 D. C. Michaels, Circ. Res. 61, 704 共1987兲. 5 S. Yamaguchi, H. Isejima, T. Matsuo, R. Okura, K. Yagita, M. Kobayashi, and H. Okamura, Science 302, 1408 共2003兲; T. M. Antonsen, Jr., R. T. Faghih, M. Girvan, E. Ott, and J. Platig, Chaos 18, 037112 共2008兲. 6 I. Z. Kiss, Y. Zhai, and J. L. Hudson, Science 296, 1676 共2002兲. 7 K. Wiesenfeld and J. W. Swift, Phys. Rev. E 51, 1020 共1995兲. 8 S. Nakamura, J. Struct. Eng. 130, 32 共2004兲. 9 T. M. Roberts, J. Bridge Eng. 156, 155 共2003兲. 10 B. Eckhardt, E. Ott, S. H. Strogatz, D. Abrams, and A. McRobie, Phys. Rev. E 75, 021110 共2007兲; S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, and E. Ott, Nature 共London兲 438, 43 共2005兲. 11 E. Ott and T. M. Antonsen, Chaos 18, 037113 共2008兲. 12 Y. Kuramoto, in International Symposium On Mathematical Problems in Theoretical Physics, Lecture Notes in Physics Vol. 39, edited by H. Araki 共Springer-Verlag, Berlin, 1975兲; Chemical Oscillations, Waves and Turbulence 共Springer, New York, 1984兲. 13 Y. Matsumoto, S. Sato, T. Nishioka, and H. Shiojiri, Trans. Jpn. Soc. Civ. Eng. 4, 50 共1972兲; H. Bachmann and W. Ammann, “Vibrations in structures induced by man and machines,” Structural Engineering Document 3e, International Association for Bridge and Structural Engineering 共IABSE兲, Chap. 2: Man-Induced Vibrations, and Appendix A: Case Reports, 1987; S. Zivanovic, A. Pavic, and P. Reynolds, J. Sound Vib. 279, 1 共2005兲.
CHAOS 19, 013130 共2009兲
The development of generalized synchronization on complex networks Shuguang Guan,1,2 Xingang Wang,1,2 Xiaofeng Gong,1,2 Kun Li,1,2 and C.-H. Lai2,3 1
Temasek Laboratories, National University of Singapore, Singapore 117508, Singapore Beijing-Hong Kong-Singapore Joint Center of Nonlinear and Complex Systems, Singapore 117508, Singapore 3 Department of Physics, National University of Singapore, Singapore 117543, Singapore 2
共Received 12 June 2008; accepted 3 February 2009; published online 13 March 2009兲 In this paper, we numerically investigate the development of generalized synchronization 共GS兲 on typical complex networks, such as scale-free networks, small-world networks, random networks, and modular networks. By adopting the auxiliary-system approach to networks, we observe that GS generally takes place in oscillator networks with both heterogeneous and homogeneous degree distributions, regardless of whether the coupled chaotic oscillators are identical or nonidentical. We show that several factors, such as the network topology, the local dynamics, and the specific coupling strategies, can affect the development of GS on complex networks. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3087531兴 The investigation of synchronization dates back to the 17th century when Huygens first discovered synchronization between two pendulum clocks. Since then, the ubiquitous phenomena of synchronization in both natural and artificial systems have attracted great research efforts. The early theoretical studies on synchronization are mainly limited in the cases of couple periodic oscillators. In the 90s of last century, breakthrough has been made when synchronization between chaotic oscillators was discovered and extensively studied later. Recently, with the booming in the field of complex networks, synchronization among interacting oscillatory elements on networks has once again become a hot topic. So far, much effort has been given to the study of complete synchronization and phase synchronization in complex networks. However, one important synchronization forms, i.e., the generalized synchronization has seldom been addressed in complex networks. In this paper, we present a numerical investigation on the occurrence and development of generalized synchronization on various complex networks, including scale-free networks, small-world networks, random networks, and modular networks. Hopefully, this work will provide us further understanding and new perspective in the field of network synchronization.
I. INTRODUCTION
Synchronization in coupled chaotic oscillators has been extensively studied in the past 20 years,1 including complete synchronization 共CS兲,2 generalized synchronization 共GS兲,3–11 phase synchronization 共PS兲,12 lag synchronization,13 partial synchronization or clustering,14 practical 共approximation兲 synchronization,15 and anticipate synchronization,16 etc. For example, in CS, the dynamics of two coupled systems totally coincide with each other; in GS, certain functional relation exists between the dynamics of two coupled systems which are usually nonidentical. Moreover, PS is a weaker synchronization form in which the phases of two oscillators can be 1054-1500/2009/19共1兲/013130/9/$25.00
locked, while their amplitudes remain uncorrelated and chaotic. Recently, the study of synchronization has been extended to the area of complex networks.17–31 For example, synchronization on small-world networks,17–20 scale-free networks,21–23 modular networks,24–27 weighted networks,28 and gradient networks29 has been investigated. These studies aim to explore the interplay between network topology and dynamics on network. They are important for us to understand the real situations in complex systems comprising interacting elements in both nature and human society. So far, most works on synchronization in complex networks study the situations of PS and CS. Specifically, PS in complex networks is mainly investigated through the generalized Kuramoto models.20,27,30–32 In this model, the node dynamics is very simple that is governed by an ideal phase ˙ = , where is the frequency. The heterogeneoscillators, ity in the node dynamics can be modeled by assigning different, usually random, frequencies to different phase oscillators. The generalized Kuramoto models in complex networks have the advantage that they can still be treated analytically in many aspects.27,30 On the other hand, CS on complex networks is often investigated through the approach of master stability function.19,33 To apply this approach, the node dynamics in complex networks must be assumed to be identical, and then the theory of master stability function provides a general mathematical framework to relate the synchronizability of a network to the spectral properties of the corresponding coupling matrix.19,22,25,28,34 Recently, a connection graph based stability method has been proposed, which is able to give the upper bounds of minimum coupling strength for achieving global synchronization of coupled oscillators on complex networks.35 Apart from CS, this method has an important advantage in dealing with approximation synchronization on networks and even synchronization on networks with time-varying coupling.35 One interesting question in studying network synchronization is that can different synchronization forms, or collective behaviors, in coupled low-dimensional dynamical sys-
19, 013130-1
© 2009 American Institute of Physics
013130-2
Chaos 19, 013130 共2009兲
Guan et al.
tems still be observed in complex networks? We notice that GS phenomenon has not been carefully investigated on networks in previous works. In fact, GS is an important and very useful concept in the analysis of coherence in biological10 and physical11 systems consisting of multiple interacting components. For example, in Ref. 11, by applying the GS detecting method, it has been successfully demonstrated that GS relation exists in He–Ne lasers and liquid crystal spatial light modulators. These important experiments thus provide direct evidence showing that different spatiotemporal dynamics could have strong coherence between them. In this paper we present a detailed numerical study on GS phenomenon in various complex networks. Our particular interest is paid to the occurrence and development of GS on networks. For typical complex networks, including scalefree networks, small-world networks, random networks, and modular networks, interestingly, we observe that GS generally occurs, regardless of whether the node dynamics are identical or nonidentical. For networked identical oscillators, we find that usually there is a GS regime before the final global CS. We further carried out extensive numerical experiments to demonstrate how the development of GS on networks can be affected by several factors, such as the network topology, the local dynamics, as well as the specific coupling strategy. In processing this paper, we noticed a recent work which reported GS phenomenon in scale-free networks.36 Compared with the results in Ref. 36, the present work is different in the following aspects. First, in our work GS is extensively investigated in various network topologies, including scalefree networks, random networks, small-world networks, and modular networks, while in Ref. 36, GS is mainly studied in a very special network, i.e., the scale-free network with treelike structure. Second, the present work characterizes a typical path for coupled identical oscillators on complex networks, i.e., from nonsynchronization to global CS via GS, while in Ref. 36, global CS has not been achieved. Third, the present work investigates the development of GS for coupled nonidentical oscillators, either parametrically different or physically different, while Ref. 36 does not consider this general setting where the occurrence of GS is naturally justified. Finally, in the present work, we discuss the effect of different coupling strategies on the development of GS on network. Especially, we analyze and explain the different observations regarding the GS evolution on networks in our work and in Ref. 36. Therefore, in many aspects the current study deepens and widens the work in Ref. 36 and can offer more thorough and comprehensive insight for understanding GS phenomenon on complex networks. This paper is organized as follows. In Sec. II, the methods and measures to characterize GS and CS on networks are described. In Sec. III, GS of coupled identical oscillators is studied on typical complex networks. In Sec. IV, GS of coupled nonidentical oscillators is considered on networks. It is shown that the development of GS on networks can be affected by both network topology and the local dynamics. In Sec. V, the effect of different coupling strategies on the development of GS on networks is analyzed. Especially, we show that GS can be observed in coupled system with hybrid
oscillators even when their local dynamics are physically different. Finally, a section with discussions and summary ends this paper. II. APPROACHES CHARACTERIZING GS AND CS ON NETWORKS
The auxiliary-system approach has been extensively used to detect GS in two coupled chaotic systems.4 Here, we can extend it to detect GS on complex networks. The key observation is that for any given node in a network, the coupling from other nodes can be regarded as a kind of “driving.” In particular, we consider the following linearly coupled identical oscillators on a network: x˙ i = Fi共xi兲 − 兺 aijH共xi − x j兲
共1兲
j
for i = 1 , . . . , N, where xi denotes the dynamical variables of node i, Fi共xi兲 is the local vector field governing the evolution of xi in the absence of interactions with other nodes, aij is the element of the network adjacency matrix A 共aij = 1 if there is a link between node i and node j, aij = 0 otherwise, and aii = 0兲, H is the output matrix, and is the coupling strength. To apply the auxiliary-system approach, we consider a replica for each oscillator in the original network, x˙ ⬘i = Fi共x⬘i 兲 − 兺 aijH共xi⬘ − x j兲
共2兲
j
for i = 1 , . . . , N. Note that the driving variable x j is identical for both Eqs. 共1兲 and 共2兲. If, for initial conditions xi共0兲 ⫽ xi⬘共0兲, we have 兩xi共t兲 − x⬘i 共t兲兩 → 0 as t → ⬁, node i then is entrained in the sense that its dynamics is no longer sensitive to the initial conditions. In other words, there is GS relation between xi and x j for j = 1 , . . . , N. Numerically, we can examine the following local distance of GS between a node and its auxiliary counterpart: t
2 1 d共,i兲 = 兩xi共t兲 − x⬘i 共t兲兩, 兺 t 2 − t 1 t1
共3兲
where t1 is chosen to be larger than the typical transient time of the local dynamics Fi共xi兲. For oscillators on complex networks, GS may be gradually developed with the increase in the coupling strength. To characterize the development of GS on the whole networks, we can define the distance of global GS as lg共兲 = 具d共 , i兲典. Here, 具·典 denotes the spatial average over all nodes. If lg = 0, global GS has been achieved between any two pairs of oscillators on the whole network. For coupled identical oscillators on complex networks, CS is generally expected to take place. To characterize CS, we can define lc共兲 as the distance of global CS, which measures the distance between the dynamics of all oscillators and their average, i.e., t
2 1 lc共兲 = 具兩x − 具x典兩典, 兺 t 2 − t 1 t1
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For individual node dynamics, most of the time in this paper we choose the chaotic Lorenz oscillator, Fi共xi兲 = 关10共y i − xi兲,rixi − y i − xizi,xiy i − 共8/3兲zi兴T ,
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where xi ⬅ 共xi , y i , zi兲 are the state variables of the Lorenz oscillator. Note that in studying GS, the local dynamics could be different from each other. This can be modeled by setting different ri values for Lorenz oscillators in the network. Without losing generality, the coupling between two nodes in Eq. 共1兲 is through the x variable, i.e., the output matrix is H = 关1 , 0 , 0 ; 0 , 0 , 0 ; 0 , 0 , 0兴. For convenience, in the present work, we use two kinds of node indices: id and ir. In the first index, we order the degrees of the network so that id = 1 denotes the node with the largest degree, id = 2 is for the node with the second largest degree, and so on. In the second index, we rank the Lorenz oscillators in the network according to their ri values, i.e., ir = 1 denotes the node with the largest parameter r, ir = 2 is for the node with the second largest parameter r, and so on. These two indices of nodes are used throughout the paper. Besides the Lorenz oscillator, other node dynamics such as the Rossler oscillator and the logistic map have also been used in our study. III. COUPLED IDENTICAL OSCILLATORS: THE OCCURRENCE AND DEVELOPMENT OF GS
Previously, in the study of synchronization of two coupled chaotic oscillators, it is found that CS generally takes place when the coupled oscillators are identical, while GS usually occurs when the coupled oscillators are nonidentical. However, this generally accepted view turns out to be not true when synchronization is studied on complex networks. In this section, we report the existence of GS for coupled identical chaotic oscillators on various complex networks, including scale-free networks, random networks, small-world networks, and modular networks. As an example, we first study the occurrence and development of GS for 300 chaotic Lorenz oscillators on a scalefree network. In this case, ri = 28 in Eq. 共5兲 for all nodes, i.e., all oscillators are identical. Intuitively, with the increase in
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coupling, we can expect that the coupled chaotic oscillators will finally achieve global CS. The interesting finding here is that before the system achieves the global CS state, there exists another synchronization regime, namely, the GS regime, which usually occurs with much smaller coupling strength compared with CS. To illustrate the GS development on scale-free network, we plot the color map of the distance matrix d共 , id兲 in Fig. 1. From Fig. 1, we can see that when the coupling strength is small, d共 , id兲 is greater than 0 for all nodes, showing that the system is in the nonsynchronous state. In addition, when the coupling strength is large enough, d共 , id兲 is 0 for all nodes, showing that all oscillators have been entrained and the coupled system is in the global GS state according to the auxiliary-system approach criterion. Between these two regimes, it is the transient regime of partial GS 共PGS兲, where part of the oscillators has been entrained but the others were not. On the other hand, for coupled identical oscillators, CS generally occurs as long as the connections of the network are dense enough, and this is the case in the above example. As the coupling strength is further increased after GS, the coupled system will finally go to the global CS state. Usually, there is a transient stage before the system achieves global CS. In this stage, part of the nodes has achieved CS in practical 共approximation兲 sense with each other. These nodes form a synchronous cluster, while the other nodes do not synchronize with them. We call this stage the partial CS 共PCS兲 regime. To characterize the development of GS for coupled Lorenz oscillators on scale-free network, in Fig. 2共a兲 we plot the distance of global GS lg versus the coupling strength. From this figure, three regimes of coupling can be clearly identified, and global GS on this specific network is found to be achieved when ⬎ 1.8. Similarly, in Fig. 2共b兲 we plot the distance of global CS lc versus the coupling strength and find that the global CS on the network can be achieved when ⬎ 3.8. Combining these results together, we can get an overall picture on the path toward global CS in the scale-free network. Specifically, five dynamical regimes can be identified as follows. I: ⱕ 0.1, the nonsynchronization regime; II:
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FIG. 4. 共Color兲 Color map of d共 , id兲 in the two-dimensional parameter space 共 , i兲, characterizing the development of GS for 500 nonidentical Lorenz chaotic oscillators on a scale-free network. The network is the BA model with m0 = m = 3 共Ref. 21兲. The development of GS in the network is similar to the situation shown in Fig. 1.
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FIG. 3. 共Color兲 Characterizing the development of GS and the path toward global CS of 300 coupled identical Lorenz oscillators on a random network with average degree of 10. 共a兲 The color map of d共 , id兲 in the twodimensional parameter space 共 , i兲. 共b兲 The number of nodes which has achieved GS vs the coupling strength. 共c兲 The distance of global CS vs the coupling strength.
0.1⬍ ⱕ 1.8, the PGS regime; III: 1.8⬍ ⬍ 3.0, the global GS regime; IV: 3.0⬍ ⬍ 3.8, the PCS regime; and V: ⬎ 3.8, the global CS regime. These regimes represent a typical path from nonsynchronization state to global CS state via GS for networked identical oscillators. Since GS is a weaker synchronization form, achieving GS on network usually requires smaller coupling strength than achieving CS. In the above we have observed GS for coupled identical chaotic oscillators on a scale-free network. How about other network topologies? In our study, we have also considered the following networks. 共1兲 A random network consisting of 300 nodes. The average degree is 10. 共2兲 A small-world network, which is obtained by rewinding 20 links in a regular network consisting of 100 nodes and each node has ten nearest neighbor connections.17 共3兲 A modular network consisting of 100 nodes which is evenly divided into five modules.27 Inside each module node is fully connected. Any two nodes in different modules have probability p = 0.01 to connect each other. 共4兲 As a special case of complex networks, a regular network consisting of 100 nodes and each node has ten nearest neighbor connections. For all these typical complex networks studied, GS has been observed. Here, we show one more example in the case of random network. In Fig. 3, the development of GS on a random network is characterized. Comparing Fig. 3共a兲 with Fig. 1, we can see that the development of GS at the beginning stage on the random network is different from the situation on the scale-free network. Mainly, most nodes in random network are entrained at approximately the same coupling strength. This is due to the fact that random network has approximately homogeneous degrees and no hubs exist as in scale-free networks. Although the development of GS on networks depends on the different network topologies, qualitatively, the path to-
ward global CS is the same as shown in Figs. 2 and 3. In fact, we have similar observations for other types of complex networks. We emphasize that this path to the global CS for networked identical chaotic oscillators is a typical one, not the complete one. One may observe different paths toward the global CS, depending on the local dynamics on networks. For example, in our study we have also observed the path to the global CS as: nonsynchronization →PS→ GS→ CS. IV. COUPLED PARAMETRICALLY DIFFERENT OSCILLATORS: TOPOLOGY VERSUS LOCAL DYNAMICS
In realistically physical or biological situations, the node dynamics are usually different from each other. For example, in neuron networks any two pairs of neurons cannot be exactly the same. This raises an important question: is it possible for nonidentical oscillators on network to achieve GS, or certain extent of coherence? If so, how do network topologies and local dynamics affect the development of GS? To address these questions, in this section we will consider coupled nonidentical oscillators in complex networks. To model different oscillators on network, we randomly set the parameter ri in Lorenz system in the interval 关28.0, 30.0兴. By applying auxiliary-system approach, we find that GS generally occurs in such coupled systems on networks. As an example, Fig. 4 illustrates the development of GS for coupled Lorenz oscillators with parameter mismatches on a scale-free network. It is shown that with the increase in coupling strength, nodes in the network are entrained first from the hubs and gradually spread to other nodes. When the coupling strength is large enough, all nodes in the network are entrained, showing that global GS has been achieved. In spite of the different local dynamics, the above example shows a similar development of GS on the scale-free network as in the case of coupled identical oscillators. This implies that here the network topology plays a dominant role in the development of GS on network. Previously, many works have revealed how network topology affects the network synchronizability under the setting of coupled identical oscillators. For networked noniden-
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FIG. 5. 共Color兲 Color maps of d共 , id兲 and d共 , ir兲 in the two-dimensional parameter space 共 , i兲 for nonidentical Lorenz oscillators on a regular network. The network has 100 nodes, and each node has six nearest neighboring connections.
FIG. 6. 共Color兲 Color maps of d共 , id兲 and d共 , ir兲 in the two-dimensional parameter space 共 , i兲 for nonidentical Lorenz oscillators on a small-world network. The network is obtained by rewiring a small part of connections on a regular network with 100 nodes 共Ref. 17兲.
tical oscillators studied in this work, apparently there are two factors affecting the development of GS on complex networks, i.e., network topology and heterogeneity in the local chaotic dynamics. In Secs. II and III, we have shown that for scale-free networks, GS typically starts from the hubs and then spreads to others nodes with relatively smaller degrees, regardless of whether the oscillators are identical or nonidentical. In these cases, heterogeneity in the degree distribution appears to be the dominant factor governing the development of GS. An interesting issue then is that for networks with homogeneous degree distributions, such as regular, smallworld, or certain modular networks, how does heterogeneity in the local dynamics affect GS? In the following, we study this question through numerical simulations. We first consider a regular network of N = 100 nodes. Each node in the network has kd = 6 connections to its nearest neighbors. As a special case, this is an exactly homogeneous network with regular degree sequence. The local dynamics in this example and in the following two examples are those of Lorenz chaotic oscillators with different parameter ri randomly distributed in the interval 关28, 38兴. Figure 5 shows the color maps of the local synchronization distances d共 , id兲 and d共 , ir兲, where id and ir are the node indices arranged according to the decreasing node degree and decreasing values of parameter ri, respectively. For the Lorenz oscillator under the parameter setting in our study, we find that the larger the value of parameter r, the larger the largest Lyapunov exponent of the chaotic attractor. Thus, in Fig. 5, the index ir actually corresponds to the decreasing value of the largest Lyapunov exponent of the local dynamics. Specifically, larger value of ir corresponds to a small value of ri so that the corresponding local dynamics is less chaotic in the sense that its largest Lyapunov exponent has a relatively smaller value. Comparing Fig. 5共a兲 with Fig. 5共b兲, we find that when transient behaviors are disregarded, nodes whose dynamics are less chaotic require smaller value of the coupling strength to
be entrained. Entrainment of more chaotic nodes requires stronger coupling. Thus the development of GS on regular network is determined by the local dynamics. We next consider the occurrence and development of GS on small-world networks. A representative example is shown in Fig. 6. We see from Fig. 6共a兲 that there is no apparent synchronization sequence of nodes according to the degree index id. However, as can be seen from Fig. 6共b兲, GS starts from nodes with smaller values of the largest Lyapunov exponent. This demonstrates that for a small-world network, heterogeneity in the local dynamics plays a dominant role in the development of GS, which is similar to the situation in regular networks. Lastly, we investigate GS in a type of modular network. A modular network is characterized by a number of sparsely connected subnetworks, each with dense internal connections. For such a network, synchronization within each individual cluster can usually be achieved readily due to the dense internal connections, so the occurrence of global synchronization is of particular interest.27 An example of the development of GS on modular network is shown in Fig. 7. We see that global GS can be achieved despite the sparse intercluster connections. An interesting phenomenon is that the development of GS does not seem to strongly depend on id or ir. Nevertheless, comparatively, it can still be found that nodes with less chaotic local dynamics are easier to be entrained with smaller coupling strength. To summarize, we have observed that the development of GS on complex networks can be affected by both the network topology and the local chaotic dynamics when they both have heterogeneity. For heterogeneous networks, such as scale-free networks, the network topology plays a leading role; while for approximately homogeneous networks, such as small-world networks and modular networks, the local dynamics is the dominant factor determining the organization and spreading of GS on networks.
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V. EFFECT OF DIFFERENT COUPLING STRATEGIES A. The linear coupling and the normalized coupling
In the above numerical experiments, we have seen that the hub nodes behave as “seeds” to develop GS in scale-free networks. Nevertheless, in a recent works,36 it is reported that only for a kind of special scale-free network with treelike structure, GS is observed to develop from the hubs and then gradually spread to other nodes in the network; while for usual scale-free networks, the heterogeneity of network seems to have little effect on the development of GS in the network. In attempting to find the reason that leads to different observations between the present work and Ref. 36, we notice that a different coupling strategy is used in Ref. 36. Take a coupled one-dimensional map network as an example, the coupled system in Ref. 36 is xn+1 = 共1 − 兲f共xni 兲 + i
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where xni is the state variable of node i at time step n, f共x兲 is the local map, and ki is the degree of node i. This coupling scheme with each node in the network driven by the local mean field is essentially the same as the following normalized coupling strategy: x˙ i = Fi共xi兲 −
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where notations are the same as those in Eqs. 共1兲 and 共6兲. In this coupling scheme, the coupling strength for each node i is normalized by its degree. Therefore, the effective coupling strength each node received from its network neighbors is of the same order of magnitude, regardless of the fact if it is a hub with very large degree or a node with very small degree. Obviously, due to the normalization, the effect of network topology on the development of GS has been suppressed. As
a consequence, there should be no distinct difference among the coupling strength when nodes in the network achieve GS. The above heuristical idea can be further understood through the local stability analysis for nodes in the network. The conditional stability of coupled system 共1兲 is equivalent to the stability of CS manifold between system 共1兲 and its corresponding auxiliary system 共2兲.4 Letting ⌬xi = xi⬘ − xi and subtracting Eq. 共1兲 from Eq. 共2兲, we obtain ⌬xi = Fi共xi⬘兲 − Fi共xi兲 − 兺 aijH⌬xi j
⬇ 关DF共xi兲 − kiH兴 · ⌬xi .
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We see that for node i, the effective coupling strength is proportional to its degree ki. For a fixed value of , the coupling between a hub node and its counterpart in the auxiliary system can be significantly larger than the coupling for nodes with smaller degree, leading to “earlier” synchronization between the hub nodes in the original and in the auxiliary systems. Therefore, for usual linear coupled oscillator system, the general observation is that, in a complex network with heterogeneous degree distribution, the set of hub nodes provides a skeleton around which synchronization is developed. The above analysis is also suitable for coupled system 共7兲. In this case, there will be no factor ki as in Eq. 共8兲. This implies that the effect of network topology on the development of GS in network is almost eliminated. In the following, we further present two examples where the oscillators in a scale-free network are coupled with the normalized coupling strategy. The first example is coupled Lorenz oscillators described by Eq. 共7兲, and the second example is the following coupled Logistic maps: xn+1 = f共xni 兲 − i
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For both cases, the local dynamics are identical, i.e., ri = 28 for all Lorenz oscillators and f共x兲 = 4x共1 − x兲 for all Logistic maps. Figure 8 illustrates the development of GS for the above two systems on a scale-free network. The effect of different coupling strategies on the development of GS can be verified by comparing Fig. 8 with Fig. 1. For system 共1兲, GS first occurs on the hubs and then gradually spread to other nodes; while for systems 共7兲 and 共9兲, GS almost simultaneously takes place on the hubs and the other nodes. Actually, there is no significant difference among all nodes on networks. B. Coupled hybrid oscillators
In Sec. IV, we have shown that the development of GS on networks can be affected by two heterogeneous factors: the network topology and the local dynamics. In the following, we further investigate the situations when the parameter mismatches of local dynamics are very large, or the local dynamics are physically different. By adopting the normalized coupling strategy, we can conveniently investigate this issue. Here we present two examples showing how GS is developed for coupled hybrid oscillator system in networks. By hybrid we mean that the oscillators in the network can be classified into different
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types which are parametrically different or physically different. In the first example, we consider a system coupled with hybrid Lorenz oscillators. To be specific, we arbitrarily select 5% Lorenz oscillators in the network and make their parameter ri be randomly distributed in the interval 关30, 40兴. For the rest oscillators, they are the same with ri = 28, which are significantly smaller than that of the 5% oscillators. In Fig. 9, the development of GS for such a hybrid system is illustrated on a scale-free network. It is seen that most oscillators in the network are entrained at ⬇ 2. However, a small number of oscillators require significantly larger coupling strength to be entrained. A careful examination of the locations of these nodes reveals that they just correspond to the 5% oscillators with larger r values. In the second example, the coupled hybrid system consists of two kinds of oscillators which have different dynamical equations. Similar to the first example, 5% nodes are randomly selected to be the identical Rossler oscillators, while the rest 95% percent nodes are the identical Lorenz oscillators with ri = 28. The dynamical equations of the Rossler oscillator read as
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FIG. 8. 共Color兲 Color maps of d共 , id兲 in the two-dimensional parameter space 共 , i兲, characterizing the development of GS for 300 identical Lorenz chaotic oscillators 共a兲 and 300 identical Logistic maps 共b兲 on a scale-free network. The network is the same as in Fig. 1. Compared with Fig. 1, the effect caused by the normalized coupling strategy is obvious.
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F共xi兲 = 关− 共y i + zi兲,xi + 0.2y i,0.2 + zi共xi − 5.7兲兴T ,
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where xi ⬅ 共xi , y i , zi兲 are the state variables of the Rossler oscillator. In Fig. 10, we plot the local distance of GS versus the coupling strength for all nodes in a scale-free network. It is clearly shown GS can be achieved for such a coupled hybrid system. The development of GS in network has two distinct stages: the Rossler oscillators are much easier to be entrained compared with the Lorenz oscillators. The latter needs much larger coupling strength to achieve GS. Note that the largest Lyapunov exponent of the Lorenz oscillator is proportional to its r value, and the Rossler oscillator has a much smaller Lyapunov exponent than that of the Lorenz oscillator; we can conclude from the above two examples that GS usually develops from the nodes in the network where the local dynamics are less chaotic in the sense that the largest Lyapunov exponents of the local dynamics have relatively smaller values. Although our numerical simulations are carried out on a scale-free network, we believe this conclusion can still hold for other network topologies when the normalized coupling strategy is applied.
VI. DISCUSSIONS AND SUMMARY
The concept of GS was first developed in studying synchronization of two coupled nonidentical oscillators with drive-response configuration.3 Since the dynamics of the drive and the response are different, strict CS is impossible to occur in such systems. However, there may be certain functional relation between the dynamics of two oscillators, which is defined as the GS. Extensive studies have shown that GS relations are usually very complicated and odd and, in principle, these relations are difficult to be identified analytically. In Ref. 5, a more rigorous definition of differentiable GS is discussed. To detect the occurrence of GS in coupled systems, Abarbanel et al.4 proposed an effective method known as auxiliary-system approach, which has been successfully applied both numerically and experimentally.11 Although GS was initially investigated in directionally coupled systems, it is natural to extend this concept to the mutually coupled systems and networks as long as there exist certain functional relations among the node dynamics. For
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example, in Ref. 10, a mutual prediction method was proposed to detect dynamical interdependence and GS in a neural ensemble; also in Ref. 6, a mutual false nearest neighbors method was used to characterize GS in bidirectionally coupled dynamical systems. Later, in Ref. 7, auxiliary method was extended to mutually coupled nonidentical chaotic systems, and in Ref. 9, the noise effect on GS was addressed. Recently, in Ref. 36 GS was reported in a treelike scale-free network. In fact, in physical systems, the mutually coupled systems are as common as directionally coupled systems. It turns out that the GS concept is also very helpful in investigating the coherence and dependence of dynamics in mutually coupled systems and networks. We emphasize that apart from GS, there are other important synchronization regimes, such as practical synchronization, pulse synchronization, phase 共frequency兲 locking, which are very useful in describing the collective behaviors in coupled dynamical systems. In summary, we have investigated the occurrence and development of GS on various complex networks, including scale-free networks, small-world networks, random networks, and modular networks. It is shown that GS generally takes place in such networks for both coupled identical oscillators and nonidentical oscillators. For coupled identical oscillators, there exists a typical path toward global CS, i.e., nonsynchronization →GS→ CS. We find that the development of GS on complex networks depends on both network topology and local dynamics when the coupled oscillators are nonidentical. Moreover, the specific coupling strategy also plays an important role during the evolution of synchronization. Under the linear dissipative coupling scheme, for heterogeneous networks, GS generally starts from a small number of hub nodes and then spreads to the rest nodes in the network; while for homogeneous networks, GS usually starts from the nodes whose local dynamics are less chaotic in the sense that the largest Lyapunov exponents have relatively smaller values. We further show that the effect of network topology on the development of GS can be suppressed if the coupling strengths of nodes in the network are normalized. Under such coupling scheme, the development of GS is essentially determined by the chaotic extent of local dynamics. We also demonstrate that GS can occur in coupled systems with hybrid oscillators on complex networks, and the development of GS has distinct stages due to physically different local dynamics. We emphasize that while there are extensive works on synchronization in complex networks, prior to this work the general existence of GS on typical networks and the factors affecting the development of GS have not been carefully investigated. Our work reveals that on complex networks, coupled oscillators may present fundamentally different synchronization regimes which deserve further study.
ACKNOWLEDGMENTS
This work was supported by Temasek Laboratories at the National University of Singapore through the DSTA Project No. POD0613356.
1
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Verlag, Berlin, 1984兲; S. H. Strogatz, Physica D 143, 1 共2000兲. L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 80, 2109 共1998兲. 34 S. Guan, X. Wang, K. Li, B.-H. Wong, and C.-H. Lai, Chaos 18, 013120 共2008兲. 33
Chaos 19, 013130 共2009兲
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V. N. Belykh, I. V. Belykh, and M. Hasler, Physica D 195, 159 共2004兲; 195, 188 共2004兲. 36 Y.-C. Hung, Y.-T. Huang, M.-C. Ho, and C.-K. Hu, Phys. Rev. E 77, 016202 共2008兲. 35
CHAOS 19, 013131 共2009兲
The effect of noise on the complete synchronization of two bidirectionally coupled piecewise linear chaotic systems Yuzhu Xiao,a兲 Wei Xu,a兲 Xiuchun Li, and Sufang Tang Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China
共Received 23 August 2008; accepted 21 January 2009; published online 13 March 2009兲 In this paper, we study the synchronization of two bidirectionally coupled piecewise linear chaotic systems when the coupling strength is disturbed by the common or different noise. Based on stochastic differential equation theory, we verify that the noise can really induce the occurrence of synchronization, and the sufficient conditions of synchronization with probability 1 are established. We also find that with the common noise it is easier to induce the synchronization than with different noise. Moreover, two examples are provided and some numerical simulations are performed to verify the theoretical results. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3080194兴 Noise is ubiquitous in natural and synthetic systems; the noise effect on synchronization is very important. One interesting thing is that random noise can induce synchronization. This phenomenon is contrary to intuition and attracts much attention. Recently, the authors Lin and Chen proposed a good analytical method to analyze the effect of multiplicative noise in two unidirectionally coupled chaotic systems. However, the communication between two coupled systems is usually mutual in most cases; many natural and synthetic systems should be described by bidirectionally coupled systems, such as laser systems, electronic circuits, and chemical and biological systems. So in this paper, we will improve the method of Lin to analyze the effect of noise on bidirectionally coupled piecewise linear chaotic systems. I. INTRODUCTION
Synchronization, as an omnipresent phenomenon in natural systems and artificial oscillators, has been well known to scientists since the first observation by Huygens in two coupled pendulum clocks.1 In the classical sense, synchronization means the adjustment or entrainment of the frequency of periodic oscillators due to weak interaction.2 In recent years, following the development of the theory of deterministic chaos, the notion of synchronization was extended to the coupled chaotic systems inspired by the work of Pecora and Carroll.3,4 It is well known that the sensitivity to the initial condition is a generic character of chaotic oscillators. Two chaotic orbits, starting from slightly different initial points, separate exponentially with time. Due to this generic feature of chaotic systems, the chaotic synchronization is deem to have a great amount of application areas, for instance, secure communications,5 biological systems,6 chemical reaction,7 physical systems,8 etc. As a consequence, a兲
Authors to whom correspondence should be addressed. Electronic mail: [email protected] and [email protected]. Telephone: ⫹86029-88495453. Fax: ⫹86-029-88495453.
1054-1500/2009/19共1兲/013131/10/$25.00
research on chaos synchronization has been greatly developed, and different types of synchronization have been proposed, including complete synchronization,9 phase synchronization,10,11 generalized synchronization,12 and lag synchronization.13 Since noise is ubiquitous in natural and synthetic systems, the synchronization of coupled systems is unavoidably affected by multiplicative and additive noises. Therefore, the investigation of the noise effect on synchronization is of great importance. One of the remarkable discoveries is that random noise can induce synchronization. This phenomenon is contrary to intuition and attracts much attention. Noise induced synchronization can only be achieved if the largest Lyapunov exponent is negative.14 Noise makes the system spend more time in the “convergence region.” At first, some authors claim that noise can induce synchronization is due to nonzero means of noise, and unbiased noise cannot lead to synchronization.15 In fact, bias noise can move the system into another regime. However the recent work in Ref. 16 shows that some chaotic maps can become synchronized by additive unbiased noise. However most of the above researches concentrate on the effect of additive noise, and most of the results are numerical results. Recently, Lin and co-workers17,18 found that multiplicative noise can induce complete synchronization in two unidirectionally coupled chaotic systems and proposed a good analytical method to prove this phenomenon based on stochastic differential equation theory. However, the communication between two coupled systems is usually mutual in most cases, so many natural and synthetic systems should be described by bidirectionally coupled systems, such as laser systems, electronic circuits, and chemical and biological systems. Inspired by the above analysis, in this paper we propose a method to analyze the effect of noise in bidirectionally coupled piecewise linear chaotic systems which is the extension of the method in Refs. 17 and 18. The rest of this paper is organized as follows. In Sec. II, a model is treated. Based on the theory of stochastic differential equation, we verify
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that the presence of the common or different noise in coupling terms can really induce synchronization of bidirectionally coupled piecewise linear systems, and the sufficient conditions of synchronization with probability one are established. In Sec. III, two examples are provided to verify the effectiveness of the theoretical results established in Sec. II. In Sec. IV, we verify that common noise is easier to induce synchronization with probability 1 than different noise by numerical simulations. Finally, some conclusions are written in Sec. V.
II. NOISE INDUCES SYNCHRONIZATION
Let us begin with two bidirectionally coupled n-dimensional piecewise linear chaotic systems, which can be described as x˙ 1 = f共x1兲 + c共x2 − x1兲, x˙ 2 = f共x2兲 + c共x1 − x2兲,
共1兲
where xi = 共xi1 , xi2 , . . . , xin兲T 苸 Rn共i = 1 , 2兲 are the state vectors, f = 共f 1 , f 2 , . . . , f n兲T : Rn → Rn is a piecewise linear vector function describing the dynamics of a single oscillator, and c is a positive constant describing the coupling strength between two subsystems. This is a classical model of two bidirectionally coupled oscillators. In this model, the coupling strength c is a constant. In fact, the coupling strength c is usually disturbed by noise, and even the coupling strength c is just determined by some kind of noise. Therefore, for system 共1兲, we add the noise in the coupling strength and have the following model: x˙ 1 = f共x1兲 + 共c + d1共t兲兲共x2 − x1兲, x˙ 2 = f共x2兲 + 共c + d2共t兲兲共x1 − x2兲,
共2兲
where positive constant d is the noise strength and 1,2共t兲 are Gaussian white noises with 具i共t兲典 = 0, 具i共t兲i共t⬘兲典 = ␦共t − t⬘兲. Obviously, the noise-coupling terms in system 共2兲 are of multiplicative case, which can be interpreted in the sense of Itoˆ. In this paper, we will study system 共2兲 in two different cases: 共a兲 1共t兲 = 2共t兲, i.e., two subsystems are affected by common noise. 共b兲 1共t兲 and 2共t兲 are statistically independent, i.e., two subsystems are affected by different noise. Compare with the work in Ref. 19, the main difference is that the noise is multiplicative noise. Moreover, we also require the piecewise linear function f satisfies the following assumption: Assumption: For any x = 共x1 , x2 , . . . , xn兲T 苸 Rn and y = 共y 1 , y 2 , . . . , y n兲T 苸 Rn, there exists a positive constant l satisfying 共x − y兲T关f共x,t兲 − f共y,t兲兴 ⱕ l共x − y兲T共x − y兲.
共3兲
Condition 共3兲 is usually called global Lipschitz condition, and l is called Lipschitz constant. For the continuous smooth chaotic systems, it is difficult to find constant l, such as Lorenz system and Rössler system. However we can find
the constant l by inequality proof for some well-known piecewise linear chaotic systems, such as the famous Chua’s circuits,20 the Rössler-like system,21 the cellular neural network 共CNN兲 neural model in Ref. 22, and so on. A. A common noise
In this part, we study the effect of common noise on the synchronization with probability 1. Actually, the two coupled systems 共2兲 are said to achieve complete synchronization if x1 = x2 → s共t兲, as t → ⬁. Here, s共t兲 苸 Rn is a solution of a single oscillator, satisfying s˙ = f共s兲. We usually call s共t兲 as synchronization manifold. However in this paper, inspired by Ref. 23, we use the S共t兲 = 共x1 + x2兲 / 2 instead of synchronization manifold s共t兲. Obviously, ¯S共t兲 satisfies the following equation: ˙ S共t兲 = 21 共f共x1兲 + f共x2兲兲 G共x兲.
共4兲
Define the synchronization errors ei共t兲 = xi共t兲 − S共t兲 共i = 1 , 2兲, then one has the error dynamics e˙ 1 = f共x1兲 − G共x兲 + c共e2 − e1兲 + d共t兲共e2 − e1兲, e˙ 2 = f共x2兲 − G共x兲 + c共e1 − e2兲 + d共t兲共e1 − e2兲.
共5兲
Here, one should notice that ei共t兲 共i = 1 , 2兲 satisfy the following condition: 共6兲
e1 + e2 = 0. This condition is very important in the following proof. The above Eq. 共5兲 can be written as a matrix form, ˙ 共t兲 = F共x兲 + cC共E兲 + H共E兲W ˙ 共t兲. E
共7兲
Here, E共t兲 =
冉冊
C共E兲 =
e1 e2
,
F共x兲 =
冉 冊 e2 − e1
e1 − e2
,
冉
f共x1兲 − G共x兲 f共x2兲 − G共x兲
H共E兲 = d
冊
,
冉 冊 e2 − e1 e1 − e2
,
and
˙ 共t兲 = 共t兲. W According to the theory of stochastic differential equation,24,25 and due to assumption 共3兲, one can easily verify that the error equation 共7兲 possesses a global unique solution, denoted by E共t ; t0 , E0兲, for any initial condition. Obviously, E共t ; 0 , 0兲 ⬅ 0 is a trivial solution of error system 共7兲. Moreover, after introducing the synchronization errors ei共t兲 共i = 1 , 2兲, the synchronization problem of two coupled systems 共2兲 can be translated into the stability problem of the trivial solution of system 共7兲, i.e., the synchronization of coupled system 共2兲 corresponds to limt→⬁储E共t兲储 = 0 with probability 1. Here, 储 · 储 simply stands for Euclidean norm. In what follows, we will give sufficient conditions for
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the complete synchronization of coupled system 共2兲 with probability 1. To this end, according to Ref. 18, we introduce the following function for system 共7兲: V共E兲 = 21 log共ET共t兲E共t兲兲.
M共t兲 =
t
t0
ET共s兲H共E兲 dW共s兲, ET共s兲E共s兲
关M共t兲,M共t兲兴 =
Here, E 共t兲 denotes the transpose of E共t兲. By applying Itoˆ formula18,24 to Eq. 共8兲 along with system 共7兲, we have T
冕
= d2
DV共E共s兲兲关F共x兲 + cC共E兲兴
冎
= d2
冕再 t
关eT1 e1 + eT2 e2兴2 关eT1 e1 + eT2 e2兴2
ds
ds + c
共12兲
冕
关eT1 共e2 − e1兲 + eT2 共e1 − e2兲兴
t
eT1 e1 + eT2 e2
t0
eT1 e1 + eT2 e2
共11兲
holds almost surely. Following, we will give the estimation of Eq. 共9兲,
eT1 e1 + eT2 e2
共e2 − e1兲T共e2 − e1兲 + 共e1 − e2兲T共e1 − eT2 兲
t0
关2eT1 e1 + 共2eT2 e2兲兴2
M共t兲 =0 t→⬁ t
eT1 关f共x1兲 − f共s共t兲兲兴 + eT2 关f共x2兲 − f共s共t兲兲兴
t0
t
ds
lim
E共t兲ET共t兲 D2V共E兲 = T − 2 T . E E 共E 共t兲E共t兲兲2 I
The last part of Eq. 共9兲 is
1 + d2 2
关eT1 共e2 − e1兲 + eT2 共e1 − e2兲兴2
Therefore, according to the strong law of large numbers,18,24 one has that
where the derivatives of the function V共E共t兲兲 are
冕
t
= 4d2共t − t0兲. 共9兲
+ M共t兲,
t
冕 冕
t0
1 + tr关HT共E共s兲兲D2共V共E共s兲兲兲H共E共s兲兲兴 ds 2
V共E共t兲兲 = V共E共t0兲兲 +
关ET共s兲H共E兲兴2 ds 关ET共s兲E共s兲兴2
t0
t0
ET共t兲 DV共E兲 = T , E 共t兲E共t兲
t
t0
t
V共E共t兲兲 = V共E共t0兲兲 +
共10兲
Also the quadratic variation in Eq. 共10兲 is
共8兲
冕再
冕
−2
关共e2 − e1兲Te1 + 共e1 − e2兲Te2兴2 共eT1 e1 + eT2 e2兲2
冎
ds
ds + M共t兲
ⱕ V共E共t0兲兲 + l共t − t0兲 − 2c共t − t0兲 − 2d2共t − t0兲 + M共t兲.
Furthermore, we have
V共E共t兲兲 ⱕ l − 2c − 2d2 . lim sup t t→⬁
共13兲
˙ 共t兲 = F共x兲 + cC共E兲 + 1 H共E兲W ˙ 共t兲. E 2
共14兲
In the above proof, we have used conditions 共3兲, 共6兲, and 共12兲. Obviously, if the coupling strength c, noise strength d, and constant l make l − 2c − 2d2 ⬍ 0, the synchronization errors ei共i = 1 , 2兲 converge to zero as t → ⬁ with probability 1. From Eq. 共14兲, we can also see that the noise really has a positive effect on the synchronization.
共15兲
Here, the definitions of E共t兲, F共x兲, and C共E兲 are same as in Sec. II A,
H共E兲 =
冉
冊
d 共e2 − e1兲 共e2 − e1兲 , 2 共e1 − e2兲 共e1 − e2兲
冉 冊
˙ 共t兲 = 1共t兲 . W 2共t兲
By applying Itoˆ formula to Eq. 共8兲 along with system 共15兲, we have
冕再 t
V共E共t兲兲 = V共E共t0兲兲 +
t0
B. Two different noises
In this part, we will study the effect of different noises on the occurrence of synchronization with probability 1. Similar to Sec. II A, we introduce synchronization errors ei共t兲 共i = 1 , 2兲, Then one derives error dynamics
DV共E共s兲兲关F共x兲 + cC共E兲兴
冎
1 + tr关HT共E共s兲兲D2共V共E共s兲兲兲H共E共s兲兲兴 ds 2 + M共t兲.
共16兲
Equation 共16兲 is the same as Eq. 共9兲. However the difference
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Xiao et al.
˙ 共t兲 will induce different estimations of in H共E兲 and W V共E共t兲兲. The quadratic variation in the last part of Eq. 共16兲 is 关M共t兲,M共t兲兴 =
冕
t
t0
Therefore, according to the strong law of large numbers,18,24 one has that
关ET共s兲H共E兲兴关ET共s兲H共E兲兴T ds 关ET共s兲E共s兲兴2
1 = d2 2
冕
t
关eT1 共e2
− e 1兲 + 关eT1 e1 +
t0
M共t兲 =0 t→⬁ t
共17兲
lim
eT2 共e1 − e2兲兴2 ds eT2 e2兴2
holds almost surely. Following, we will give the estimation of Eq. 共16兲,
= 2d 共t − t0兲. 2
V共E共t兲兲 ⱕ V共E共t0兲兲 + l共t − t0兲 − 2c共t − t0兲 1 + d2 4
冕再 t
关共e2 − e1兲T共e2 − e1兲 + 共e1 − e2兲T共e1 − e2兲兴2
t0
eT1 e1 + eT2 e2
−2
关共e2 − e1兲Te1兴 + 关共e1 − e2兲Te2兴2 关eT1 e1 + eT2 e2兴2
冎
ds
= V共E共t0兲兲 + l共t − t0兲 − 2c共t − t0兲 − d2共t − t0兲.
Furthermore, we have h
av error共d兲 = lim sup t→⬁
V共E共t兲兲 ⱕ l − 2c − d2 . t
共18兲
Obviously, if the coupling strength c, noise strength d, and constant l make l − 2c − d2 ⬍ 0, the synchronization errors ei共i = 1 , 2兲 converge to zero as t → ⬁ with probability 1, i.e., the two coupled systems synchronize with probability 1. Moreover, we can see that the presence of different noises in coupling terms can really induce the synchronization.
III. NUMERICAL EXAMPLES
1 兺 h i=1
冕
T2
T1
兺k=1 兩x1k共t, i,d兲 − ¯Sk共t, i,d兲兩/n n
T2 − T1
, 共20兲
where i 苸 ⍀, ⍀ denotes the set of all elementary events, x j共t , i , d兲 represents the solution of Eq. 共2兲, the definition of S共t , i , d兲 is the same as in Sec. II. It is obvious that sa error共t兲 is the average of h sample paths of synchronization error. This can be used as the scale of synchronization with probability 1 at a fix noise strength. av error共t兲 is the average of h sample paths of time average synchronization error. This can be used to show the effect of synchronization in coupled system 共2兲 with the noise strength change. Example 1: In this example, we use a three-dimensional neural network22 to describe the dynamics of a single oscillator, which can be described as
In Sec. II, based on the theory of stochastic differential equation, we have verified that the noise can really induce synchronization in two bidirectionally coupled piecewise linear chaotic systems, and the sufficient conditions of synchronization with probability 1 are established. In this section, two examples are provided and some numerical simulations are performed to verify the theoretical results. In the numerical simulations, we use Euler–Maruyama scheme for stochastic differential equation26 to solve the equations, and the random numbers are generated by IMSL, the library of FORTRAN. In order to measure the synchronization, we define the following quantities:
h
sa error共t兲 =
n
1 兺 兺 兩x1k共t, i兲 − x2k共t, i兲兩, h i=1 k=1
共19兲
FIG. 1. Chaotic attractors generated by three-dimensional neural network 共21兲 after transient time T = 1000 has been removed.
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−5
10
(a) Common noise
−10
10
−15
log(av error)
10
−20
10
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
−10
10
(b) different noise −12
10
FIG. 2. This shows the evolution of av error共d兲 in two coupled systems 共20兲 with h = 1000, T0 = 5, and T1 = 105 when the coupling strength is perturbed by common noise or different noise.
−14
10
−16
10
−18
10
−20
10
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
d
共21兲
x˙ = − Dx + Tg共x兲, where x = 共x1 , x2 , x3兲T 苸 R3 is the state vector,
冢
1.25
− 3.2 − 3.2
T = − 3.2 1.1 − 3.2 4.4
冣
− 4.4 , 1.0
D is an identity matrix, and g共x兲 = 共g共x1兲 , g共x2兲 , g共x3兲兲T, where g共s兲 = 共兩s + 1兩 − 兩s − 1兩兲 / 2. According to Ref. 22, system 共21兲 has a double-scrolling chaotic attractor 共Fig. 1兲. It is easier to verify that g共s兲 satisfies 兩g共s1兲 − g共s2兲兩 ⱕ 兩s1 − s2兩. For any x, y 苸 R3, we have 共x − y兲T − 共Dx + Tg共x兲 + Dy − Tg共y兲兲 ⱕ − 共x − y兲TD共x − y兲 + 兩共x − y兲TT共g共x兲兲 − g共y兲兩 ⱕ 兩共x − y兲兩T共兩T兩 − D兲兩共x − y兲兩 ⱕ m共x − y兲T共x − y兲 where
兩T兩 =
冢
1.25 3.2 3.2 3.2
1.1 4.4
3.2
4.4 1.0
冣
and m ⬇ 7.339 is the maximum eigenvalue of 兩T兩 − D. So we
can acquire the constant l ⬇ 7.339 in condition 共3兲. The main purpose of this paper is to study the effect of noise on the synchronization. Therefore, in the numerical simulations, we let coupling strength c = 0, i.e., two systems just bidirectionally couple by multiplicative noise. According to the theoretical results in Sec. II, the two coupled systems are synchronized with probability 1, if the noise strengths d ⬎ 1.916 when noise is common, or the noise strengths d ⬎ 2.709 when noise is different. In order to show that the conditions are sufficient, we compute the av error共d兲 from d = 1.92 to d = 2.92 when noise is common, and from d = 2.71 to d = 3.71 when noise is a different noise. The numerical simulation results are shown in Figs. 2共a兲 and 2共b兲. From these figures, we can see that the av error共d兲 is very small and decreases with noise strength increasing. That means that the coupled systems are synchronized during these intervals. We also compute the sa error共t兲 at small noise strengths which just reach conditions 共14兲 and 共18兲, and at the large noise strength d = 20. The numerical simulation results are shown in Figs. 3共a兲–3共d兲. From these figures, we can see that sa error共t兲 converges to zero with time increasing. The individual systems maintain the original dynamical after two coupled systems achieve complete synchronization because the coupled term vanishes. Some chaotic attractors after synchronization achievement are shown in Figs. 4共a兲–4共d兲. Example 2: In this example, we use the Rössler-like system21 to describe the dynamics of a single oscillator, which can be described as
x˙ = Dx + g共x兲,
共22兲
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10 (a) d=1.92, common noise
(b) d=2.71, different noise
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−5
10
10
−10
−10
log(sa error(t))
10
10
0
1
2
3
4
5
1
2
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FIG. 3. This shows the temporal evolution of sa error共t兲 in two coupled systems 共21兲 with h = 1000 when the coupling strength is perturbed by common noise or different noise.
5
0
0
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10
(c) d=20.0, common noise
(d) d=20..0, different noise
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10 −15
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0.2
t
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(b) d=20.0
1
1
0.5
0.5 X22
x12
1.5
0
0
−0.5
−0.5
−1
−1
−1.5 −2
−1
0 x11
1
−1.5 −2
2
1.5
1
0 x21
1
2
FIG. 4. Chaotic attractors of system 共21兲 after synchronization achievement: 关共a兲 and 共b兲兴 for common noise and 关共c兲 and 共d兲兴 for different noise.
(d) d=20.0
1
1
0.5
0.5 x22
x12
0 x21
1.5 (c) d=20.0
0
0
−0.5
−0.5
−1
−1
−1.5 −2
−1
−1
0 x11
1
2
−1.5 −2
−1
2
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Noise induced synchronization
where x = 共x1 , x2 , x3兲T 苸 R3 is the state vector,
D=␣
冢
−⌫ − − 1 0
␥ 0
0 −
冣
f共s兲 =
再
s ⬍ 2.56
0,
共s − 2.56兲, s ⱖ 2.56.
冎
Chaos 19, 013131 共2009兲
Here, we set positive constants ␣ = 0.03,  = 1.5, ␥ = 0.2, = 1.5, ⌫ = 0.075, = 0.75, and = 21.43. According to Ref. 21, under the above parameters system 共22兲 is chaotic 共Fig. 5兲. For any x, y 苸 R3, g共x兲 satisfies
,
1 1 共x − y兲T共g共x兲 − g共y兲兲 ⱕ 2 ␣共x1 − y1兲2 + 2 ␣共x3 − y3兲2 .
g共x兲 = 共0 0 ␣ f共x1兲 兲T ,
So we have
共x − y兲T共Dx + g共x兲 − Dy − g共y兲兲 ⱕ 0.5␣共x − y兲T共D + DT兲共x − y兲 + 0.5␣共x1 − y 1兲2 + 0.5␣共x3 − y 3兲2 = 共x − y兲
T
冢
− ␣⌫ + 0.5␣ 0.5共␣ − ␣兲 0.5共␣ − ␣兲 − 0.5␣
␣␥ 0
− 0.5␣ 0 − ␣ + 0.5␣
冣
共x − y兲
共x − y兲 F共x − y兲. T
It is obviously that F is a real symmetric matrix. According to the theory of matrix, the estimation of Eq. 共10兲 could be further given by 共x − y兲T共Dx + g共x兲 − Dy − g共y兲兲 ⱕ m共x − y兲T共x − y兲, Where m = 0.465 is the maximum eigenvalue of F. So we acquire the constant l ⬇ 0.465 in condition 共3兲. In this part, we also set coupling strength c = 0, so the two coupled systems are synchronized with probability 1 if the noise strengths d ⬎ 0.482 when noise is common, or the noise strengths d ⬎ 0.682 when noise is different. Similar to example one, we compute the quantities av error共d兲 and sa error共t兲. The numerical results are show in Figs. 6共a兲, 6共b兲, 7共a兲–7共d兲, and 8共a兲–8共d兲. From these figures, we can see that the numerical results are consistent with theoretical results acquired in Sec. II. IV. COMPARISON OF TWO CASES
In Sec. II, we have attained the conditions of synchronization with probability 1 in two bidirectionally coupled piecewise linear chaotic systems. From these conditions, we can see that there exists a critical value for common and different noise, respectively; if the noise strength is larger than that value, the two coupled systems can reach synchronization, and that value for common noise is smaller than the value for different noise. However we should note that the conditions are sufficient, the coupled systems may synchronize at small noise strength than these values. What is the real relation between two cases? In this part, we verify this by numerical simulations. In order to show the effect of noise on synchronization with the noise strength change, we compute the av error共d兲 with h = 1000, T1 = 0, and T2 = 1000. In Figs. 9 and 10, we plot the evolution of av error共d兲 of two coupled systems 共21兲 and 共22兲, respectively. In each figure, the dot line corre-
sponds to the case of different noise, and the real line corresponds to the case of common noise. From these two figures, we can see that the av error共d兲 decreases with the coupling strength increasing, and the av error共d兲 of common noise is always smaller than the av error共d兲 of different noise. These imply that the noise strength of common noise makes coupled systems synchronize with probability 1, which is smaller than the strength of different noise. In other words, the common noise is easier to induce synchronization with probability 1 than different noise. V. CONCLUSION
In this paper, we developed the method proposed in Ref. 18 to study the synchronization of two bidirectionally coupled piecewise linear chaotic systems when coupling strength is perturbed by common or different white noises. Based on the theory of stochastic differential equation, we verify that noise can induce the synchronization of bidirec-
FIG. 5. Chaotic attractor of system 共22兲 after transition T = 1000 has been removed.
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Chaos 19, 013131 共2009兲
FIG. 6. This shows the evolution of av error共d兲 in two coupled systems 共22兲 with h = 1000, T0 = 70, and T1 = 170 when the coupling strength is perturbed by common noise or different noise.
FIG. 7. This shows the temporal evolution of sa error共t兲 in two coupled systems 共22兲 with h = 1000 when the coupling strength is perturbed by common noise or different noise.
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Noise induced synchronization
Chaos 19, 013131 共2009兲
FIG. 8. Chaotic attractor of system 共22兲 after synchronization achievement: 共a兲 and 共b兲 for common noise and 共c兲 and 共d兲 for different noise.
FIG. 9. This shows the evolution of av error共d兲 in two coupled systems 共21兲. The dotted line corresponds to the case of different noise and the solid line corresponds to the case of common noise.
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FIG. 10. This shows the evolution of av error共d兲 in two coupled systems 共22兲. The dotted line corresponds to the case of different noise and the solid line corresponds to the case of common noise.
tionally coupled piecewise linear chaotic system, and the sufficient conditions of synchronization with probability 1 are established. Moreover, we also find that the common noise is easier to induce synchronization with probability 1 than different noise. In this paper, we just study two bidirectionally coupled piecewise linear chaotic systems, but our method can be extended to multicoupled piecewise linear oscillators. Furthermore, maybe this method can be extended to complex network; this is our further work.
ACKNOWLEDGMENTS
The authors are grateful for the support of the National Natural Science Foundation of China 共Grant Nos. 10502042 and 10872165兲. C. Huygens, Philos. Trans. R. Soc. London 4, 937 共1669兲. I. I. Blekman, Synchronization of Dynamical Systems 共Nauka, Moscow, 1971兲. 3 L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 共1990兲. 4 L. M. Pecora and T. L. Carroll, Phys. Rev. A 44, 2374 共1991兲. 5 C. D. Li, X. F. Liao, and K. Wong, Physica D 194, 187 共2004兲. 6 X. J. Wang and J. Rinzel, Neural Comput. 4, 84 共1992兲. 7 Y. Kuramoto, Chemical Oscillations, Waves and Turbulence 共Springer, Berlin, 1984兲. 1 2
8
L. O. Chua, M. Itah, L. Kosarev, and K. Eckert, J. Circuits Syst. Comput. 3, 93 共1993兲. 9 K. Pyragas, Phys. Rev. E 54, R4508 共1996兲. 10 G. M. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 共1996兲. 11 G. M. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 78, 4193 共1997兲. 12 X. Huang and J. D. Cao, Nonlinearity 19, 2797 共2006兲. 13 Y. H. Sun and J. D. Cao, Phys. Lett. A 364, 277 共2007兲. 14 A. S. Pikovsky, Phys. Rev. Lett. 73, 2931 共1994兲. 15 H. Herzel and J. Freund, Phys. Rev. E 52, 3238 共1995兲. 16 C. H. Lai and C. S. Zhou, EPL 43, 376 共1998兲. 17 W. Lin and Y. B. He, Chaos 15, 023705 共2005兲. 18 W. Lin and G. Chen, Chaos 16, 013134 共2006兲. 19 C. S. Zhou and J. Kurths, Phys. Rev. Lett. 88, 230602 共2002兲. 20 L. O. Chua, C. W. Wu, A. Huang, and G. Q. Zhong, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 40, 732 共1993兲. 21 I. A. Heisler, T. Braun, Y. Zhang, G. Hu, and H. A. Cerdeira, Chaos 13, 185 共2003兲. 22 F. Zou and J. A. Nossek, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 40, 166 共1993兲. 23 W. L. Lu and T. P. Chen, Physica D 213, 214 共2006兲. 24 L. Arnold, Stochastic Differential Equation and Application 共Academic, New York, 1972兲. 25 A. Friedman, Stochastic Differential Dquations and Applications 共Academic, New York, 1975兲. 26 P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations 共Springer, Berlin, 1995兲.
CHAOS 19, 013132 共2009兲
Invariant submanifold for series arrays of Josephson junctions Seth A. Marvela兲 and Steven H. Strogatz Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA
共Received 18 December 2008; accepted 2 February 2009; published online 17 March 2009兲 We study the nonlinear dynamics of series arrays of Josephson junctions in the large-N limit, where N is the number of junctions in the array. The junctions are assumed to be identical, overdamped, driven by a constant bias current, and globally coupled through a common load. Previous simulations of such arrays revealed that their dynamics are remarkably simple, hinting at the presence of some hidden symmetry or other structure. These observations were later explained by the discovery of N − 3 constants of motion, the choice of which confines the resulting flow in phase space to a low-dimensional invariant manifold. Here we show that the dimensionality can be reduced further by restricting attention to a special family of states recently identified by Ott and Antonsen. In geometric terms, the Ott–Antonsen ansatz corresponds to an invariant submanifold of dimension one less than that found earlier. We derive and analyze the flow on this submanifold for two special cases: an array with purely resistive loading and another with resistive-inductive-capacitive loading. Our results recover 共and in some instances improve兲 earlier findings based on linearization arguments. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3087132兴 Josephson junctions are superconducting devices with many practical applications, ranging from voltage standards to ultrasensitive detectors. From a mathematical perspective, their nonlinear dynamics are fascinating, especially when many junctions are coupled together in an array. For about the past 20 years, theorists have been intrigued by the strange collective behavior seen in numerical experiments on arrays of identical junctions in series. The behavior began to make sense when it was eventually realized that despite the presence of dissipation in the underlying circuits, the equations possess an enormous number of constants of motion. These constants restrict the dynamics to low-dimensional manifolds in phase space. In this paper, we show that in certain cases the reduced dimensionality can be even more severe than previously realized. For example, by making use of an ansatz recently introduced by Ott and Antonsen, we demonstrate that a resistively loaded array can behave exactly as if its phase space was two dimensional, even when the array consists of infinitely many junctions. I. INTRODUCTION
Forty years ago, Winfree pioneered the study of synchronization in large populations of coupled limit-cycle oscillators.1 Since then, the field has expanded considerably, thanks in large part to Kuramoto’s elegant reformulation2,3 of Winfree’s intuitive model. Both of their models were originally motivated by biological phenomena4 such as the alpha rhythm of brain waves,5–7 the collective firing of cardiac pacemaker cells,8–10 the coordinated flashing of Southeast Asian fireflies,11–15 menstrual synchrony among close female friends,16–18 and biochemical oscillations in cell populations.19–24 However, the techniques developed in anaa兲
Electronic mail: [email protected].
1054-1500/2009/19共1兲/013132/9/$25.00
lyzing these systems soon proved relevant to physical problems, including the dynamics of laser arrays,25–31 chargedensity waves,32–34 and coherence among sites of electrochemical dissolution.35–37 For simplicity, the individual oscillators in such models have often been assumed to be coupled equally strongly to each other, a form of interaction known variously as global, infinite-range, or mean-field coupling. While this form of coupling is a crude approximation in most cases, series arrays of Josephson junctions constitute a notable exception. In these systems, exact global coupling between the Josephson junctions emerges naturally from Kirchhoff’s laws and the physical properties of weakly coupled superconductors.38,39 In the early 1990s, numerical simulations of Josephson junction arrays revealed that they were prone to a large degree of neutral stability.38–42 This peculiar phenomenon was first seen in a simple array in which N identical junctions were connected in series and coupled by a resistor in parallel with all of them. The numerics suggested that the trajectories of the system were always trapped on two-dimensional tori, no matter how many junctions were included in the array.38 Later studies showed that arrays of identical junctions coupled through other kinds of loads displayed similarly nongeneric forms of behavior.39,41 These and other puzzling observations were partially explained by the subsequent discovery that the equations of motion could be reduced to a dynamical system of much lower dimension.43–45 Specifically, Watanabe and Strogatz43,44 found a time-dependent trigonometric transformation that expressed the junction phases in terms of N constant phases and three collective time-dependent variables, each obeying a suitable differential equation. The transformation took the form
19, 013132-1
© 2009 American Institute of Physics
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冋
tan
Chaos 19, 013132 共2009兲
S. A. Marvel and S. H. Strogatz
册冑
j共t兲 − ⌽共t兲 = 2
冋
1 + ␥共t兲 j − ⌰共t兲 tan 2 1 − ␥共t兲
册
共1兲
for Josephson junctions j = 1 , . . . , N. Here, the variables j共t兲 denote the junction phases, while the constants j denote the fixed phases on which the transformation operates, and ⌽共t兲, ␥共t兲, and ⌰共t兲 denote the collective variables. In the limit of infinitely many Josephson junctions, another result could be extracted from Eq. 共1兲. The transformation maps old phases j to new phases j in such a way that a uniform distribution in , with phases spread evenly around the circle, is transformed into a nonuniform distribution in , with phases symmetrically clumped about some mean phase and distributed according to a Poisson kernel. This result implies that the space of Poisson kernels is dynamically invariant. The reasoning is as follows: if the phase distribution takes the form of a Poisson kernel at any time, it can be viewed as having arisen 共via the transformation兲 from an initially uniform phase distribution and hence will remain distributed as a Poisson kernel for all time. Recently, Ott and Antonsen46 showed by explicit calculation that the submanifold of Poisson-kernel phase distributions, henceforth called the Poisson submanifold, is invariant for a much wider class of models, including the Kuramoto model and other systems in which the oscillators are nonidentical. For the Kuramoto model in which the oscillators have Lorentzian-distributed natural frequencies, Ott and Antonsen46 derived an exact differential equation for the evolution of the complex order parameter 共the centroid of the phase distribution around the unit circle兲. As it happens, the amplitude and phase of the order parameter completely characterize the Poisson kernel. Hence, by extracting the dynamics of the order parameter, Ott and Antonsen46 simultaneously unveiled the dynamics on the Poisson submanifold. Our goal in this paper is to use Ott and Antonsen’s ansatz46 to investigate the dynamics of series arrays of identical Josephson junctions. Before turning to this specific task, however, we first consider the scope of the ansatz itself. What algebraic form do the governing equations need to have in order for the ansatz to work? In Sec. II, we pinpoint the family of equations that can be simplified in this way, a family that includes Josephson junction series arrays as a special case. Sections III and IV then apply the ansatz to two particular arrays, one with a resistive load and another with an RLC load. The analysis illuminates the order parameter dynamics on the entire Poisson submanifold. In this way, we clarify for the first time how the associated arrays behave both near their equilibria and periodic orbits and far from those special states. Finally, Sec. V discusses what the ansatz does—and does not—imply about the dynamics of the original Josephson junction arrays. Although the ansatz provides powerful insights, it does not tell the whole story; it misses certain important dynamical states that lie off the submanifold of Poisson kernels. These more general states can be handled by considering Eq. 共1兲 within the formalism of Möbius transformations, as was first suggested by Goebel45 and recently demonstrated elsewhere.47,48
II. REDUCIBLE SYSTEMS
The most extensively studied systems of phase oscillators, from the Kuramoto model to Josephson junction arrays, involve purely sinusoidal interactions. This single-harmonic structure is the key to the success of the Ott–Antonsen ansatz,46 as the following calculation shows. Consider a system of N identical phase oscillators governed by
˙ j = fei j + g + ¯f e−i j
共2兲
for j = 1 , . . . , N. Here f is any smooth, complex-valued, 2-periodic function of the phases 1 , . . . , N. The function f is allowed to depend on time and any other auxiliary state variables in the system 共for example, the charge on a load capacitor or the current through a load resistor for the Josephson junction arrays discussed in Sec. IV兲. What is crucial, however, is that f must not depend on the oscillator index j; it must be the same function for all j. Likewise, the function g must be independent of j. Note that g has to be ˙ j is real. real valued since Intuitively, the functions f and g can be regarded as common fields felt by all the oscillators. These fields might involve averages over all the phases, as in models with meanfield coupling, but this is not necessary. In fact, Eq. 共2兲 need not even have permutation symmetry. That is, the equations need not stay the same under an arbitrary interchange of indices, because the functions f and g need not respect such a symmetry. The only requirement is that f and g must be the same for all j. The class of systems 共2兲 was studied previously by Watanabe and Strogatz,44 who showed that all equations of this form are solved by the transformation 共1兲, where the evolution of ⌽, ␥, and ⌰ is governed by the forms of f and g 关although f and g in our paper would be written 共g − ih兲 / 2 and f in theirs兴. We now show that the newly discovered Ott–Antonsen ansatz also works on this same class of systems, suggesting some intimate relationship between the two reduction methods, as addressed in Refs. 47 and 48. The Ott–Antonsen ansatz is restricted to the infinite-N limit of Eq. 共2兲. In this limit, one describes the system not in terms of the motion of individual oscillators but rather in terms of the evolution of the phase density 共 , t兲, defined such that 共 , t兲d gives the fraction of phases that lie between and + d at time t. Then satisfies the continuity equation
˙ +
共v兲 = 0,
共3兲
where the velocity field is v共,t兲 = fei + g + ¯f e−i
共4兲
from Eq. 共2兲. Here, in the case of infinite N, our assumptions about the coefficient functions f and g take the form that f and g may depend on t but not . The time dependence of f and g can arise either explicitly 共through external forcing, say兲 or implicitly 共through the time dependence of the harmonics of or any auxiliary state variables in the system兲.
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Invariant submanifold of Josephson array
Next, following Ott and Antonsen,46 suppose is of the form
再
⬁
1 1 + 兺 共¯␣共t兲nein + ␣共t兲ne−in兲 共,t兲 = 2 n=1
冎
RJ
共5兲
Ib RJ
for some unknown function ␣ that is independent of with modulus no greater than unity. Note that Eq. 共5兲 is just an algebraic rearrangement of the usual form for the Poisson kernel 1 − r2 1 = , 2 1 − 2r cos共 − 兲 + r2
RJ 共6兲
where r and are defined via
␣ = rei .
共7兲
In geometrical terms, ansatz 共5兲 defines a submanifold in the infinite-dimensional space of density functions . This submanifold is two dimensional and is parametrized by the complex number ␣ 共or equivalently, by the polar coordinates r and 兲. We now find the flow on the invariant submanifold of Poisson kernels. To do so, we substitute the velocity field 共4兲 and ansatz 共5兲 into the continuity equation 共3兲. After reindexing where appropriate, we obtain ⬁
关␣˙ − i共f ␣2 + g␣ + ¯f 兲兴 兺 n␣n−1e−in + c.c. = 0,
共8兲
n=1
where c.c. denotes the complex conjugate of the first term in Eq. 共8兲. Note the coincidence here: the expression in brackets is a common factor for each term in the sum. Hence, although Eq. 共8兲 constitutes an infinite set of amplitude equations, with one for each harmonic, all of them are satisfied simultaneously if the bracketed expression vanishes. This condition is also necessary, since ⬁
␣−1 兺 n共␣e−i兲n = n=1
e−i ⫽ 0. 共1 − ␣e−i兲2
共9兲
Thus Eq. 共8兲 is satisfied for all if and only if ␣ satisfies the Riccati equation
␣˙ = i共f ␣2 + g␣ + ¯f 兲.
共10兲
It proves convenient to re-express this result in terms of the complex order parameter z, defined as usual by the centroid of the phase distribution z共t兲 =
冕
2
i
e 共,t兲d .
共11兲
0
By substituting Eq. 共5兲 into Eq. 共11兲, we find that z = ␣. Hence, z also must satisfy Eq. 共10兲, z˙ = i共fz2 + gz + ¯f 兲.
R
共12兲
This equation gives the flow on the Poisson submanifold. When f and g are functions of z and t alone, as in the case that they can be expressed in terms of the harmonics of , Eq. 共12兲 constitutes a closed two-dimensional system.
FIG. 1. A series of Josephson junctions in parallel with a resistive load. Ib is the constant bias current, R is the load resistance, and RJ is the internal resistance of a single Josephson junction.
III. RESISTIVELY LOADED CIRCUIT
We now apply Eq. 共12兲 to obtain the reduced dynamics of the circuit shown in Fig. 1. This circuit consists of N Josephson junctions wired in series and placed in parallel with a resistive load R and a constant current source Ib. Let Ic denote the critical current of each junction and RJ denote its internal resistance. For simplicity, the junctions are assumed to be heavily overdamped so that we can neglect their internal capacitance. As demonstrated by Tsang et al.,38 the time-dependent dynamics of this circuit can be converted via Kirchhoff’s laws and the physical properties of Josephson junctions to the dimensionless system N
1 ˙ j = ⍀ + a cos j + 兺 cos k , N k=1
共13兲
for j = 1 , . . . , N. Here, j = ␦ j − / 2, where ␦ j is the phase difference across the jth oscillating Josephson junction. We write the system in these unusual variables to highlight its reversibility symmetry; the equations stay the same if we change j → − j and t → −t. The dimensionless groups a and ⍀ in Eq. 共13兲 are given in terms of the original circuit parameters by a = −R / 共NRJ兲 − 1 and ⍀ = IbR / 共NIcRJ兲. Taking the limit N → ⬁ of Eq. 共13兲, we obtain the velocity field a a v共,t兲 = ei + 共⍀ + r cos 兲 + e−i , 2 2
共14兲
where r and denote the amplitude and phase of the complex order parameter z, as before. Equation 共14兲 has the form of Eq. 共4兲, so by Eq. 共12兲, the order parameter dynamics are
z˙ = i
冋
册
a a 2 z + 共⍀ + r cos 兲z + . 2 2
共15兲
Taking the real and imaginary parts of Eq. 共15兲 yields the two-dimensional nonlinear system
013132-4
r˙ =
Chaos 19, 013132 共2009兲
S. A. Marvel and S. H. Strogatz
TABLE I. Existence of the fixed points for ⍀ ⬎ 0.
1+b 2 共r − 1兲sin , 2
Fixed point
共16兲 1+b 共r + r−1兲cos + r cos , ˙ = ⍀ − 2 where we have introduced the parameter b = −a − 1 = R / 共NRJ兲, which simplifies the parameter space and fixed points of Eq. 共16兲. Note that b ⬎ 0 for any real circuit, although in what follows we will also allow negative values of b, since our main interest is with Eq. 共13兲 as a dynamical system, not as a model of a real device. A. Fixed points
This section analyzes the fixed points of Eq. 共16兲 with respect to their dependence on the parameters b and ⍀. There are many cases, so we organize the calculations as a series of simple claims. Readers who prefer to bypass the algebra can skip ahead to Sec. III D which summarizes the findings. First we convert Eq. 共16兲 to Cartesian coordinates. This removes the coordinate singularity at r = 0 and permits simpler proofs of where the fixed points of Eq. 共16兲 exist in the b-⍀ parameter space. Let x = r cos and y = r sin . Then Eq. 共16兲 becomes x˙ = bxy − ⍀y, 共17兲 1−b 2 1+b 2 1+b y˙ = x + ⍀x + y − . 2 2 2 Note that Eq. 共17兲 remains identical under the transformation ⍀ → −⍀, x → −x, which represents the symmetry of the resistively loaded circuit in Fig. 1 under a sign reversal of Ib and reflection of the coordinate system on which phase synchrony is measured. Hence, the number and stability of the fixed points remain unchanged for each point in b-⍀ space reflected across the b axis. Without loss of generality, we therefore consider only positive values of ⍀ from here on. If b ⫽ 0 , ⫾ 1, Eq. 共17兲 has four fixed points: xⴱ = ⍀/b,
y ⴱ = 冑1 − ⍀2/b2 ,
共18a兲
xⴱ = ⍀/b,
y ⴱ = − 冑1 − ⍀2/b2 ,
共18b兲
xⴱ =
− ⍀ + 冑⍀2 − b2 + 1 , 1−b
y ⴱ = 0,
共18c兲
xⴱ =
− ⍀ − 冑⍀2 − b2 + 1 , 1−b
y ⴱ = 0.
共18d兲
Points 共18a兲 and 共18b兲 lie on the boundary r = 1 of the unit disk and represent synchronized rest states: equilibrium states in which all the Josephson junctions are perfectly in phase and do not oscillate. In this case, all the individual phases j equal the same constant, and hence equal the constant phase of the centroid. Physically, such states would be superconducting. The source current tunnels through each of the N junctions without developing any voltage across the load.
Conditions
Eqs. 共18a兲 and 共18b兲 Eq. 共18c兲 Eq. 共18d兲
⍀ ⱕ 兩b兩 ⍀2 − b2 + 1 ⱖ 0 and ⍀ ⱖ b ⍀2 − b2 + 1 ⱖ 0 and ⍀ ⱕ −b
In contrast, fixed points 共18c兲 and 共18d兲 have r ⬍ 1 and lie inside the unit disk, meaning that the Josephson junctions are not all in phase. This type of fixed point is known as a splay state.38–42 It represents a periodic collective state in which the junctions oscillate out of phase but in such a highly organized way that the overall phase distribution remains stationary. In particular, the macroscopic order parameters r and stay constant even though individual junctions change their state nonuniformly, hesitating at some phases and accelerating at others. The stationary distribution of phases takes the form of a Poisson kernel, as expected.
B. Partitioning the parameter space
We now determine where in the b-⍀ parameter space the fixed points exist. By inspection, the synchronized rest states 共18a兲 and 共18b兲 exist if and only if ⍀ ⱕ 兩b兩. Meanwhile, for the splay states 共18c兲 and 共18d兲 to exist, xⴱ must be real 共i.e., ⍀2 − b2 + 1 ⱖ 0兲 and 兩xⴱ兩 ⱕ 1. The condition 兩xⴱ兩 ⱕ 1 places additional restrictions on the b and ⍀ values at which 共18c兲 and 共18d兲 exist. We now derive these restrictions explicitly, with the key results summarized in Table I. Claim: On ⍀ , b ⬎ 0, Eq. 共18c兲 exists if and only if ⍀ ⱖ b. Proof: Let h共b , ⍀兲 = ⍀2 − b2 + 1. Then by algebraic rearrangement,
冉
− ⍀ + 冑h 1−b
冊
2
ⱕ 1 ⇔ ⍀冑h ⱖ ⍀2 − b2 + b.
共19兲
When the discriminant of Eqs. 共18c兲 and 共18d兲 is nonnegative, ⍀2 − b2 + b ⱖ b − 1. Furthermore, b − 1 ⱖ 0 for b ⱖ 1 and b共1 − b兲 ⱖ 0 for b on 关0, 1兴, so both sides of the second inequality in Eq. 共19兲 are non-negative for all b ⬎ 0. Thus, we can square both sides of this inequality to remove the square root ⍀2h ⱖ 共⍀2 − b2 + b兲2 ⇔ ⍀ ⱖ 兩b兩.
共20兲
We can also see that Eq. 共20兲 implies Eq. 共19兲 when h ⱖ 0, because there the larger quantity of the first inequality in Eq. 共20兲 is non-negative and remains so upon removing the square in going from Eq. 共20兲 to Eq. 共19兲. Hence, the claim is proved. 䊐 Claim: On ⍀ , −b ⬎ 0, Eq. 共18c兲 exists if and only if h ⱖ 0. Proof: In the forward direction, we again obtain Eq. 共19兲, which we split into two cases:
013132-5
Chaos 19, 013132 共2009兲
Invariant submanifold of Josephson array 2
TABLE II. Linearization of Eq. 共17兲 at Eq. 共18a兲 for ⍀ ⬎ 0.
b>Ω
1
2
τ
4Δ = 0
τ 0
Interval of b
Signs of ⌬ , +
共−⬁ , min兵−1 , −⍀其兲 共−1 , min兵− 21 , −⍀其兲 共− 21 , −⍀兲 共⍀ , + ⬁兲
⌬⬎0, ⌬⬍0, ⌬⬍0, ⌬⬎0,
Stability classification
+ ⬍ 0 + ⬍ 0 + ⬎ 0 + ⬎ 0
Stable node Saddle point Saddle point Unstable node
b< Ω -1
Similarly, the determinant and trace of the linearization in Eqs. 共18c兲 and 共18d兲 are
-2 -1
-0.5
0
0.5
1
⌬⫾ = ⫾
Δ FIG. 2. 䉭 and of the linearization of Eq. 共18a兲 plotted parametrically as a 1 function of b for ⍀ = 8 .
⍀冑h ⱖ 兩⍀2 − b2 + b兩 ⇔ ⍀ ⱖ − b,
共21a兲
⍀冑h ⱕ 兩⍀2 − b2 + b兩 ⇔ ⍀ ⱕ − b.
共21b兲
Since regions of ⍀ ⱖ −b, ⍀ ⱕ −b both exist on ⍀ , −b , h ⱖ 0, this direction does not yield any additional restrictions. In the reverse direction, we can drop the absolute value signs from Eq. 共21a兲 since the larger quantity remains positive when h ⱖ 0. Equation 共21b兲 also implies Eq. 共19兲 if ⍀2 − b2 + b ⱕ 0. Since ⍀ ⱕ −b implies ⍀2 − b2 + b ⱕ 0 for b ⬍ 0, Eq. 共18c兲 exists everywhere its discriminant is non-negative 共on the quadrant ⍀ , −b ⬎ 0兲. 䊐 Claim: On ⍀ ⬎ 0, Eq. 共18d兲 exists if and only if h ⱖ 0 and ⍀ ⱕ −b. Proof: Again by algebra,
冉
− ⍀ − 冑h 1−b
冊
2
ⱕ 1 ⇔ ⍀冑h ⱕ − ⍀2 + b2 − b.
共22兲
We can square both sides of Eq. 共22兲 to obtain ⍀2h ⱕ 共⍀2 − b2 + b兲2 ⇔ ⍀ ⱕ 兩b兩.
共23兲
In the reverse direction, removing the square from the first inequality of Eq. 共23兲 requires that ⍀2 − b2 + b ⱕ 0. However, b共1 − b兲 ⱖ 0 for b on 关0, 1兴 and ⍀2 − b2 + b ⱕ 0 implies h ⬍ 0 for b ⬎ 1, so this inequality is not satisfied for b ⬎ 0. Nevertheless, ⍀ ⱕ −b implies ⍀2 − b2 + b ⱕ 0 for b ⬍ 0, so Eq. 共18d兲 exists if and only if h ⱖ 0 and ⍀ ⱕ −b. 䊐 C. Stability of the fixed points
In Eqs. 共18a兲 and 共18b兲, the linearization of Eq. 共17兲 has the determinant and trace: ⌬ = b共b + 1兲共1 − ⍀2/b2兲,
⫾ = ⫾ 共2b + 1兲冑1 − ⍀2/b2 ,
⍀ 冑h − b h, 1−b 1−b
共25a兲
= 0,
共25b兲
where Eq. 共18c兲 has determinant ⌬+ and Eq. 共18d兲 has determinant ⌬−. We now consider how ⌬⫾ in Eq. 共25a兲 takes signs as a function of b and ⍀. The results are summarized in Table III. Claim: On ⍀ ⬎ 0, Eq. 共18c兲 is a center. Proof: Clearly, ⌬+ ⬎ 0 for b ⬍ 0. On b ⬎ 1, ⌬+ ⬎ 0 ⇔
兩b兩 ⍀ 冑h ⇔ ⍀ ⬎ b. h⬎ 兩1 − b兩 兩1 − b兩
共26兲
Likewise, for b on 共0, 1兲, ⌬+ ⬎ 0 ⇔
⍀ 冑h ⬎ 兩b兩 h ⇔ ⍀ ⬎ b. 兩1 − b兩 兩1 − b兩
共27兲
Since ⍀ ⱖ b for all 共b , ⍀兲 where Eq. 共18c兲 exists, ⌬+ ⬎ 0 and Eq. 共18c兲 is a center on b ⬎ 0, as well. 䊐 Claim: On ⍀ ⬎ 0, Eq. 共18d兲 is a center for b ⬎ −1 and a saddle for b ⬍ −1. Proof: We need only be concerned with the negative b axis, since Eq. 共18d兲 does not exist where ⍀ , b ⬎ 0. For b on 共⫺1, 0兲, ⌬− ⬎ 0 ⇔
兩b兩 ⍀ 冑h ⇔ ⍀ ⬍ − b, h⬎ 兩1 − b兩 兩1 − b兩
共28兲
while for b ⬍ −1, ⌬− ⬍ 0 ⇔
⍀ 冑h ⬎ 兩b兩 h ⇔ ⍀ ⬍ − b. 兩1 − b兩 兩1 − b兩
共29兲
Since ⍀ ⱕ −b for all 共b , ⍀兲 where Eq. 共18d兲 exists, ⌬− ⬎ 0 and Eq. 共18d兲 is a center for b on 共⫺1,0兲, while ⌬− ⬍ 0 and Eq. 共18兲 is a saddle for b ⬍ −1. 䊐 TABLE III. Linearization of Eq. 共17兲 at Eqs. 共18c兲 and 共18d兲 for ⍀ ⬎ 0.
共24兲
where Eq. 共18a兲 has trace + and Eq. 共18b兲 has trace −. By careful consideration of Eq. 共24兲 and Fig. 2, we find that ⌬ and + take signs according to the four cases in Table II. The case of ⌬ and − is analogous.
Fixed point Eq. 共18c兲 Eq. 共18d兲
Interval of b 共−冑⍀2 + 1 , ⍀兲 冑 共− ⍀2 + 1 , min兵−1 , −⍀其兲 共−1 , −⍀兲
Sign of ⌬⫾
Stability classification
⌬+ ⬎ 0 ⌬− ⬍ 0 ⌬− ⬎ 0
Center Saddle point Center
013132-6
3
3
2
2
Ω 1 0
(A) 3
Ω
saddle point
stable node
1
unstable node
2
1
b
0
1
(B) 3
stable node
2
1
b
0
1
2
1
2
(a)
(b)
(c)
(d)
(e)
(f)
3
2
saddle point
2
center
Ω
Ω
1 0
saddle point
unstable node
0
2
3
(C) 3
Chaos 19, 013132 共2009兲
S. A. Marvel and S. H. Strogatz
center
1
2
1
b
0
1
0
2
(D) 3
2
1
b
0
FIG. 3. Regions of existence in the upper-half b-⍀ plane for each of the four fixed points. 共a兲 corresponds to fixed point 共18a兲, 共b兲 to fixed point 共18b兲, 共c兲 to 共18c兲, and 共d兲 to 共18d兲. The existence and stability of Eq. 共18a兲 at a given point 共b , −⍀兲 of parameter space for ⍀ ⬎ 0 are given by the existence and stability of Eq. 共18a兲 at 共b , ⍀兲, and likewise for Eq. 共18b兲. By contrast, the existence and stability of Eq. 共18c兲 at 共b , −⍀兲 are equivalent to the existence and stability of Eq. 共18d兲 at 共b , ⍀兲, and vice versa.
D. Parameter space and phase portraits
Figure 3 summarizes our findings regarding the regions of the b-⍀ parameter plane where each fixed point exists, as well as the stability classifications of the fixed points on these regions. If we let ⍀ → −⍀, then 兩xⴱ兩 of Eq. 共18c兲 becomes 兩xⴱ兩 of Eq. 共18d兲 and vice versa. Hence, the existence and stability of Eq. 共18c兲 at a given point 共b , −⍀兲 of parameter space for ⍀ ⬎ 0 are given by the existence and stability of Eq. 共18d兲 at 共b , ⍀兲, and vice versa. Figure 4 combines the four separate panels of Fig. 3 into a single image. The various regions in Fig. 3 yield six qualitatively distinct regions of the b-⍀ plane in Fig. 4. We distinguish between regions 共b兲 and 共e兲 in Fig. 4, because the xⴱ at which the single center of these regions is located changes sign upon crossing b = −1. Figure 5 plots the phase portraits for the order parameter dynamics governed by Eq. 共16兲. The panels show the qualitatively different behavior that occurs in the six regions of Fig. 4. Figure 5共a兲 depicts what happens in the region where b ⬎ 0 and ⍀ ⬍ b. There, all trajectories are attracted to a stable fixed point on the unit circle, representing a synchronized rest state. Notice that an invariant vertical line seems to join the repelling fixed point with the attracting one. To
3 (e)
2
(b)
Ω 1
(d)
(f)
FIG. 5. Representative phase portraits for the six regions of the upper-half b-⍀ plane with qualitatively distinct phase plane behaviors. The trajectories are plotted in polar coordinates r and on the unit disk. Solid dots denote Lyapunov stable fixed points, while open dots denote unstable fixed points. The letter labels 共a兲–共f兲 match with the labels in Fig. 4.
prove that this vertical line truly is invariant, observe from Eq. 共17兲 that x˙ = 0 whenever x = ⍀ / b. Hence a solution that starts on this line stays there forever. Figure 5共b兲 shows the case where b ⬎ −1 and ⍀ ⬎ 兩b兩. The fixed points that previously existed on the unit circle have disappeared. They annihilated each other when ⍀ = 兩b兩, thereby creating the periodic orbit on the unit circle seen in Fig. 5共b兲. Physically, this orbit represents a synchronized oscillation with all the junctions moving in phase. However, this synchronous state is not attracting; it is neutrally stable. In fact, the entire unit disk is filled with neutrally stable periodic orbits, all of which surround a neutrally stable fixed point, the splay state mentioned earlier. The remaining panels show examples of additional cases when b ⬍ 0. We do not dwell on these, as they correspond to a negative resistance in either the load or the Josephson junctions and hence are physically unrealistic. The main features to observe are the saddle connection and the coexistence of two neutrally stable splay states in Fig. 5共c兲 and the saddle and center splay states in Fig. 5共f兲. IV. RLC-LOADED CIRCUIT
We turn now to a more complicated kind of Josephson junction array. Instead of the purely resistive load assumed earlier, we allow a load comprised of a resistor, inductor, and capacitor in series. This load is placed in parallel with N identical, overdamped Josephson junctions wired in series. The whole circuit is driven by a constant current source Ib, as shown in Fig. 6. Consider the infinite-N limit of this system. As shown by Strogatz and Mirollo,39 the time-dependent dynamics of the array can be written in dimensionless form as
(a) (c)
0 -3
-2
-1
0
1
2
b FIG. 4. The six qualitatively distinct regions of the b-⍀ parameter plane. The partition is symmetric about the b axis.
i ˙ 兲 − i e−i , v共,t兲 = ei + 共Ib − Q 2 2
共30兲
where the state variable Q is governed by a dimensionless version of Kirchhoff’s voltage law:
013132-7
Chaos 19, 013132 共2009兲
Invariant submanifold of Josephson array
Q through the load capacitor. The control parameters are the bias current Ib, load inductance L, 共rescaled兲 resistance S, and capacitance C. A complete analysis of Eq. 共35兲 is beyond the scope of this paper. From previous numerical experiments, we know that there would be many attractors and other complicated features to consider.39–42 Rather than try to enumerate and analyze all of these, our aim will be to merely list a few basic facts about the fixed points of the system. In particular, we show that an earlier result—an eigenvalue equation whose roots give the four nontrivial Floquet multipliers of the splay state—has a more straightforward derivation within the present framework.
C Ib L
R
FIG. 6. A series of Josephson junctions in parallel with an RLC load. Ib represents a dimensionless version of the constant bias current, C a dimensionless load capacitance, L a dimensionless load inductance, and R a dimensionless load resistance.
¨ + 共R + 1兲Q ˙ + C−1Q = I − LQ b
冕
2
共,t兲sin d .
共31兲
A. Fixed points
We first convert r and of Eq. 共35兲 to Cartesian coordinates x and y as in the purely resistive case. The result is x˙ = − xy + 共Ib − L−1 P兲y,
0
In Eqs. 共30兲 and 共31兲, Q共t兲 is the dimensionless charge on the capacitor, while the other new quantities are as indicated in the caption of Fig. 6. Equation 共30兲 has the special trigonometric form v共 , t兲 = fei + g +¯f e−i required by the Ott–Antonsen method and therefore is reducible by the method of Sec. II. By reading off the f and g implied by Eq. 共30兲 and substituting them into the Riccati equation 共12兲 for the order parameter z, we obtain z˙ =
1 − z2 ˙ 兲z, + i共Ib − Q 2
共32兲
which has real and imaginary parts r˙ =
1 − r2 cos , 2 共33兲
r + r−1 ˙. sin + Ib − Q ˙ = 2
˙ and computing the integral, Eq. 共31兲 can By defining P = Q also be split into a two-dimensional system: LP˙ = − 共R + 1兲P − C−1Q + Ib − r sin ,
˙ = P. Q
共34兲
Now let P⬘ = LP, Q⬘ = LQ, S = −共R + 1兲 / L, and ⬘ = / 2 − . If we substitute these definitions into Eqs. 共33兲 and 共34兲 and drop the primes, we obtain 1 − r2 sin , r˙ = 2
y˙ = 21 共x2 − y 2 + 1兲 − 共Ib − L−1 P兲x, P˙ = Ib + SP − 共LC兲−1Q − x,
˙ = P. Q
There are four fixed points of Eq. 共36兲: x ⴱ = I b,
y ⴱ = −,
x ⴱ = I b,
y ⴱ = − −,
Pⴱ = 0,
共LC兲−1Qⴱ = 0,
Pⴱ = 0,
共37a兲
共LC兲−1Qⴱ = 0,
共37b兲
x ⴱ = I b + +,
y ⴱ, Pⴱ = 0,
Qⴱ = +LC,
共37c兲
x ⴱ = I b − +,
y ⴱ, Pⴱ = 0,
Qⴱ = − +LC,
共37d兲
where ⫾ = 冑⫾I2b ⫿ 1. Once again, Eqs. 共37a兲 and 共37b兲 represent synchronous fixed points in which the Josephson junctions are in phase and not oscillating, while Eqs. 共37c兲 and 共37d兲 represent splay-state periodic orbits. The derivation of Eq. 共37兲 makes no assumptions about the parameters except that they are real scalars, so we have no need to disregard certain parameter values as we did for b in Eq. 共18兲. Note that Eqs. 共37a兲 and 共37b兲 exist if and only if 兩Ib兩 ⱕ 1. Similarly, the requirement 兩xⴱ兩 ⱕ 1 implies that Eq. 共37c兲 only exists for Ib 苸 共−⬁ , −1兴 and Eq. 共37d兲 only exists for Ib 苸 关1 , ⬁兲. B. Stability of the splay state
Since the splay states are the primary concern in much of the existing literature, we compute the linearization of Eq. 共36兲 at Eq. 共37c兲,
r + r−1 cos − Ib + L−1 P, ˙ = 2 共35兲
P˙ = Ib + SP − 共LC兲−1Q − r cos ,
共36兲
˙ = P. Q
This is the low-dimensional system that governs the flow on the Poisson submanifold. Observe that the state variables of Eq. 共35兲 are the order parameter amplitude r and phase , along with the dimensionless rescaled current P and charge
J=
冢
⫿ +
0
⫾ +
0
L 共Ib ⫾ +兲
0
−1
0
− 共LC兲−1
0
0
s 1
0
−1
0
0
冣
.
The Jacobian 共38兲 has the characteristic polynomial
共38兲
013132-8
Chaos 19, 013132 共2009兲
S. A. Marvel and S. H. Strogatz
L4 + 共R + 1兲3 + 共C−1 + L2+兲2 + 共Ib+ + R2+兲 + 2+C−1 = 0,
共39兲
where we have substituted back in the definition of S and multiplied through by L. The characteristic polynomial 共39兲 was first derived 15 years ago 关see Eq. 共13兲 of Ref. 39兴. At the time, its derivation gave the first explanation for why there are just four nonneutral Floquet multipliers for the splay state of the RLC system.41 It also allowed analytical predictions of those multipliers.39 However, the earlier derivation39 involved Fourier expansions of infinitesimal perturbations about the splay states, a procedure more complicated than the one given here. In retrospect, we can see now that perturbations tangent to the Poisson submanifold are precisely those responsible for the non-neutral directions; perturbations transverse to this manifold are the neutral ones. V. DISCUSSION
It is important to understand both the successes and the limitations of our analysis. The systems we have studied comprise a special class of Josephson junction arrays, namely, those in which all the junctions are identical and heavily overdamped, meaning that we can ignore their internal capacitance. The junctions are connected in series, driven by a constant bias current and coupled through a load in parallel. For this class of arrays, it has been known since 1994 that the governing equations specify a family of lowdimensional invariant manifolds.44 For resistively loaded arrays, these invariant manifolds are three dimensional, while for arrays with an RLC load, they are five dimensional. These results hold for any number of junctions and extend to infinite N. In this paper, we have shown that in the infinite-N limit, the equations can be reduced even more dramatically. In other words, a strictly smaller invariant submanifold exists as a degenerate case of the invariant manifolds known previously. We have called it the Poisson submanifold; it consists of all phase distributions taking the form of a Poisson kernel. Using the Ott–Antonsen ansatz, we explicitly calculated the flow on this manifold and found it to take the form of a Riccati equation. The resulting low-dimensional dynamical systems shed new light on the behavior of Josephson junction series arrays. For example, earlier local arguments38–42 showed that the synchronous periodic state and splay states for arrays with a resistive load exhibit neutral stability to linear order over a wide range of parameters. But do they exhibit neutral stability in nonlinear reality? This has been a long-standing open question. We can now observe that the answer is yes. Figure 5共b兲 shows that the splay state is surrounded by neutrally stable periodic orbits that fill the Poisson submanifold, continuing all the way out to the synchronized orbit on the boundary of the disk. The other phase portraits in Fig. 5 similarly provide new information regarding the global structure. For example, they elucidate how splay states bifurcate with the synchronous
periodic states and reveal global features such as the vertical heteroclinic orbit joining the in-phase rest states in several panels of Fig. 5. Previous global results regarding Josephson junction arrays were confined to the averaged versions of such systems,43,44 which were derived via perturbation methods in the limit of weak coupling or high bias current.49 Our results, by contrast, hold for all values of the circuit parameters. The trade-off is that they require infinite N. However, the most serious drawback of our analysis is that it focuses on a thin slice of phase space which is unrepresentative of the dynamics of the full system. For instance, the Poisson submanifold for the resistively loaded array is a degenerate, two-dimensional leaf in the foliation of phase space by three-dimensional invariant manifolds. Could attractors or other interesting dynamical states exist off the Poisson submanifold? Numerical simulations say yes: chaos occurs in the original equations for resistively loaded arrays.42,44 The remaining challenge is then to show that the reduced equations faithfully capture this chaos on the larger invariant manifolds. We will address this issue in a subsequent paper48 in which we use Möbius transformations to simplify the circuit equations for Josephson junction arrays. Essentially the same idea has been developed independently by Pikovsky and Rosenblum,47 who obtained new results for the Kuramoto model as well as more complex hierarchies of oscillators. ACKNOWLEDGMENTS
This research was supported in part by the National Science Foundation Grant No. CISE-0835706. A. T. Winfree, J. Theor. Biol. 16, 15 共1967兲. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence 共Springer, Berlin, 1984兲. 3 J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, J. Theor. Biol. 77, 137 共2005兲. 4 S. H. Strogatz, in Frontiers in Mathematical Biology, Lecture Notes in Biomathematics Vol. 100, edited by S. A. Levin 共Springer, New York, 1994兲, pp. 122–138. 5 N. Wiener, Nonlinear Problems in Random Theory 共MIT Press, Cambridge, MA, 1958兲. 6 N. Wiener, Cybernetics, 2nd ed. 共MIT Press, Cambridge, MA, 1961兲. 7 T. D. Frank, A. Daffertshofera, C. E. Pepera, P. J. Beeka, and H. Hakenb, Physica D 144, 62 共2000兲. 8 C. S. Peskin, Mathematical Aspects of Heart Physiology 共Courant Institute of Mathematical Sciences, New York, 1975兲. 9 D. C. Michaels, E. P. Matyas, and J. Jalife, Circ. Res. 58, 706 共1986兲. 10 D. C. Michaels, E. P. Matyas, and J. Jalife, Circ. Res. 61, 704 共1987兲. 11 J. Buck and E. Buck, Science 159, 1319 共1968兲. 12 F. E. Hanson, Fed. Proc. 37, 2158 共1978兲. 13 J. Buck, Q. Rev. Biol. 63, 265 共1988兲. 14 G. B. Ermentrout, J. Math. Biol. 29, 571 共1991兲. 15 D. Kim, Biosystems 76, 7 共2004兲. 16 M. K. McClintock, Nature 共London兲 229, 244 共1971兲. 17 M. J. Russell, G. M. Switz, and K. Thompson, Pharmacol., Biochem. Behav. 13, 737 共1980兲. 18 H. C. Wilson, Psychoneuroendocrinology 17, 565 共1992兲. 19 A. K. Ghosh, B. Chance, and E. K. Pye, Arch. Biochem. Biophys. 145, 319 共1971兲. 20 D. Njus, V. D. Gooch, and J. W. Hastings, Cell Biophys. 3, 223 共1981兲. 21 J. W. Hastings, H. Broda, and C. H. Johnson, in Temporal Order, edited by L. Rensing and N. I. Jaeger 共Springer, Berlin, 1985兲, pp. 213–221. 22 S. Dano, F. Hynne, S. D. Monte, F. d’Ovidio, P. G. Sorensen, and H. Westerhoff, Faraday Discuss. 120, 261 共2002兲. 1 2
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D. B. Murray, S. Roller, H. Kuriyama, and D. Lloyd, J. Bacteriol. 183, 7253 共2001兲. 24 J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, Proc. Natl. Acad. Sci. U.S.A. 101, 10955 共2004兲. 25 H. G. Winful and S. S. Wang, Appl. Phys. Lett. 53, 1894 共1988兲. 26 K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, Phys. Rev. Lett. 65, 1749 共1990兲. 27 R.-D. Li and T. Erneux, Phys. Rev. A 46, 4252 共1992兲. 28 L. Fabiny, P. Colet, R. Roy, and D. Lenstra, Phys. Rev. A 47, 4287 共1993兲. 29 S. Y. Kourtchatov, V. V. Likhanskii, A. P. Napartovich, F. T. Arecchi, and A. Lapucci, Phys. Rev. A 52, 4089 共1995兲. 30 G. Kozyreff, A. G. Vladimirov, and P. Mandel, Phys. Rev. Lett. 85, 3809 共2000兲. 31 T. Heil, I. Fischer, W. Elsasser, J. Mulet, and C. R. Mirasso, Phys. Rev. Lett. 86, 795 共2001兲. 32 S. H. Strogatz, C. M. Marcus, and R. M. Westervelt, Phys. Rev. Lett. 61, 2380 共1988兲. 33 S. H. Strogatz and R. M. Westervelt, Phys. Rev. B 40, 10501 共1989兲.
A. A. Middleton, Phys. Rev. Lett. 68, 670 共1992兲. W. Wang, I. Z. Kiss, and J. L. Hudson, Chaos 10, 248 共2000兲. 36 I. Z. Kiss, Y. Zhai, and J. L. Hudson, Phys. Rev. Lett. 88, 238301 共2002兲. 37 I. Z. Kiss, Y. Zhai, and J. L. Hudson, Science 296, 1676 共2002兲. 38 K. Y. Tsang, R. E. Mirollo, S. H. Strogatz, and K. Wiesenfeld, Physica D 48, 102 共1991兲. 39 S. H. Strogatz and R. E. Mirollo, Phys. Rev. E 47, 220 共1993兲. 40 K. Y. Tsang and I. B. Schwartz, Phys. Rev. Lett. 68, 2265 共1992兲. 41 S. Nichols and K. Wiesenfeld, Phys. Rev. A 45, 8430 共1992兲. 42 D. Golomb, D. Hansel, B. Shraiman, and H. Somopolinsky, Phys. Rev. A 45, 3516 共1992兲. 43 S. Watanabe and S. H. Strogatz, Phys. Rev. Lett. 70, 2391 共1993兲. 44 S. Watanabe and S. H. Strogatz, Physica D 74, 197 共1994兲. 45 C. J. Goebel, Physica D 80, 18 共1995兲. 46 E. Ott and T. M. Antonsen, Chaos 18, 037113 共2008兲. 47 A. Pikovsky and M. Rosenblum, Phys. Rev. Lett. 101, 264103 共2008兲. 48 R. Mirollo, S. A. Marvel, and S. H. Strogatz 共unpublished兲. 49 J. W. Swift, S. H. Strogatz, and K. Wiesenfeld, Physica D 55, 239 共1992兲. 34 35
CHAOS 19, 013133 共2009兲
„ , p , q…-vulnerabilities: A unified approach to network robustness Regino Criado, Javier Pello, Miguel Romance, and María Vela-Pérez Departamento de Matemática Aplicada, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
共Received 16 October 2008; accepted 3 February 2009; published online 17 March 2009兲 We define a new general framework, the family of 共 , p , q兲-vulnerabilities, as a tool that produces many new vulnerability functions that measure the capacity of a network to maintain its functional performance under random damages or malicious attacks. This new framework comprises most of the vulnerability definitions appearing in the literature and allows to calculate some relationships between the different 共 , p , q兲-vulnerabilities in terms of their function or their parameters p , q that improve several known results for the vulnerability functions. Some graphics of simulations are provided in order to show the sharpness of these relationships. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3087314兴 Many relevant properties of the real world, from Internet to protein folding, going through fields as disparate as sociology or technology, may be described in terms of network properties.1–12 The study of the structural properties of the underlying network can be very important in the understanding of the functions of a complex system as well as in quantifying the strategic importance of a node (or a set of nodes) in order to preserve the best functioning of the network as a whole. The concept of vulnerability in a network quantifies the capacity of a network to maintain its functional performance under random damages, malicious attacks, or dysfunctions of any type. Several different approaches have been introduced to measure the vulnerability of a complex network (see, for instance, Refs. 9 and 13–20). In this paper we present a more general framework that comprises most of the vulnerability definitions appearing in literature and allows calculating some relationships between them.
I. INTRODUCTION
Considering that different types of networks and different applications suggest different approaches to the concept of a network’s vulnerability, it is obvious that there exist several ways of measuring the drop of performance of a network under malicious attacks or random damages depending on the aspect we focus on. Some of these approaches are related to the following contexts: • Connectivity loss 共see, for example, Refs. 21 and 22兲. This approach measures vulnerability in terms of the loss of connectivity when we remove some nodes and links. Under this point of view, the more homogeneous a network is 共i.e., with all the nodes and links playing a similar role兲, the more robust that network is. This approach is particularly interesting for military purposes and military networks or for civilian networks facing possibly terroristic activity. An alternative approach under this point of view is given in Ref. 4. • Variation of the network performance 共see, for example, Refs. 8 and 18–20兲. This approach measures vulnerability 1054-1500/2009/19共1兲/013133/11/$25.00
of a network in terms of the fall of its efficiency when damage occurs. • Betweenness measures.14,22–25 This approach focuses on the strategic importance of specific links and nodes in order to preserve the functioning and performance of the network as a whole. • Centrality measures26,27 or percolation theory.3,13,28 Each of these approaches has its advantages and disadvantages, and the most suitable approach for a specific case may depend on the problem under investigation and the size of the network. We show that almost all of these approaches 关with the exception of those which include probabilistic notions, verbi gratia 共v.gr.兲, percolation theory兴 can be submerged in a general framework which gives us a new perspective and formalism to the concept of a network’s vulnerability. We do it by using two auxiliary numerical parameters, p and q, and a specific auxiliary function and establishing the concept of 共 , p , q兲-vulnerability of a network G. Furthermore, we establish some bounds between the different measures for the vulnerability. In the following, we will consider a complex network G = 共X , E兲 of n nodes and m links, where X is the set of nodes and E is the set of edges. If i , j 苸 X are nodes of G, dij will denote the geodesic distance between i and j in the network G and nij is the number of different geodesic paths that join i and j. If ᐉ 苸 E is a link and v 苸 X is a node, then nij共v兲 and nij共ᐉ兲 will denote the numbers of geodesic paths that join i and j and go through v or ᐉ, respectively. II. „ , p , q…-VULNERABILITIES: DEFINITIONS AND MAIN EXAMPLES
Many of the vulnerability functions introduced in the literature are actually some kind of aggregation of local parameters 共such as the variation of performance, connectivity properties, or betweenness兲; therefore, we can introduce a general framework that gives a new family of vulnerability functions that extends those known functions. If G is a complex network, Y is a subset of ordered pairs of nodes or links, Z is a subset of nodes or links, and : Y ⫻ Z → 关0 , + ⬁兲 is a
19, 013133-1
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function, we can introduce the (,p,q)-vulnerability of G for any p , q 苸 关0 , ⬁兴 as the value V,p,q共G兲 =
冋 冉
1 1 兺 兺 共i, j,z兲p 兩Z兩 z苸Z 兩Y兩 共i,j兲苸Y
冊册
i⫽j苸X
,
A. Fall of efficiency-type vulnerabilities
One of the main approaches to a network’s vulnerability relates this concept to the fall of the network’s efficiency when a node fails.9 Following this idea, the vulnerability of a network G due to failure of a single node can be defined by several aggregation techniques 共see Ref. 9兲 mainly as Vmax共G兲 = max兵E共G兲 − E共G \ 兵v其兲; v 苸 X其
共1兲
¯V共G兲 = 1 兺 共E共G兲 − E共G \ 兵v其兲兲, n v苸X
共2兲
or
where E共G兲 denotes the efficiency of G defined8 by 1 1 兺 n共n − 1兲 i⫽j苸X dij
共3兲
and G \ 兵v其 is the network G without node v and its incident links. These two definitions of vulnerabilities are particular cases of 共 , p , q兲-vulnerabilities simply by considering
Z = X and taking 1 : Y → 关0 , 1兴 defined for every i , j , v 苸 X 共i ⫽ j兲 by
冦
1 1 1 1 − if i ⫽ v ⫽ j n共n − 1兲 dij 共n − 1兲共n − 2兲 d⬘ij 1 1 n共n − 1兲 dij
otherwise,
冧
where d⬘ij is the geodesic distance in G \ 兵v其. By using these settings it is easy to check that ¯V共G兲 is the 共1 , 1 , 1兲-vulnerability of G and Vmax共G兲 is the 共1 , 1 , ⬁兲-vulnerability of G. Then V1,1,q共G兲 interpolates between ¯V共G兲 and Vmax共G兲 in the range q 苸 关1 , ⬁兴. Another parameter that measures the vulnerability as the fall of efficiency 共in a size-independent way兲 was introduced in Ref. 23. In this paper the author proposed two sizeindependent vulnerability functions given by Wmax共G兲 = max兵W共G, v兲; v 苸 X其
1 1 − dij d⬘ij
冊
共5兲
and d⬘ij is the distance between i and j in the network G \ 兵v其. ¯ 共G兲 can be exIt is easy to check that both Wmax共G兲 and W pressed as 共 , p , q兲-vulnerabilities taking the same sets Y and Z as before and 2 : Y → 关0 , 1兴 defined for every i , j , v 苸 X 共i ⫽ j兲 by
冦
1 1 − if i ⫽ v ⫽ j 2共i, j, v兲 = dij d⬘ij otherwise. 0
共4兲
¯ 共G兲 = 共1 / n兲兺 W共G , v兲, inspired by Eqs. 共1兲 and 共2兲, and W v苸X respectively, where
冧
¯ 共G兲 = V By using these settings we get that W 2,1,1共G兲, Wmax共G兲 = V2,1,⬁共G兲 and V2,1,q共G兲 interpolates between both parameters when q 苸 关1 , ⬁兴. Note that these definitions can be stated for more general networks 共such as weighted or directed networks兲 and for other distances in the graph involving distances based on the functionality of the network instead of its topological properties. The approach to vulnerability based on the fall of efficiency also considered the case of failures in multiple nodes or links 共see, for example, Ref. 23兲. These measures can be stated in terms of 共 , p , q兲-vulnerabilities simply by replacing the set Z by a set of subsets of nodes or links. From this point of view, all the 共 , p , q兲-vulnerabilities introduced in this section can be extended to measures involving multiple node or link failures. It can be proved that there is a strong correlation between the 共1 , p , q兲-vulnerabilities and the 共2 , p , q兲 vulnerabilities. This is because if i , j , v 苸 X are three different nodes, then we can prove that
Y = 兵共i, j兲;i ⫽ j 苸 X其,
1共i, j, v兲 =
冉
i,j⫽v
q/p 1/q
that is, V,p,q共G兲 is an aggregation of the function 共i , j , z兲 through all the possible values of 共i , j兲 苸 Y and z 苸 Z. Let us notice that most of the different definitions for the vulnerability of a complex network are particular cases for the 共 , p , q兲-vulnerability, as we can see in the following examples.
E共G兲 =
兺
W共G, v兲 =
1共i, j, v兲 =
冉
冊
1 2 1 2共i, j, v兲 − . 共n − 1兲共n − 2兲 n dij
共6兲
By using this expression, we can prove that for any complex network G, V1,1,1共G兲 =
1 V ,1,1共G兲. 共n − 1兲共n − 2兲 2
共7兲
This relationship was proved in Ref. 23 but without using the language of 共 , p , q兲-vulnerabilities. Note that by combining expression 共6兲 with the relationship between the p-mean functions 共i.e., Hölder’s inequalities, see, for example, Ref. 29兲, we can produce sharp estimates that prove the correlation between the 共1 , p , q兲-vulnerabilities and the 共2 , p , q兲vulnerabilities. The flexibility of the 共 , p , q兲 vulnerabilities allows us to introduce other useful vulnerability measures based on the variation of the geodesic distance structure when failure occurs. If we take Y = 兵共i , j兲 ; i ⫽ j 苸 X其, Z = X, as before, and we consider 3 : Y → 关0 , + ⬁兴 defined for every i , j , v 苸 X 共i ⫽ j兲 by
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3共i, j, v兲 =
再
d⬘ij − dij if i ⫽ v ⫽ j otherwise,
0
Chaos 19, 013133 共2009兲
冎
we have introduced a size-independent measure of the vulnerability of a complex network based on the variation of its diameter or its mean geodesic distance, while if we take 4 : Y → 关0 , + ⬁兴 given by
冦
d⬘ij − dij if i ⫽ v ⫽ j dij 4共i, j, v兲 = otherwise, 0
冧
we have introduced another size-independent normalized measure of the robustness based in the relative variation of its diameter or mean geodesic distance.
It was proven in Ref. 31 that there are several upper and lower bounds for VX,q共G兲 in terms of VE,q共G兲 and therefore it is natural to think that there should be some kind of bounds for V6,p,q共G兲 in terms of V5,p,q共G兲, extending the results of Ref. 31. In the last part of this section we will improve these estimates and we will give sharp results for these 共 , p , q兲vulnerabilities. Let us start with the relationship between V5,1,q共G兲 and V6,1,q共G兲, which actually are the link-based and node-based multi-scale vulnerabilities. If we rewrite the results proven in Ref. 31, we have that
V6,1,q共G兲 ⱕ 21/q−1
冉冊
V6,1,q共G兲 ⱖ 21/q−1
冉冊
m n
1/q
1 共grmax兲1−1/qV5,1,q共G兲 + , 共8兲 n
B. Multi-scale node and link-based vulnerabilities
Another approach to a network’s vulnerability is based on the concentration of the geodesic structure throughout the network. In Ref. 14, a link-based multi-scale vulnerability of a complex network G was introduced as
冉
1 VE,q共G兲 = 兺 bq m ᐉ苸E ᐉ
冊
1/q
for any q 苸 关1 , + ⬁兲, where bᐉ is the betweenness of the link ᐉ 苸 E 共see, for example, Refs. 10 and 30兲 given by bᐉ =
1 n 共ᐉ兲 兺 ij . n共n − 1兲 i,j苸X nij i⫽j
Another related concept of vulnerability was introduced in Ref. 31, where node-based multi-scale vulnerability of a complex network given for any q 苸 关1 , + ⬁兲 as VX,q共G兲 = =
冉
1 兺 bq n v苸X v
冊
冉 冋 1 兺 n v苸X
1/q
1 nij共v兲 兺 n共n − 1兲 i,j苸X nij i⫽j
册冊 q
1/q
was considered. These two measures of vulnerability based on the concentration of the geodesic structure throughout the network are particular cases of 共 , p , q兲 vulnerabilities. On the one hand, if we take Y = 兵共i , j兲 ; i ⫽ j 苸 X其, Z = E and we consider 5 : Y → 关0 , 1兴 defined for every i , j 苸 X 共i ⫽ j兲 and ᐉ 苸 E by
5共i, j,ᐉ兲 =
nij共ᐉ兲 , nij
then VE,q共G兲 = V5,1,q共G兲. On the other hand if we consider the same set Y, Z = X and we take 6 : Y → 关0 , 1兴 given for every i , j , v 苸 X 共i ⫽ j兲 by
6共i, j, v兲 =
nij共v兲 , nij
then VX,q共G兲 = V6,1,q共G兲 for every q 苸 关1 , + ⬁兲.
m n
1/q
V5,1,q共G兲,
共9兲
where grmax is the maximal degree of G. The first inequality is sharp for all q simply by taking G = Kn 共the complete graph兲. In the rest of this subsection we will improve the second estimate by using techniques coming from the geometry of finite-dimensional convex geometry. First of all we should need a lemma that relates the ᐉq norm of a vector with the ᐉq norm of one of its translations. Lemma 1: Let x = 共x1 , . . . , xn兲 苸 Rn such that xi ⱖ 1 for all 1 ⱕ i ⱕ n . If we take ¯x = 共x1 + 1 , . . . , xn + 1兲 苸 Rn , then for every 1 ⱕ q ⬍ ⬁
¯ 储qq ⱖ 共储x储q + 1兲q + 共n − 1兲, 储x where 储x储q = 共兺ni=1兩xi兩q兲1/q. Proof: We prove it by induction on the dimension n of the space. For n = 1, the result is trivial since the inequality is 共x + 1兲q ⱖ 共x + 1兲q. The case n = 2 is the hardest part of the proof since we have to prove that if x1 , x2 ⱖ 1 and 1 ⱕ q ⬍ ⬁, then
共x1 + 1兲q + 共x2 + 1兲q ⱖ 共共xq1 + xq2兲1/q + 1兲q + 1.
共10兲
If we denote = 储v储q = 共xq1 + xq2兲1/q and take ␥ : 关0 , 1兴 哫 R2 given by ␥共t兲 = 共共k − t兲1/q , t1/q兲, where k = q and t 苸 关0 , k / 2兴, then Eq. 共10兲 is equivalent to showing that
储␥共t兲 + 共1,1兲储q ⱖ 储共,0兲 + 共1,1兲储q . It is easy to check that
储␥共0兲 + 共1,1兲储q = 储共,0兲 + 共1,1兲储q . Hence, if we take 共t兲 = 储␥共t兲 + 共1 , 1兲储qq = 共共k − t兲1/q + 1兲q + 共t1/q + 1兲q, then we have that
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1 1 ⬘共t兲 = q共共k − t兲1/q + 1兲q−1 共k − t兲1/q−1共− 1兲 + q共t1/q + 1兲q−1 t1/q−1 = 共t1/q + 1兲q−1t1/q−1 − 共k − t兲1/q−1共共k − t兲1/q + 1兲q−1 q q = 共t1/q + 1兲q−1共t1/q兲1−q − 共共k − t兲1/q兲1−q共共k − t兲1/q + 1兲q−1 =
冉
= 1+
冊 冉 q−1
1 t1/q
− 1+
1 共k − t兲1/q
冊
2t ⱕ k ⇔ t ⱕ k − t ⇔ t
ⱕ 共k − t兲
t1/q + 1 t1/q
冊 冉 q−1
−
nij共v兲 =
1/q
Hence 共1 + t−1/q兲q−1 ⱖ 共1 + 共k − t兲−1/q兲q−1 ⇒ ⬘共t兲 ⱖ 0. Therefore, 共t兲 = 储␥共t兲 + 共1 , 1兲储qq is a nondecreasing function, so we have that
1 2
冉兺
共x1 + 1兲q + 共x2 + 1兲q ⱖ 共共xq1 + xq2兲1/q + 1兲q + 1,
¯ 储qq = 储x ¯ 0储qq + 储xn+1 + 1储qq ⱖ 共储x0储q + 1兲q + 共n − 1兲 + 共xn+1 + 1兲q , 储x
共储x0储q + 1兲q + 共n − 1兲 + 共xn+1 + 1兲q
which proves the result. 䊐 The second ingredient comes from mathematical analysis and it is a well-known consequence of Hölder’s inequality 共see, for example, Ref. 29兲. Lemma 2: (Ref. 29) If a = 共a1 , . . . , ak兲 苸 Rk is a vector and 1 ⱕ p ⱕ q ⬍ + ⬁ , then 1/p−1/q
冉兺 冊 冉兺 冊 兩ai兩
i=1
1/q
V5,1,q共G兲 +
ᐉ苹v
1 n1+1/q
冊
q
+
兺 共i,j兲苸Y
nij共ᐉ兲 = 兺 5共i, j,ᐉ兲, nij 共i,j兲苸Y
¯y v =
兺 共i,j兲苸Y
nij共v兲 = 兺 6共i, j, v兲, nij 共i,j兲苸Y
共n − 1兲nm1/qV5,1,q共G兲 =
冉兺 冊
共n − 1兲n1+1/qV6,1,q共G兲 =
冉兺 冊
储a储q ,
k
ⱕ
冉兺
冊
nij共ᐉ兲 ,
n−1 nq+1
册
1/q
¯ ᐉ兲ᐉ苸E 苸 Rm, ¯y = 共y ¯ v兲v苸X 苸 Rn. Hence, we have that and ¯x = 共x
= 共储x储q + 1兲q + n,
兩ai兩
2
¯xᐉ =
q ⱖ 共共储x0储qq + xn+1 兲1/q + 1兲q + 1 + 共n − 1兲
i=1
␦ iv + ␦ j v
Proof: We will employ the following notation. For every ᐉ 苸 E and v 苸 X
where we have used the induction hypothesis. Applying the bidimensional case, inequality 共10兲 shows that
1/q
冊
nij共ᐉ兲 +
ⱕ V6,1,q共G兲.
i.e., the formula for the case n = 2. Finally, if x 苸 Rn+1, we can set this vector as x = 共x0 , xn+1兲 where x0 = 共x1 , . . . , xn兲 and ¯x0 = 共x1 + 1 , . . . , xn + 1兲. Hence, we have that
q
q−1
where ␦ab = 1 if a = b and ␦ab = 0 otherwise, and ᐉ 苹 v means that the link ᐉ is incident to v. By combining these lemmas we can prove the following result. Theorem 1: Let G be a network with n nodes and m links. If 1 ⱕ q ⬍ ⬁ , then 1 2m 2 n
which is equivalent to
k
ᐉ苹v
冋冉 冉 冊
储␥共t兲 + 共1,1兲储q ⱖ 储共,0兲 + 共1,1兲储q ,
i.e.,
冊
= 共1 + t−1/q兲q−1 − 共1 + 共k − t兲−1/q兲q−1 .
⇔ t−1/q ⱖ 共k − t兲−1/q ⇔ 1 + t−1/q ⱖ 1 + 共k − t兲−1/q .
储a储q ⱕ 储a储 p ⱕ k
共k − t兲1/q + 1 共k − t兲1/q
q−1
Since t ⱕ k / 2 we get that 1/q
冉
p
1/p
冉兺 冊 k
ⱕk
1/p−1/q
兩ai兩
q
ᐉ苸E
v苸X
¯xqᐉ
¯y qv
1/q
1/q
¯ 储q , = 储x
共11兲
¯ 储q , = 储y
共12兲
since all ¯xᐉ and all ¯y v are non-negatives. By Lemma 3, for every v 苸 X
1/q
.
i=1
Finally, the last tool needed to show the relationship between V5,1,q共G兲 and V6,1,q共G兲 is the following lemma that shows the strong relationship between the number of geodesics that contain a node v and the number of geodesics that contain some links incident to v 共see Ref. 31兲. Lemma 3: (Ref. 31) Let G be a network with n nodes and m links. If we fix v 苸 X , then for every i , j 苸 X
¯y v = =
␦ iv + ␦ j v 1 nij共ᐉ兲 nij共ᐉ兲 + 兺 兺 兺 兺 2 共i,j兲苸Y ᐉ苹v nij 2 共i,j兲苸Y ᐉ苹v nij 1 兺 ¯xᐉ + ᐉ苹 兺 2 ᐉ苹v v
冉兺
共i,j兲苸Y
␦iv + ␦ jv nij共ᐉ兲 2
nij
冊
.
Let us notice that by the symmetry of the arguments in the last sum, for every v 苸 X
共 , p , q兲 vulnerabilities
013133-5
兺 共i,j兲苸Y
␦iv + ␦ jv nij共ᐉ兲 2
nij
Chaos 19, 013133 共2009兲
兺 ␦ iv 共i,j兲苸Y
=
nij共ᐉ兲 = nij
nv j共ᐉ兲 nv j
兺
j⫽v
whose matrix representation is given by the incidence matrix of G. ¯ ᐉ兲ᐉ苹v can be seen as a vector of gr共v兲 coordiSince 共x ¯ ᐉ兲ᐉ苹v储1 ⱖ 储共x ¯ ᐉ兲ᐉ苹v储q. Therenates, Lemma 2 ensures that 储共x fore, changing the order of summation,
j苸X
兺 5共v, j,ᐉ兲,
=
j⫽v
j苸X
and combining this result with the last expression of ¯y v we find that 1 ¯y v = 兺 ¯xᐉ + 兺 2 ᐉ苹v ᐉ苹v
¯储q = 储z
1 2
=
1 2
nv j共ᐉ兲 nv j
兺
j⫽v j苸X
1 = 兺 ¯xᐉ + 2 ᐉ苹v
nv j共ᐉ兲 1 = 2 nv j
兺兺
j⫽v ᐉ苹v
¯xᐉ + 共n − 1兲,
冐
冐 冐
冐
q 1 ¯z + 共1, . . . ,1兲 , = n−1 q q
冐
共13兲
1 2
冉兺 冉兺 冊 冊 v苸X
ᐉ苹v
¯xᐉ
q 1/q
=
1 2
冉兺
v苸X
¯ ᐉ兲ᐉ苹v储q1 储共x
冊
1/q
=
1 n共n − 1兲
冊
冉
1 n共n − 1兲
冊
冉
冊
q
n
m n
q
ⱖ
q
1 q 1 ¯ 储q ⱖ 储y n n共n − 1兲
q
1 1/q−1 1/q 关共2 m n共n − 1兲V5,1,q共G兲 + 共n − 1兲兲q + 共n − 1兲q+1兴 n
冉 冉冊
1/q
V5,1,q共G兲 +
1 n
1+1/q
¯xqᐉ 兺 1 兺 ᐉ苸E v苸X v苸ᐉ
冊
冊
1/q
1/q
共15兲
1 ¯z + 1 n−1
q
+ 共n − 1兲,
where we have used Eq. 共15兲 in the last inequality. Finally, by combining the last inequality with Eqs. 共11兲 and 共12兲, we get that if q ⱖ 1
1 1/q−1 ¯ 储q + 共n − 1兲兲q + 共n − 1兲1+q兴 关共2 储x n
1/q−1 1/q m nV5,1,q共G兲 + 1兲q + 共n − 1兲兴 = q+1 关共2
= 21/q−1
v苸X
¯ ᐉ兲ᐉ苹v储qq 储共x
¯ 储q , = 21/q−1储x
冐 冉冐 冐 冊
q
1
=
1/q
冉
冉兺
¯ 储q + 共n − 1兲兲q + 共n − 1兲1+q , ⱖ 共21/q−1储x
Note that the vector ¯z 苸 R can be seen as the bisector plus the projection of ¯y if we consider the projection ␥ : Rm → Rn
冉
¯xqᐉ
1 2
1 2
¯ 储qq ⱖ 共储z ¯储q + 共n − 1兲兲q + 共n − 1兲1+q 储y
n
V6,1,q共G兲q =
1 ¯y n−1
共14兲
.
冉兺 冊
=
ⱖ
and hence
where ¯z = 共共1 / 2兲兺ᐉ苹v¯xᐉ兲v苸X is a vector in Rn such that ¯储q = 储z
v苸X ᐉ苹v
1/p
¯xqᐉ
1/q
where we have used that, for any ᐉ 苸 E, when we add 1 to all nodes v 苸 X such that v 苸 ᐉ, we only have two summands 共the two vertices of ᐉ兲, and therefore this sum is always equal to 2. By applying Lemma 1 to Eq. 共13兲 we have that
since the value 兺ᐉ苹v共nv j共ᐉ兲 / nv j兲 is the count of all the links incident to v and hence 兺ᐉ苹vnv j共ᐉ兲 = nv j. In order to get the estimates for V5,1,q共G兲 and V6,1,q共G兲 ¯ 储q and 储y ¯ 储q. In order we only have to give some results for 储x to do so, we will employ the last equality, which implies that q
¯ ᐉ兲ᐉ苹v储q1 储共x
ᐉ苸E
j苸X
1 ¯y n−1
v苸X
= 21/q−1
冉兺 冊 ᐉ苹v
冉兺 冊 冉兺 兺 冊
冊
q
+
1 n
冋冉
21/q−1m1/qV5,1,q共G兲 +
冉冉 冊
n−1 1 2m q+1 = 2 n n
1/q
V5,1,q共G兲 +
1 n
冊
q
1 n
1+1/q
+
冊
n−1 nq
q
+
册
n−1 , nq+1
which gives us the result. 䊐 By combining the bound obtained in Theorem 1 and the upper estimate 共8兲 共which was proven in Ref. 31兲, we can obtain some bounds that relate V5,p,q共G兲 with V6,p,q共G兲 for any p , q 苸 关1 , ⬁兲. A lower bound for V6,p,q共G兲 is
冋冉 冉 冊 冉 1 2m 2 n
1/q
1 n共n − 1兲
冊
1−1/p
V5,p,q共G兲 +
1 n1+1/q
冊
q
+
n−1 nq+1
册
1/q
,
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Criado et al.
Relationships between Vuln. psi5 and Vuln. psi6 (pref. attach.), p=2
Relationships between Vuln. psi and Vuln. psi (ER), p=2 5
6
0.065
0.22
0.06
0.2 0.18
0.055
0.16
0.05 Vuln. psi6
Vuln. psi6
0.14
0.045 0.04
0.12 0.1
0.035
0.08
0.03
0.06
0.025 0.02 0.5
0.04
1
1.5
2
2.5
3
3.5
Vuln. psi
4 x 10
5
关n共n − 1兲兴
1−1/p
冋冉
grmax共G兲 2
冊 冉冊 1−1/q
m n
1/q
V5,p,q共G兲 +
0
0.005
0.01
0.015
0.02 Vuln. psi
0.025
0.03
0.035
0.04
5
FIG. 2. 共Color online兲 V5 vs V6 for preferential attachment.
FIG. 1. 共Color online兲 V5 vs V6 for ER.
and an upper bound for V6,p,q共G兲 is
0.02
3
册
1 , n
where grmax共G兲 is the maximal degree of the network G. Note that both these estimates and the bounds in Theorem 1 improve the lower bounds proved in Ref. 31 and recover the estimates of Theorem 1 by taking p = 1. These relationships can be tested by means of numerical examples. To this purpose, we make two numerical testings. For the first one, we consider networks with a number of vertices n = 50. We make 100 simulations of two different topological structures: an Erdős–Rényi 共ER兲32 random graph with probabilities varying from pmin = 0.4 to pmax = 0.9 with a step of 0.1 共Fig. 1兲 and a scale-free network constructed by preferential attachment rules introduced by Barabási and Albert,33 where we construct a network of n nodes sequentially by adding in each step a node linked with d other random nodes according to their degree 共see also Ref. 34兲. In our simulations we vary from d = 2 to d = 46 with a step of 1 共Fig. 2兲.
The criterium considered in Refs. 24 and 25 for choosing such a leader in a complex network is related to spotting the node vo that minimizes the expected number of disconnected nodes from vo under a random breakdown of a node other than vo. In order to compute these values the concept of bottleneck was introduced. If we take three nodes x , y , z 苸 X, we say that y is a bottleneck from x to z if every path from x to z goes through y. Note that if we denote by ⌸共x , z兲 the set of all bottlenecks from x to z, then it was proved in Ref. 25 that the expected number of disconnected nodes from v under a random breakdown of a node other than v is exactly D共v兲 =
1 兺 共兩⌸共v,w兲兩 − 1兲, n − 1 v⫽w苸X
where 兩⌸共vo , w兲兩 is the number of bottleneck from vo to w and n is the number of nodes of the complex network. It is clear that this function D共·兲 gives a criterium for measuring the vulnerability, since the lower the average value of D共·兲, the higher robustness of the network we get. Therefore, we can define the bottleneck vulnerability of a complex network G as
C. Bottleneck-type vulnerabilities
In this subsection we will present a concept of vulnerability based on the change of the geodesic structure when one of the nodes fails. This approach was introduced in Refs. 24 and 25 where the problem of locating a leader node on a complex network was considered. This problem is interesting due to its practical applications, such as the key transfer protocol design for multiparty key establishment 共see Ref. 35兲, where it is not always adequate to consider all nodes in the network as equally important, for instance, to distribute a message from a central server to a group of users, or a key for private communication, which requires secure communication between the leader 共or the initiator兲 and all other members in order to initially establish the key. In any case, the keys 共or a part兲 must be sent through a communication network which holds up the flow of information.
B共G兲 =
1 1 D共v兲 = 兺 兺 兺 共兩⌸共v,w兲兩 − 1兲. n v苸X n共n − 1兲 v苸X v⫽w苸X 共16兲
This vulnerability function can be also expressed in terms of a 共 , p , q兲 vulnerability simply by taking Y = 兵共i, j兲;i ⫽ j 苸 X其, Z = X and 7 : Y → 关0 , 1兴 defined for every i , j , v 苸 X 共i ⫽ j兲 as
7共i, j, v兲 = ⌸共i,j兲共v兲 =
再
1 if v 苸 ⌸共i, j兲 0 otherwise.
冎
Let us see that V7,1,1共G兲 = 共1 / n兲共B共G兲 + 1兲. By changing the order of summations we get that
共 , p , q兲 vulnerabilities
013133-7
Chaos 19, 013133 共2009兲
V7,1,1共G兲 =
1 兺 兺 7共i, j,z兲 n 共n − 1兲 v苸X i⫽j
Bg共G兲 =
=
1 兺 兺 ⌸共i,j兲共v兲 n2共n − 1兲 i⫽j v苸X
=
=
1 兺 兩⌸共i, j兲兩 n 共n − 1兲 i⫽j
=
2
1 兺 D g共 v 兲 n v苸X 1 兺 兺 共兩⌸g共v,w兲兩 − 1兲, n共n − 1兲 v苸X v⫽w苸X
共17兲
2
冉
1 1 1+ 兺 共兩⌸共i, j兲兩 − 1兲 n n共n − 1兲 i⫽j
冊
1 = 共B共G兲 + 1兲. n The concepts of bottleneck and bottleneck vulnerability B共·兲 are very useful when we are dealing with complex networks which are tree shaped 共see Ref. 24兲, but they are very restrictive when we consider general networks 共see Ref. 25兲 since the set ⌸共i , j兲 is usually trivial. This is due to the fact that a node v is a bottleneck from i to j if and only if all the possible paths from i to j go through v, which is too restrictive. In order to avoid this inconvenience in a general network, we can introduce a relative concept 共geodesic bottleneck兲 which gives a qualitative perspective of the geodesic structure of the complex network. We say that a node y is a geodesic bottleneck from node x to node z if every geodesic path from x to z necessarily goes through y. We denote by ⌸g共x , z兲 the set of all geodesic bottlenecks from x to z. If v 苸 X and we denote by Dg共v兲 the expected number of nodes that change their distance 共inside the graph兲 to node v when a failure at some other node occurs, then we can prove that Dg共v兲 is related to the geodesic bottlenecks for v. In fact, we have the following equality:
which is a vulnerability measure of the network also related to the 共 , p , q兲 vulnerabilities. If we take again
Y = 兵共i, j兲;i ⫽ j 苸 X其, Z = X and 8 : Y → 关0 , 1兴 defined for every i , j , v 苸 X 共i ⫽ j兲 as
8共i, j, v兲 = ⌸g共i,j兲共v兲 =
再
1 if v 苸 ⌸g共i, j兲 0 otherwise,
冎
then, by using the same ideas as before, it is straightforward to check that V8,1,1共G兲 = 共1 / n兲共Bg共G兲 + 1兲. The bottleneck vulnerability and the geodesic bottleneck vulnerability of a complex network G coincide if G is a tree, and they are closely correlated in a general network, as the following result shows. Proposition 1: Let G be a complex network with n nodes and let 1 ⱕ p , q ⱕ ⬁ . Then
V7,p,q共G兲 ⱕ V8,p,q共G兲 ⱕ ␥共p,q,n兲V7,p,q共G兲, D g共 v 兲 =
1 兺 共兩⌸g共v,z兲兩 − 1兲. n − 1 z苸X
This formula is a direct consequence of the fact that, if we fix v 苸 X, then the number of nodes that change their distance to node v when a failure in a node y ⫽ v occurs is the number of nodes z such that y 苸 ⌸g共v , z兲 and hence
1 D g共 v 兲 = 兺 兩兵z 苸 X,y 苸 ⌸g共v,z兲其兩 n − 1 y⫽v =
1 兺 兺 ⌸ 共v,z兲共y兲 n − 1 y⫽v z苸X g
=
1 兺 兺 ⌸ 共v,z兲共y兲 n − 1 z苸X y⫽v g
=
1 兺 共兩⌸g共v,z兲兩 − 1兲. n − 1 z苸X
By using this concept we can introduce the geodesic bottleneck vulnerability of a complex network G as
where
␥共p,q,n兲 =
冦
冉冊 冉冊 n 2
1/p
n 2
2/q−1/p
共n − 1兲1/p−1/q
if p ⱕ q
共n − 1兲1/q−1/p , q ⬍ p.
冧
Proof: On the one hand, if i , j 苸 X then ⌸共i , j兲 債 ⌸g共i , j兲 and therefore 7共i , j , v兲 ⱕ 8共i , j , v兲, which leads us to the lower estimate. On the other hand, in order to prove the upper estimate we should consider two different cases. First, if p ⱕ q, note that
兩X兩兩Y兩q/pVq
8,p,q
共G兲 =
兺 冉兺 ⌸ 共i,j兲共v兲冊 苸X i⫽j
v
g
q/p
,
and therefore, by using Lemma 2 with p⬘ = 1, q⬘ = q / p and since 2 ⱕ 兩⌸共i , j兲兩 , 兩⌸g共i , j兲兩 ⱕ n, we have that
013133-8
Chaos 19, 013133 共2009兲
Criado et al.
兩Y兩q/p兩X兩Vq
8,p,q
兺 冉兺 ⌸ 共i,j兲共v兲冊 苸X i⫽j
共G兲 =
q/p
g
v
兺 ⌸ 共i,j兲共v兲 = 兩Y兩q/p−1兺 兺 ⌸ 共i,j兲共v兲 i⫽j 苸X
ⱕ 兩Y兩q/p−1 兺
v苸X i⫽j
g
v
g
n n n = 兩Y兩q/p−1 兺 兩⌸g共i, j兲兩 ⱕ 兩Y兩q/p−1 兺 兩⌸共i, j兲兩 = 兩Y兩q/p−1 兺 兺 ⌸共i,j兲共v兲 ⱕ 兩Y兩q/p−1 兺 2 2 2 i⫽j i⫽j v苸X i⫽j v苸X
冉兺
⌸共i,j兲共v兲
i⫽j
冊
q/p
n = 兩Y兩2q/p−1兩X兩Vq ,p,q共G兲, 7 2
冉 冊 N
which shows that
冉冊
n V8,p,q共G兲 ⱕ 2
兺 兩ai兩 i=1
1/p
共n − 1兲
1/p−1/q
V7,p,q共G兲.
8,p,q
兺 冉兺 ⌸ 共i,j兲共v兲冊 苸X i⫽j
共G兲 =
g
v
=
q/p
ⱕ
冉冊 n 2
冉 冊 兺 兩ai兩
r
i=1
n
v
n n ⌸共i,j兲共v兲 ⱕ 兩Y兩1−q/p 兺 兺 兺 2 v苸X i⫽j 2 v苸X
g
冉兺
g
⌸共i,j兲共v兲
i⫽j
2/q−1/p
共n − 1兲1/q−1/pV7,p,q共G兲.
䊐 As a particular case of the last result, by taking p = q = 1 we get the following analytical estimates that relate B共G兲 with Bg共G兲:
冉
i=1
N
兩⌸g共i, j兲兩 ⱕ 兺 兩⌸共i, j兲兩 兺 ⌸ 共i,j兲共v兲 = 兺 兺 ⌸ 共i,j兲共v兲 = 兺 兺 2 i⫽j 苸X i⫽j i⫽j v苸X i⫽j
which leads us to ensure that V8,p,q共G兲 ⱕ
N
ⱕ 兺 兩ai兩r ⱕ N1−r
for all a1 , . . . , aN 苸 R 共see, for example, Ref. 29兲. If we use these inequalities for r = q / p and since 2 ⱕ 兩⌸共i , j兲兩 , 兩⌸g共i , j兲兩 ⱕ n, then we prove that
Finally, if q ⬍ p, we will use that if 0 ⬍ r ⬍ 1 then
兩Y兩q/p兩X兩Vq
r
冊
n−2 n . B共G兲 ⱕ Bg共G兲 ⱕ B共G兲 + 2 n Note that these lower estimates are sharp, since if G is a tree then ⌸共i , j兲 = ⌸g共i , j兲 for every i , j 苸 X and therefore V8,p,q共G兲 = V7,p,q共G兲 for every 1 ⱕ p , q ⱕ ⬁. III. RELATIONSHIPS BETWEEN „ , p , q…-VULNERABILITIES
In the previous section we have introduced a method for obtaining a wide variety of vulnerability measures. Roughly speaking, if we want to consider a vulnerability function that fits a particular problem, we should tune three parameters. First we have to fix a weight function : Y ⫻ Z → 关0 , + ⬁兲 and then we have to consider two exponents 1 ⱕ p , q ⱕ ⬁ in order to obtain the 共 , p , q兲-vulnerability measure. In this section we give some relationships between these 共 , p , q兲 vulnerabilities and we show how these measures behave when we change one of their ingredients 共i.e., the function or one
冊
q/p
n = 兩Y兩兩X兩Vq ,p,q共G兲, 7 2
of the exponents 1 ⱕ p , q ⱕ ⬁兲. Our first result of this section shows some sharp estimates for the 共 , p , q兲-vulnerability functions when we tune one of their exponents p or q. Proposition 2: Let G be a complex network, Y be a subset of ordered pairs of nodes or links, Z be a subset of nodes or links, and : Y ⫻ Z → 关0 , + ⬁兲 . Then we have the following. • If 1 ⱕ p ⱕ p⬘ ⬍ ⬁ and 1 ⱕ q ⱕ ⬁ , then 1 兩Y兩
1/p−1/p⬘
V,p⬘,q共G兲 ⱕ V,p,q共G兲 ⱕ V,p⬘,q共G兲.
• If 1 ⱕ p ⱕ ⬁ and 1 ⱕ q ⱕ q⬘ ⬍ ⬁ , then 1 兩Z兩
1/q−1/q⬘
V,p,q⬘共G兲 ⱕ V,p,q共G兲 ⱕ V,p,q⬘共G兲.
Proof: Note that if 1 ⱕ p ⬍ ⬁ and we take for every z 苸 Z the 兩Y兩-dimensional vector v共z兲 = 共共i, j,z兲兲共i,j兲苸Y 苸 R兩Y兩 ,
then V,p,q共G兲 =
1 兩Y兩 兩Z兩1/q 1/p
冉兺
z苸Z
储v共z兲储qp
冊
1/q
.
Hence, by Lemma 2, if we consider 1 ⱕ p ⱕ p⬘ ⬍ ⬁, since v共z兲 苸 R兩Y兩 we get that
共 , p , q兲 vulnerabilities
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储v共z兲储 p⬘ ⱕ 储v共z兲储qp ⱕ 兩Y兩q/p−q/p⬘储v共z兲储 p⬘ . q
q
Therefore, by summing up over all z 苸 Z, we get that 1 兩Y兩
1/p−1/p⬘
V,p⬘,q共G兲 ⱕ V,p,q共G兲 ⱕ V,p⬘,q共G兲.
In addition to this, if we consider the 兩Z兩-dimensional vector v = 共v共z兲兲z苸Z, then 兩Y兩1/p兩Z兩1/qV,p,q共G兲 = 储v储q. Hence, by using again Lemma 2 but now in R兩Z兩, we get that if 1 ⱕ q ⱕ q⬘ ⬍ ⬁, then 储v储q⬘ ⱕ 储v储q ⱕ 兩Z兩1/q−1/q⬘储v储q⬘ , which proves that 1 兩Z兩
1/q−1/q⬘
V,p,q⬘共G兲 ⱕ V,p,q共G兲 ⱕ V,p,q⬘共G兲.
䊐 If we take a complex network G, a subset of ordered pairs of nodes or links Y, a subset of nodes or links Z, and a function : Y ⫻ Z → 关0 , + ⬁兲, then for every 1 ⱕ p , q ⬍ ⬁, 1 V,⬁,q共G兲 ⱕ V,p,q共G兲 ⱕ V,⬁,q共G兲, 兩Y兩1/p 1 V,p,⬁共G兲 ⱕ V,p,q共G兲 ⱕ V,p,⬁共G兲 兩Z兩1/q simply by taking p⬘ → ⬁ or q⬘ → ⬁, respectively, in the last proposition. Note that either the estimates obtained in Proposition 2 or the estimates stated in this remark are sharp in general, since in order to state them we mainly use Hölder’s inequality which becomes an equality for some vectors in Rn. At this point we know how the 共 , p , q兲 vulnerabilities evolve when we tune one of their exponents p or q. In the following subsections we will show how the
兩Y兩兩X兩Vp
2,p,p
共 , p , q兲-vulnerability functions behave when we change the weight function , from 1 to 8, and we will get several estimates that relate the families of measures presented in the previous section. A. Fall of efficiency-type vulnerabilities versus multi-scale vulnerabilities
In the first section we have proved that the fall of efficiency-type vulnerabilities can be stated as 共 , p , q兲vulnerabilities simply by considering the weight functions 1, 2, 3, or 4, while the multi-scale vulnerabilities are also 共 , p , q兲-vulnerabilities by considering 5 or 6. We have proved that there is a strong correlation between the 共5 , p , q兲-vulnerability and the 共6 , p , q兲-vulnerability functions and there are also some correlations between the 共 , p , q兲-vulnerabilities defined by 1, 2, 3, or 4. In this subsection we will see some relationships between the 共2 , p , q兲-vulnerability function and the 共6 , p , q兲vulnerability function, but these estimates can be also extended to other vulnerabilities of both families. The main result of this subsection shows that if G is a complex network, then V2,p,q共G兲 ⱕ V6,p,q共G兲 for all 1 ⱕ p , q ⱕ ⬁. This inequality is a consequence of the fact that 2共i , j , v兲 ⫽ 0 if and only if all the geodesics from i to j go through v, i.e., 6共i , j , v兲 = 1 and therefore for every i , j , v 苸 X 共i ⫽ j兲
2共i, j, v兲 = 2共i, j, v兲6共i, j, v兲. Thus, since 2共i , j , v兲 苸 关0 , 1兴, we get that 2共i , j , v兲 ⱕ 6共i , j , v兲 and hence we obtain the result. If we consider the particular case p = q, by using Hölder’s inequality for every r , s 苸 共1 , ⬁兲 such that 1 / r + 1 / s = 1, we get that
兺 2p共i, j, v兲 = 兺 兺 2p共i, j, v兲6p共i, j, v兲 ⱕ 冉 兺 兺 2pr共i, j, v兲冊 冉 兺 兺 6ps共i, j, v兲冊 兺 苸X i⫽j 苸X i⫽j 苸X i⫽j 苸X i⫽j 1/r
共G兲 =
v
ⱕ
v
2pr共i, j, v兲冊 兺 冉 兺 6ps共i, j, v兲冊 兺 兺 冉 苸X i⫽j v苸X i⫽j 1/r
v
and therefore V2,p,p共G兲 ⱕ V2,pr,p共G兲V6,ps,p共G兲, which improves the previous result simply by taking s → 1+. These relationships can also be tested by means of numerical examples. To do so, we make two numerical testing as we did before. For the first one, we consider networks with a number of vertices n = 50. We make 100
v
1/s
= 兩Y兩兩X兩Vp
1/s
v
2,pr,p
共G兲Vp
6,ps,p
共G兲,
simulations of Erdős–Rényi random graphs with probabilities varying from pmin = 0.4 to pmax = 0.9 with a step of 0.1 共Fig. 3兲. For the second one, we consider a scale-free network constructed by preferential attachment rules varying from d = 2 to d = 46 with a step of 1 共Fig. 4兲.
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Criado et al. Relationships between psi2 and psi6 (ER), p=2
Relationships between Vuln. psi2 and Vuln. psi6 (pref. attac.), p=2 0.14
0.06
0.12
0.05
0.1
Vuln Psi6
Vuln Psi6
0.04
0.03
0.08
0.06
0.02 0.04
0.01
0
0.02
0
0
1
2
3
4
5
Vuln Psi
2
6
7 x 10
B. Multi-scale vulnerability versus geodesic bottleneck vulnerability
In this section we will prove that we can interpolate between the node-based multi-scale vulnerability and the geodesic bottleneck vulnerability by using an appropriate weight function . Remember that the node-based multiscale vulnerability is inspired on the betweenness balance of the network and can be stated as the 共6 , p , q兲-vulnerability function given for every i , j , v 苸 X by nij共v兲 6共i, j, v兲 = , nij
8共i, j, v兲 = ⌸g共i,j兲共v兲 for all i , j , v 苸 X. It is clear that roughly speaking both quantities measure how the geodesic structure is distributed through the network, but they do so in different levels. This naive idea can be analytically formulated since we can interpolate these measures. If we consider for every 0 ⱕ ␣ ⱕ 1 the function ␣ : 关0 , 1兴 → 关0 , 1兴 given by x if ␣ ⱕ x ⱕ 1 0 otherwise,
冎
and we denote the 共␣ ⴰ 6 , p , q兲-vulnerability function 共i.e., = ␣ ⴰ 6兲 by V␣,p,q共·兲, then this function interpolates between the 共6 , p , q兲-vulnerability and the 共8 , p , q兲vulnerability function. Let us start stating some basic properties of this measure. First of all, note that if G is a complex network and 0 ⱕ ␣ ⱕ  ⱕ 1, then V␣,p,q共G兲 ⱖ V,p,q共G兲
0.01
0.015 Vuln Psi
0.02
0.025
0.03
2
between V6,p,q共·兲 and V8,p,q共·兲. On the one hand, it is straightforward to check that 0共·兲 is the identity in 关0,1兴, and therefore V0,p,q共·兲 = V6,p,q共·兲.
共18兲
for all 1 ⱕ p , q ⱕ ⬁, since for every x 苸 关0 , 1兴, ␣共x兲 ⱖ 共x兲. In addition to this, note that V␣,p,q共G兲 interpolates
共19兲
On the other hand, V1,p,q共G兲 = V8,p,q共G兲 for every 1 ⱕ p , q ⬍ ⬁, since
1
while the geodesic bottleneck vulnerability function measures the average number of nodes that change their distance when a node fails and it is also the 共8 , p , q兲-vulnerability function given by
再
0.005
FIG. 4. 共Color online兲 V2 vs V6 for preferential attachment.
FIG. 3. 共Color online兲 V2 vs V6 for ER.
␣共x兲 =
0
5
冉 冊
再
1 if nij共v兲 = nij nij共v兲 = 0 otherwise nij
共20兲
冎
= ⌸g共i,j兲共v兲,
which is a consequence of the fact that nij共v兲 = nij if and only if v 苸 ⌸g共i , j兲. By using these properties of the functions V␣,p,p共·兲 we can give some relationships between the 共5 , p , q兲vulnerability and the 共8 , p , q兲-vulnerability function. If G is a complex network and 1 ⱕ p , q ⬍ ⬁, then it is easy to check that V8,p,q共G兲 ⱕ V6,p,q共G兲, and, in addition to this, if G is a tree, then V6,p,q共G兲 = V7,p,q共G兲 = V8,p,q共G兲 = V␣,p,q共G兲 for all 0 ⱕ ␣ ⱕ 1. The inequality is an easy consequence of Eqs. 共18兲–共20兲 since V8,p,q共G兲 = V1,p,q共G兲 ⱕ V0,p,q共G兲 = V6,p,q共G兲, while if G is a tree, then for every pair of nodes i ⫽ j 苸 X, there is at most one geodesic path joining i and j, and therefore 0 ⱕ nij , nij共v兲 ⱕ 1, which makes that
6共i, j, v兲 =
再
1 if nij共v兲 = nij nij共v兲 = 0 otherwise nij
冎
= ⌸g共i,j兲共v兲
= ⌸g共i,j兲共v兲, i.e., 6共i , j , v兲 = 7共i , j , v兲 = 8共i , j , v兲 for every v 苸 X and moreover,
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共 , p , q兲 vulnerabilities
再
1 if nij共v兲 = nij nij共v兲 = 0 otherwise nij
冎 冉 冊 = 1
nij共v兲 . nij
As a consequence of this fact we conclude that V6,p,q共G兲 = V7,p,q共G兲 = V8,p,q共G兲 = V␣,p,q共G兲. C. Geodesic bottleneck vulnerability versus fall of efficiency-type vulnerabilities
In previous subsections we showed that the geodesic bottleneck vulnerability of a complex network G is the 共8 , p , q兲-vulnerability function and one of the fall of efficiency-type vulnerabilities is the 共2 , p , q兲-vulnerability. Note that if we take i , j , v 苸 X then 2共i , j , v兲 ⫽ 0 if and only if 8共i , j , v兲 = 1 and therefore
2共i, j, v兲 = 2共i, j, v兲8共i, j, v兲. Combining this expression with the techniques used in the last subsection, we can give some estimates that relates both types of vulnerabilities obtaining that if G is a complex network, then V2,p,q共G兲 ⱕ V8,p,q共G兲 for all 1 ⱕ p , q ⱕ ⬁ and V2,p,p共G兲 ⱕ V2,pr,p共G兲V8,ps,p共G兲, for all r , s 苸 共1 , ⬁兲 such that 1 / r + 1 / s = 1. IV. CONCLUSIONS
Several different approaches have been introduced in the literature to measure the capacity of a network to maintain its functional performance under random damage, malicious attacks, or dysfunctions of any type. Depending on the size of the network, the nature of the problem, the type of applications we are analyzing, or even the target we are pursuing, we will have to decide which vulnerability function is best suited for that analysis. Almost all of these approaches 共with the exception of those which include probabilistic notions, v.gr., percolation theory兲 are particular cases for specific values of the parameters p and q and for a specific function of the general framework given by the concept of 共 , p , q兲 vulnerability. So, inside this general framework, we have more information to decide which vulnerability function 共the particular values of p and q and the specific function 兲 is best suited to analyze a specific problem. We also obtain some bounds and relationships among these vulnerability functions and we show their sharpness through some relevant simula-
Chaos 19, 013133 共2009兲
tions. This general framework gives us a relevant and useful tool to be applied to real-world complex networks. Y. Bar-Yam, Dynamics of Complex Systems 共Addison-Wesley, Reading, 1997兲. 2 S. Boccaletti, M. Ivanchenko, V. Latora, A. Pluchino, and A. Rapisarda, Phys. Rev. E 75, 045102 共2007兲. 3 S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, Phys. Rep. 424, 175 共2006兲. 4 R. Criado, A. García del Amo, B. Hernández-Bermejo, and M. Romance, J. Comput. Appl. Math. 192, 59 共2006兲. 5 R. Criado, B. Hernández-Bermejo, and M. Romance, Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2289 共2007兲. 6 P. Holme and J.-H. Kim, Phys. Rev. E 65, 066109 共2002兲. 7 H. Jeong, S. Mason, A. L. Barabási, and Z. N. Oltvai, Nature 共London兲 411, 41 共2001兲. 8 V. Latora and M. Marchiori, Phys. Rev. Lett. 87, 198701 共2001兲. 9 V. Latora and M. Marchiori, New J. Phys. 9, 188 共2007兲. 10 M. E. J. Newman and M. Girvan, Phys. Rev. E 69, 026113 共2004兲. 11 O. Sporns, Complexity 8, 56 共2002兲. 12 S. H. Strogatz, Nature 共London兲 410, 268 共2001兲. 13 R. Albert, H. Jeong, and A. L. Barabási, Nature 共London兲 406, 378 共2000兲. 14 S. Boccaletti, J. Buldú, R. Criado, J. Flores, V. Latora, J. Pello, and M. Romance, Chaos 17, 043110 共2007兲. 15 R. Cohen, K. Erez, D. ben-Avraham, and S. Havril, Phys. Rev. Lett. 85, 4626 共2000兲. 16 R. Cohen, K. Erez, D. ben-Avraham, and S. Havril, Phys. Rev. Lett. 86, 3682 共2001兲. 17 R. Criado, J. Flores, B. Hernández-Bermejo, J. Pello, and M. Romance, J. Math. Model. Algorithms 4, 307 共2005兲. 18 P. Crucitti, V. Latora, M. Marchiori, and A. Rapisarda, Physica A 320, 622 共2003兲. 19 P. Holme, B. J. Kim, C. N. Yoon, and S. K. Han, Phys. Rev. E 65, 056109 共2002兲. 20 V. Latora and M. Marchiori, Chaos, Solitons Fractals 20, 69 共2004兲. 21 A. H. Dekker and B. D. Colbert, Proceedings of ACSC04, the 27th Australasian Computer Science Conference, Dunedin, New Zealand, 18–22 January 2004 共unpublished兲. 22 M. E. J. Newman and G. Goshal, Phys. Rev. Lett. 100, 138701 共2008兲. 23 S. Boccaletti, R. Criado, J. Pello, M. Romance, and M. Vela-Pérez, Int. J. Bifurcation Chaos Appl. Sci. Eng. 19, 2 共2009兲. 24 R. Criado, J. Flores, M. I. Gonzalez-Vasco, and J. Pello, J. Comput. Appl. Math. 204, 10 共2007兲. 25 R. Criado, J. Flores, J. Pello, and M. Romance, Eur. Phys. J. Spec. Top. 146, 145 共2007兲. 26 E. Estrada and J. Rodríguez-Velázquez, Phys. Rev. E 71, 056103 共2005兲. 27 J. Rodríguez, E. Estrada, and A. Gutiérrez, Linear Multilinear Algebra 55, 293 共2007兲. 28 M. E. J. Newman, SIAM Rev. 45, 167 共2003兲. 29 W. Rudin, Real and complex analysis, 3rd ed. 共McGraw-Hill, New York, 1987兲. 30 S. Wasserman and K. Faust, Social Networks Analysis 共Cambridge University Press, Cambridge, 1994兲. 31 R. Criado, J. Pello, M. Romance, and M. Vela-Pérez, Int. J. Bifurcation Chaos Appl. Sci. Eng. 19, 2 共2009兲. 32 P. Erdős and A. Rényi, Publ. Math. 共Debrecen兲 6, 290 共1959兲. 33 A. L. Barabási and R. Albert, Science 286, 15 共1999兲. 34 V. Batagelj and U. Brandes, Phys. Rev. E 71, 036113 共2005兲. 35 C. Boyd and A. Mathuria, Protocols for Authentication and Key Establishment (Information Security and Cryptography) 共Springer, New York, 2003兲. 1
CHAOS 19, 013134 共2009兲
Onset of synchronization in weighted scale-free networks Wen-Xu Wang,1 Liang Huang,1 Ying-Cheng Lai,1,2 and Guanrong Chen3 1
Department of Electrical Engineering, Arizona State University, Tempe, Arizona 85287, USA Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287, USA 3 Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China 2
共Received 13 October 2008; accepted 3 February 2009; published online 17 March 2009兲 We investigate Kuramoto dynamics on scale-free networks to include the effect of weights, as weighted networks are conceivably more pertinent to real-world situations than unweighted networks. We consider both symmetric and asymmetric coupling schemes. Our analysis and computations indicate that more links in weighted scale-free networks can either promote or suppress synchronization. In particular, we find that as a parameter characterizing the weighting scheme is varied, there can be two distinct regimes: a normal regime where more links can enhance synchronization and an abnormal regime where the opposite occurs. A striking phenomenon is that for dense networks for which the mean-field approximation is satisfied, the point separating the two regimes does not depend on the details of the network structure such as the average degree and the degree exponent. This implies the existence of a class of weighted scale-free networks for which the synchronization dynamics are invariant with respect to the network properties. We also perform a comparison study with respect to the onset of synchronization in Kuramoto networks and the synchronization stability of networks of identical oscillators. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3087420兴 Understanding synchronization in complex networks of large numbers of interacting oscillators is important for areas ranging from communication to biology. How network properties affect the synchronization dynamics has been an issue of active research in recent years. In realworld networks, the interactions among oscillators are often asymmetric and nonuniform. This has motivated research on synchronization in weighted complex networks. Previous studies on this topic focused on the synchronizabilities of networks of identical oscillators, for which the approach of eigenvalue analysis is applicable. However, in realistic networked systems, node dynamics can be nonidentical. There is thus a need to study synchronization in weighted complex networks with heterogeneous node dynamics. Here, we use the Kuramoto model to address this problem. We consider both symmetric and asymmetric weighting schemes and obtain analytic predictions concerning the synchronization properties of the network. Our analysis and computations indicate that more links in weighted scale-free networks can either promote or, counterintuitively, suppress synchronization. A finding is that, regardless of the coupling scheme, two regimes with the opposite synchronization behaviors are separated by a point, at which the onset of network synchronization does not depend on the link density. This then implies the existence of a class of weighted scale-free networks for which the synchronization dynamics are invariant with respect to network connectivity. I. INTRODUCTION
Synchronization in large networked systems has been an active area of research in statistical and nonlinear physics1,2 1054-1500/2009/19共1兲/013134/8/$25.00
since the pioneering work of Kuramoto.3 Given a network of coupled nonlinear oscillators, Kuramoto focused on the phase variables and derived a model of phase-coupled oscillators, N
˙ i = i + 兺 C ji sin共 j − i兲,
共1兲
j=1
where i and i are the phase and the natural frequency of oscillator i, respectively, N Ⰷ 1 is the total number of oscillators, is a global coupling parameter that is identical for all oscillators, and 兵Cij其 is the coupling matrix. In the model, each oscillator, when isolated, is characterized by a uniform rotation of a given frequency. The frequencies of all oscillators are drawn from a probability distribution g共兲 with a single maximum at zero. The interactions among the oscillators are described by nonlinear coupling terms. The model is thus relatively simple but highly nontrivial in the sense that it captures many generic features of the dynamics of realistic oscillator networks. A particularly appealing feature of the model is that it is amenable to analysis partly because of the simple node dynamics. In addition, because of the different intrinsic frequencies associated with the oscillators, the isolated node dynamics are not identical but heterogeneous. The model thus enables the problem of synchronization of large networks of heterogeneous elements to be addressed. Indeed the Kuramoto model has been a paradigm for obtaining analytic insights into a variety of synchronization phenomena in physics and biology, as well as in engineering and technology.4,5 The coupling scheme in the original Kuramoto paradigm is all-to-all.3,4 The network is thus densely connected, so the standard mean-field theory can be used to analyze the tran-
19, 013134-1
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sition to synchronization. More specifically, let be a global coupling parameter and R ⱖ 0 be an order parameter, where a nonzero value of R indicates certain degree of coherence among the dynamics of the coupled oscillators. A mean-field treatment usually gives c, the critical coupling value at which R starts to increase from zero, or the transition to partial synchronization. Recently, the Kuramoto model has been adopted to networks of complex topologies.6,7 For example, transition to synchronization in scale-free networks8 has been investigated with predictions for c under both mutual and directed coupling schemes.6 At the onset, partial synchronization emerges in the form of small synchronous clusters. Mean-field theory predicts that c is determined by the frequency distribution of the phase oscillators and the first two moments of the node-degree distribution. The onset of global synchronization where all oscillators begin to synchronize in complex clustered networks has also been considered.7 In this paper, we investigate the transition to synchronization in weighted scale-free networks under the Kuramoto paradigm. Our goal is to understand, quantitatively, the influence of weighting on the onset of synchronization. Such a network model takes into account the nonuniform interactions among nodes in the network and is therefore believed to better describe large networked systems in reality. Usually, there is a correlation between the weight distribution and the network topology.9 For scale-free networks, due to the algebraic degree distribution, the weights can be highly heterogeneous, far beyond those in the Boolean representation.9 Synchronization in weighted scale-free networks has been investigated recently where all existing works assume identical node dynamics and are mostly based on the eigenvalue analysis.2 Our approach is to use the formula for transition to synchronization in Ref. 6 as a theoretical tool to derive explicit expressions for c for different weighting schemes. To be as general as possible, we shall consider both symmetric and asymmetric couplings. The main results of this paper can be stated in terms of the dependence of c on some parameter ␣ that characterizes the weighted coupling scheme and controls the degree of homogeneity in link weights. We find that for both symmetric and asymmetric weighted networks, the dependence of c on ␣ is approximately exponential but with different rates for different densities of links even when the network size and the topological parameter are the same. As a result, there exists a critical point ␣c such that for ␣ ⬍ ␣c, networks of sparser linkage are more synchronizable than networks of denser linkage in the sense that the critical values c required for partial synchronization in the former case are less than those in the latter case, whereas the opposite holds for ␣ ⬎ ␣c 关Figs. 3 and 4兴. The striking phenomenon is that the synchronization dynamics of networks with values of ␣ in the vicinity of ␣c are invariant with respect to changes in the network link density, and the value of c at ␣c is independent of the network structural details. In both the ␣ ⬍ ␣c and the ␣ ⬎ ␣c regimes, the network connectivity changes little in the sense that links associated with larger-degree nodes are assigned with smaller weights, but the synchronization dynamics are qualitatively different in the two regimes. In general,
this phenomenon can be attributed to the complex interplay between network structure and dynamics. The phenomenon is predicted analytically with solid numerical support. A result in Ref. 6 is that for random networks with some prescribed degree distribution, the critical coupling strength c for transition to synchronization is determined by the largest eigenvalue of the adjacency matrix. We will also use this eigenvalue approach to investigate the onset of synchronization in weighted networks to provide additional support for our results. In addition, we will compare results for the onset of synchronization in the Kuramoto network with those from the analysis of synchronization stability in networks of identical oscillators with the same topology. In Sec. II, we describe our network model with both symmetric and asymmetric weighted coupling schemes. In Sec. III, we provide theoretical and numerical results with respect to the order parameter near the onset of synchronization and c. An eigenvalue analysis for the critical coupling strength is presented in Sec. IV, providing additional support for the generality of our findings. In Sec. V, the dynamics of weighted Kuramoto networks are compared with the dynamics of networks of identical oscillators with the same coupling scheme. Conclusions and discussions are offered in Sec. VI.
II. MODEL DESCRIPTION
We construct scale-free networks by using the standard preferential-attachment model.8 At each time step, a new node with m links is added and preferentially attached to m existing nodes with probability proportional to the degrees of the existing nodes. Degree distributions of the networks are given by P共k兲 ⬃ k−3, the minimum node degree of the networks is m, and the average degree is 2m. We then assign weights to links in terms of the node degrees. Two types of schemes are considered. 共1兲 Symmetric coupling. We assume the weight associated with the link between nodes i and j to be wij = 共kik j兲␣, where ki and k j are the degrees of i and j, respectively, and ␣ is a control parameter. The coupling strength Cij between nodes i and j becomes Cij = Aijwij, where 兵Aij其 is the adjacency matrix of the network. As a result, the coupling strength is symmetric: Cij = C ji. This weighted scheme is supported by empirical data of some realworld weighted networks.9 共2兲 Asymmetric coupling. If the directional couplings between two connected nodes are asymmetric, the network will typically be weighted and directed. The weight from node i to node j is wij = k␣i , but wij ⫽ w ji = k␣j for ki ⫽ k j. The coupling matrix element from node i to node j is Cij = Aijwij. This choice of the weighting scheme is motivated by the fact that in certain real-world networks, the weight of a node has a nonlinear correlation with its degree in the form si ⬃ k␣i , where s is the node weight.9 The asymmetric coupling scheme takes into account both the nonlinear correlation and the fact that the influences from a node to its neighbors are the same.
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A global order parameter can be defined to quantify the degree of coherence in the network,6
兺n=1 rn r= N , 兺n=1 dn
onset of synchronization, we set out to calculate the various averaged quantities in Eqs. 共5兲–共8兲.
N
共2兲
To obtain the critical coupling strength from Eq. 共5兲 requires that the total local coupling strength be calculated, which then yields 1. We start by rewriting di as
where the local order parameter rn is defined as N
r ne in =
Cmn具ei 典t . 兺 m=1
共3兲
m
In Eq. 共3兲, n is the phase associated with the order parameter and 具·典t denotes the average over time t. The local total coupling strength of node n is given by N
dn =
兺 Cmn ,
A. Symmetric coupling scheme
共4兲
m=1
where, for a directed network, the subscript mn indicates that the coupling is from node m to node n. When the phase oscillators are not synchronized, the order parameter r has near-zero values; whereas if the oscillators are fully synchronized, r becomes unity. In between the two extreme states, oscillators are partially synchronized. The onset of partial synchronization can be identified by a sudden and rapid increase in the value of r from zero. The order parameter r thus quantifies the degree of phase coherence among the oscillators in the network.
N
N
j=1
j=1
di = 兺 Cij = 兺 Aij共kik j兲␣ N
= k␣i 兺 Aijk␣j = k␣i +1 j=1
再
symmetric coupling 1具d典/具d2典, c =  1 1 = in in out 1具d 典/具d d 典, asymmetric coupling,
冎
共5兲 where 1 = 2 / 关g共0兲兴, 具·典 stands for the average over nodes in the network, d is the total local coupling strength, and din n and dout are total local incoming and outgoing coupling n strengths, respectively. The value of the order parameter r is given by
冉 冊冉 冊
2 −1 r2 = 2  1 2 c
c
di = k␣i +1
=
k␣i +1
具d2典3 2 = 4 2 . 具d 典具d典
共7兲
具dindout典3 . 具共din兲3dout典具din典2
kmax
兺
P共k⬘兩ki兲k⬘␣
kmax
k⬘␣+1 P共k⬘兲 k␣i +1具k␣+1典 = , 具k典 具k典
k⬘=kmin
兺
k⬘=kmin
共9兲
兺
k⬘=kmin
k⬘␣+1 P共k⬘兲 = 具k␣+1典
has been used. Taking the ensemble average of di, we obtain k␣i +1具k␣+1典 具k␣+1典2 1 = . 具d典 = 兺 具k典 N i=1 具k典 N
共10兲
Similarly, the ensemble average of d2i is 具d 典 = 2
=
=
冓冉
兺 j=1
冊冔 2
N
Aijk␣i k␣j
冊冔 冓冉 冓 冔 k␣i +1
kmax
兺
k⬘=kmin
k2i ␣+2具k␣+1典2 具k典2
k⬘␣+1 P共k⬘兲 具k典 =
2
具k2␣+2典具k␣+1典2 . 具k典2
共11兲
The critical coupling parameter can be written as
For asymmetric coupling, we have
2 =
kmax
共6兲
where for ⱖ c, 2 = −g⬙共0兲1 / 16. For symmetric coupling, 2 is given by
k⬘=kmin
P共k⬘兩ki兲k⬘␣ ,
where the identity
−3
,
兺
where P共k⬘ 兩 ki兲 is the conditional probability that a node of degree ki has a neighbor of degree k⬘. For a network without degree-degree correlation, we have P共k⬘ 兩 ki兲 = k⬘ P共k⬘兲 / 具k典, where P共k⬘兲 is the degree distribution of the network. This can be understood by noting that a node of degree k⬘ will be counted k⬘ times as a neighbor of some other nodes in the network. We thus have
III. EFFECT OF WEIGHTS ON SYNCHRONIZATION
For the Kuramoto model, a general formula for c, the critical coupling parameter for the onset of synchronization, has been obtained previously,6,10 as follows:
kmax
共8兲
For heterogeneous coupling, can be regarded as a nominal coupling parameter. In this case, formulas 共5兲–共8兲 are still applicable. To analyze the effect of weighted coupling on the
c = 1
具d典 具k典 2 = 1 2␣+2 , 具d 典 具k 典
共12兲
where for any given value of , 具k典 can be approximated by kmax k P共k兲dk. For a scale-free network, the degree distri兰k=k min bution can be written as P共k兲 = ck−␥, where c is given by
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FIG. 1. For an ensemble of standard scale-free networks of N = 2000 nodes and average degree 具k典 = 60, squared order parameter r2 as a function of global coupling strength for different values of the weighting parameter ␣. The coupling scheme is symmetric. Each data point is obtained by averaging over 20 network realizations. Curves are theoretical predictions given by Eq. 共17兲. Due to the finite-size effect, the algebraic exponent ␥ in the degree distribution is slightly lower than 3. Here, we use ␥ ⬇ 2.9 in Eq. 共17兲 to draw the theoretical curves.
r2 = − 2
␥−1 c = 共␥ − 1兲kmin ,
as a result of the normalization condition 兰k⬁ P共k兲dk = 1 min 共assuming ␥ ⫽ 1兲. The maximum degree of the network is determined by 兰k⬁ P共k兲dk = 1 / N, from which we obtain max kmax = kminN1/共␥−1兲. The ensemble-averaged value of k can thus be calculated as 具k典 =
冕
kmax
k P共k兲dk =
kmin
␥−1 k 共N共−␥+1兲/共␥−1兲 − 1兲, − ␥ + 1 min 共13兲
which is valid for − ␥ ⫽ −1. Inserting this result into Eq. 共12兲, we get 2 2␣ + 3 − ␥ 1 N共2−␥兲/共␥−1兲 − 1 . 2␣+1 共2␣+3−␥兲/共␥−1兲 g共0兲 2 − ␥ kmin N −1
⑀c =
共14兲
The order parameter can be calculated in a similar way. In particular, from Eq. 共7兲, we have
2 = = =
具d2典3 具d4典具d典2
冉
具k2␣+2典具k␣+1典2 具k典2
冊 冒冋 3
冉
具k4␣+4典具k␣+1典4 具k␣+1典2 具k典4 具k典
具k2␣+2典3 . 具k 典具k␣+1典 4␣+4
冊册 2
共15兲
Using Eq. 共13兲, we can express 2 as 共4␣ − ␥ + 5兲共␣ − ␥ + 2兲2 共2␣ − ␥ + 3兲3 ⫻
共N共2␣−␥+3兲/␥−1 − 1兲3 . 共N共4␣−␥+5兲/␥−1 − 1兲共N共␣−␥+2兲/␥−1 − 1兲2
Finally, we obtain
FIG. 2. 共Color online兲 Nodal order parameter rn / dn as a function of node degree k for 共a兲 ␣ = 0, 共b兲 ␣ = −0.4, 共c兲 ␣ = −0.6, and 共d兲 ␣ = −1.0 for different values of close to the onset of partial synchronization. The network size is 2000 and the average degree is 具k典 = 20.
共16兲
冉 冊冉 冊
g⬙共0兲 −1 22g3共0兲 c
c
−3
,
共17兲
where c is given by Eq. 共14兲. We now provide numerical support for our theoretical predictions 关Eqs. 共14兲 and 共17兲兴. The frequency distribution of the phase oscillators is chosen 共somewhat arbitrarily兲 to be g共兲 = 3共1 − 2兲 / 4 for 兩兩 ⱕ 1 and g共兲 = 0 otherwise. Thus 1 = 8 / 共3兲 and g⬙共0兲 = −3 / 2. Four curves of r2 versus , corresponding to four different values of the weighting parameter ␣, are shown in Fig. 1 for in the vicinity of c. For each value of ␣, the order parameter r is approximately zero for ⬍ c. At c, r starts to increase from zero, signifying a transition to partial synchronization. The transition point depends on the weighting parameter ␣. We see that c increases as ␣ decreases, indicating that partial synchronization is more difficult to achieve as ␣ becomes more negative. A larger negative value of ␣ stipulates that links between larger-degree nodes be less weighted, effectively making the network more homogeneous. The results in Fig. 1 thus suggest that homogeneous weights actually hinder synchronization. Equivalently, a heterogeneous weighting scheme tends to facilitate the transition to synchronization. This is somewhat different from the result of eigenvalue analysis of global synchronization in oscillator networks of identical node dynamics, where heterogeneity has been found to suppress synchronization.1 This seeming “paradox” can be resolved by noting that our analysis yields information about the transition to partial synchronization only, while the eigenvalue analysis is for the global synchronizability of the underlying network. In all four cases, we observe satisfactory agreement between the numerical and theoretical values of c. For ⲏ c, the agreement between the numerical and theoretical values of the order parameter is also good. It should be remarked that the empirical results of Ref. 9 suggest a value of ␣ = 0.5, which is different from the crossover point ␣ = −0.5 for the onset of synchronization on weighted networks. The results in Fig. 2 raise the question as to which nodes
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FIG. 3. 共Color online兲 For symmetric coupling scheme, critical global coupling strength c as a function of the weighting parameter ␣ for different values of the average degree 具k典. The common intersecting point for different cases occurs at ␣ = −0.5 and is marked by the vertical dash line. This point defines two regimes: 共i兲 ␣ ⬍ −0.5, the abnormal synchronization regime, where networks with more links are less synchronizable, and 共ii兲 ␣ ⬎ −0.5, the normal regime, where the opposite that more links tend to facilitate synchronization. Data points are from simulations and the curves are analytical results given by Eq. 共14兲. Other parameters are the same as those in Fig. 1.
contribute more significantly to the emergence of partial synchronization. To address this question, we study the dependence of the nodal order parameter rn / dn on degree k at the onset of synchronization. As shown in Fig. 2, for different values of ␣ and for close to the onset of synchronization, lower-degree nodes tend to be more responsible for the emergence of partial synchronization, regardless of whether node strengths are heterogeneous or homogeneous. This is somewhat counterintuitive for the case of heterogeneous nodal strengths because high-strength nodes and their neighbors are expected to first synchronize. This phenomenon can be explained by noting that the nodal order parameter is normalized by the node strength dn. Although high-degree nodes can synchronize with some of their neighbors more easily,11 their contributions to the global order parameter are smaller due to the normalization over their strengths. Another phenomenon observed from Fig. 1 is that heterogeneous strength distribution 共corresponding to higher values of ␣兲 tends to facilitate the onset of partial synchronization since lower values of c are required. This can be explained by noting that the total coupling strength among all nodes, 兺Ni=1di = N具k␣+1典2 / 具k典, is a decreasing function of ␣. This means that a larger value of the total coupling strength is associated with more heterogeneous strength distribution, which generally requires smaller values of the global coupling c to achieve partial synchronization. Thus, that heterogeneous strength distribution favors the emergence of partial synchronization can be ascribed to the behavior of the total coupling strength. The critical coupling strength c as a function of the weighting parameter ␣ for different values of the average degree is shown in Fig. 3 on a semilogarithmic scale. We observe that the dependence of c on ␣ is approximately exponential but the exponential rate is different for different values of 具k典. As a result, two different curves will intersect
at some value of ␣. A remarkable phenomenon is that all curves apparently intersect at the same point! In Fig. 3, the intersecting point is ␣c = −0.5. There are two consequences. First, for ␣ = ␣c, networks with different values of 具k典 share the same critical point of transition to synchronization. Second, the dependence of c on 具k典 shows opposite trends for ␣ ⬍ ␣c and ␣ ⬎ ␣c. In particular, for ␣ ⬍ ␣c, networks with smaller values of 具k典 exhibit smaller values of c and thus are more synchronizable. This is quite counterintuitive, as the results suggest that networks with more links are less synchronizable. We call ␣ ⬍ ␣c the abnormal synchronization regime. For ␣ ⬎ ␣c, the values of c required for synchronization are smaller for larger values of 具k典, indicating that networks with more links are more synchronizable. This is then the normal synchronization regime. The existence of a common intersecting point among the c ⬃ ␣ curves for different values of 具k典 can be explained by Eq. 共14兲. For scale-free networks, the minimum degree kmin is positively correlated with the average degree 具k典. For example, for the standard scale-free network model we use ␣+1 = 1, indicating here, 具k典 = 2kmin. In Eq. 共14兲, for ␣ = −0.5, k2min that c is independent of both kmin and 具k典. Because the intersecting point is common for ␣c, the two regimes 共i.e., ␣ ⬍ −0.5 and ␣ ⬎ −0.5兲 exhibit a reverse relationship between c and 具k典. The common intersecting point c共␣ = −0.5兲 can actually be calculated by inserting ␣ = −0.5 into Eq. 共14兲. We have c共␣ = − 0.5兲 =
2 . g共0兲
共18兲
This value depends only on the frequency distribution of oscillators not on values of quantities such as the weighting parameter ␣, the exponent of the degree distribution, and the average degree 具k典.
B. Asymmetric coupling scheme
For the asymmetric coupling scheme, the coupling matrix is given by Cij = Aijki␣. The value of c for the onset of synchronization can be derived from Eq. 共5兲. The average incoming coupling value becomes
冓 冔冓
kmax
N
具din典 =
=
ki具k␣+1典 具k典
= 具k␣+1典.
冓 冔 j=1
=
兺
兺 A jik␣j
ki
k⬘=kmin
k⬘ P共k⬘兲k⬘␣ 具k典
冔
共19兲
The average outgoing coupling strength of node i can be written as N
N
j=1
j=1
diout = 兺 Aijk␣i = k␣i 兺 Aij = k␣i ki = ki␣+1 . We have
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IV. EIGENVALUE ANALYSIS FOR CRITICAL COUPLING STRENGTH
A result from Ref. 6 is that the critical coupling strength c at which the transition occurs is determined by the largest eigenvalue of the adjacency matrix for random networks with a prescribed degree distribution,6 c =
FIG. 4. 共Color online兲 For asymmetric coupling scheme, critical global coupling strength c vs the weighting parameter ␣ for different average degree 具k典. As for the case of symmetric coupling, two distinct regimes exist: 共i兲 ␣ ⬍ −1.0, the abnormal synchronization regime and 共ii兲 ␣ ⬎ −1.0, the normal regime. Data points are from simulations and the curves are from analytical predictions Eq. 共22兲. Other network parameters are the same as those in Fig. 1.
具dindout典 = =
冓 冓
ki具k␣+1典 ␣+1 · ki 具k典 ki␣+2具k␣+1典 具k典
冔
冔
=
具k␣+2典具k␣+1典 . 具k典
共20兲
The quantity 1 in Eq. 共5兲 is given by
1 =
具k典 具din典 . in out = 具d d 典 具k␣+2典
共21兲
Inserting Eqs. 共19兲–共21兲 into Eq. 共5兲, we obtain
1 , N
共23兲
where 1 is determined by the frequency distribution g共兲. It is thus useful to investigate the dependence of 1 / N on the weighting parameter ␣ to gain more support for our results from mean-field analysis. As we will show, the eigenvalue approach suggests strongly the existence of the abnormal and normal synchronization regimes and a crossover point separating the two regimes. The eigenvalue approach for the onset of synchronization is, however, restricted to cases where the mean-field treatment is meaningful as the underlying networks are randomly constructed with some prescribed degree distribution. For standard scale-free networks generated by the preferential-attachment mechanism, although nodes are not randomly connected, the degree-degree correlation among nodes is weak, so the mean-field approximation is still valid. In this case, results from the eigenvalue analysis and from the mean-field theory agree with each other. The elements of the weighted adjacency matrix C are given by Cij = Aijwij for i ⫽ j and Cii = 0 otherwise. For the symmetric coupling case, for ␣ ⬎ −0.5, the corresponding weighted networks are highly heterogeneous. In this case, N can be obtained by considering the square of the weighted adjacency matrix C2. The largest eigenvalue N⬘ of the matrix C2 is determined by the largest diagonal term, according to the nondegenerate perturbation theory.12,13 The diagonal terms of C2 are given by N
␥−1 −␥ 兰 kkmax共␥ − 1兲kmin k kdk 具k典 min c = 11 = 1 ␣+2 = 1 k ␥−1 −␥ ␣+2 max 具k 典 兰 k 共␥ − 1兲kmin k k dk
C2ii = 兺 AijwijAijw ji = k2i ␣+1 j=1
min
=
2 ␣ − ␥ + 3 1 N共2−␥兲/共␥−1兲 − 1 . ␣+1 共␣−␥+3兲/共␥−1兲 g共0兲 2 − ␥ kmin N −1
共22兲
Comparison between the theoretical prediction 关Eq. 共22兲兴 and simulation results is given in Fig. 4. Behaviors similar to those for the symmetric coupling case are observed except that the value of ␣c separating the abnormal and the normal synchronization regimes becomes ␣c = −1. For cases of relatively large average degrees, there is a good agreement between numerics and theory. The slight difference for the case of 具k典 = 20 is due to the requirement of reasonably dense connectivity in the mean-field framework, which is better satisfied when the average degree of the network is larger. The value of c for ␣ = ␣c is still 2 / 关g共0兲兴, which depends only on the natural frequency distribution of the oscillators. The value of c共␣c兲 is thus identical for both the symmetric and the asymmetric coupling cases. It can be regarded as a general quantity characterizing the transition to synchronization in weighted scale-free networks, as it is apparently independent of structural details such as the degree-distribution exponent, the average degree, and the coupling scheme.
=
kmax
兺
k⬘=kmin
P共k⬘兩ki兲k⬘2␣
k2i ␣+1具k2␣+1典 . 具k典
共24兲
We thus have N⬘ = maxi兵C2ii其 and N = 冑N⬘ . For ␣ ⬍ −0.5, the total local coupling strength d is more homogeneous and, hence, N can be obtained by using the random matrix theory.14 In particular, N is determined by the maximum value of d to which nodes of the lowest degree contribute, i.e., N
␣ N ⬇ dmax = 兺 Aijkmin k␣j .
共25兲
j=1
再
冎
Summarizing results for both cases, we have N ⬇
␣+1 2␣+1 冑k2max 具k 典/具k典,
␣ ⬎ − 0.5
␣+1 ␣+1 kmin 具k 典/具k典,
␣ ⬍ − 0.5,
共26兲
where kmax can be obtained by averaging over different network realizations, and 具k␣+1典 and 具k典 are given by Eq. 共13兲. A comparison of results from Eq. 共26兲 and from simulation is shown in Fig. 5. We see that, except for the region around
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FIG. 5. 共Color online兲 For an ensemble of weighted, symmetrically coupled scale-free networks of N = 1500 nodes, reciprocal of the largest eigenvalue of the weighted adjacency matrix as a function of ␣ for different values of 具k典. Note that the average degrees are quite small. Data points are obtained by numerical simulations, where each point is the result of averaging over 20 random network realizations, and curves are predicted by Eq. 共26兲.
FIG. 6. 共Color online兲 For weighted, asymmetrically coupled scale-free networks, the quantity 1 / N vs ␣ for different values of the average degree 具k典. Data points are obtained by numerical simulations and curves are predicted by Eq. 共27兲. Other parameters are the same as those in Fig. 5.
N
␣ = −0.5, there is a reasonable agreement between analytical and numerical results. The interesting feature is the persistent existence of the abnormal and normal synchronization regimes, and the crossover point separating them at which the network synchronizability is apparently independent of, among others, the average degree of the network. For the asymmetric coupling case, N can be obtained in a similar way. In particular, for ␣ ⬎ −1.0, we have N
C2ii = 兺 Aijki␣k␣j = k␣i +1具k␣+1典/具k典. j=1
For ␣ ⬍ −1.0, we have N
di = 兺 Aijk␣j = ki具k␣+1典/具k典. j=1
We thus have N =
再
␣+1 ␣+1 冑kmax 具k 典/具k典,
␣ ⬎ − 1.0
␣+1
␣ ⬍ − 1.0.
kmin具k
典/具k典,
冎
共27兲
Figure 6 shows 1 / N as a function of ␣ for the asymmetric coupling scheme for weighted, sparse scale-free networks. There is a reasonable agreement between theoretical and numerical results. The key feature of Fig. 6 is similar to that of Fig. 5, i.e., the existence of the abnormal and normal synchronization regimes and a crossover point separating the two regimes.
V. COMPARISON WITH SYNCHRONIZATION IN NETWORKS OF IDENTICAL OSCILLATORS
For comparison with the results concerning the onset of synchronization in the Kuramoto networks, we consider synchronization in networks of identical oscillators1 for both coupling schemes, which can, in general, be described by
dxi = F共xi兲 + 兺 GijH共x j兲, dt j=1
共28兲
where H共x兲 is a linear coupling function, is the global coupling parameter, and G is the coupling matrix underlying the network topology. The matrix G satisfies the condition 兺Nj=1Gij = 0 for any i, where N is the network size. The variational equations are N
d␦xi = DF共s兲 · ␦xi − 兺 GijDH共s兲 · ␦x j , dt j=1
共29兲
where DF共s兲 and DH共s兲 are the Jacobian matrices of the corresponding vector functions evaluated at the synchronization state s共t兲. Diagonalizing the coupling matrix G yields a set of eigenvalues 兵⌳i , i = 1 , . . . , N其 and the corresponding normalized eigenvectors are denoted by e1 , e2 , . . . , eN. The eigenvalues are real and non-negative and can be sorted as 0 = ⌳1 ⬍ ⌳2 ⱕ ¯ ⱕ ⌳N. The smaller the ratio ⌳N / ⌳2, the stronger the synchronizability of the network.1 Figure 7 shows the eigenratio ⌳N / ⌳2 as a function of ␣ for both symmetric and asymmetric coupling schemes. One can see that the synchronizability of identical oscillators is different from that of the Kuramoto network. In particular, there are no crossover behavior and abnormal synchronization regime for both coupling schemes. Instead, the synchronizability is optimized for ␣ = −1, and higher connection density 具k典 leads to stronger synchronizability. In this case, our coupling schemes are equivalent to those studied in Ref. 2, and our results are consistent with previously obtained results. The different behaviors between the onset of synchronization in the Kuramoto network and the stability of synchronization state in networks of identical oscillators can partly be explained by the eigenvalue approach. As we have seen, the onset of synchronization in the Kuramoto network is determined by the largest eigenvalue of the adjacency matrix with zero diagonal elements, whereas the synchronizability of networks of identical oscillators is determined by the ratio of the largest and the second lowest eigenvalues of the coupling matrix. From a physical point of view, at the
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at the separating point does not depend on details such as the link density and the degree exponent. In fact, the value of c is determined only by the natural frequencies of oscillators, which are fixed when the networked dynamical system is initially defined. These results indicate that weighting can play a quite counterintuitive role in network synchronization. Understanding the effects of coupling weights on synchronization in large-scale complex networks has been an active area of ongoing research. Our work has provided some insights into the issue that may be useful for understanding synchronization-related phenomena in various complex physical, biological, and technological systems. ACKNOWLEDGMENTS
FIG. 7. 共Color online兲 For 共a兲 symmetric coupling scheme and 共b兲 asymmetric coupling scheme, ratio of the largest eigenvalue and the second lowest eigenvalue of the coupling matrix as a function of ␣ for different average degree 具k典 of scale-free networks of N = 2000 nodes. Each data point is the results of averaging over 20 random network realizations.
We thank X.-G. Wang for helpful discussions. W.X.W., L.H., and Y.C.L. were supported by AFOSR under Grant No. FA9550-07-1-0045, by ONR through WVHTC 共West Virginia High Technology Consortium Foundation兲, and by an ASU-UA Collaborative Program on Biomedical Research. G.R.C. was supported by the Hong Kong Research Grants Council under Grant No. GRF CityU 1117/08E. 1
onset of synchronization, partial synchronization emerges from a completely disordered state, while the synchronizability in networks of identical oscillators describes the stability of synchronization state with respect to perturbation. Hence, it is not unreasonable that the emergence of partial synchronization can exhibit quite distinct behaviors. VI. CONCLUSIONS
In this paper, we have investigated the onset of synchronization in weighted scale-free networks using the Kuramoto model as the platform. We have considered both symmetrical and asymmetrical coupling schemes. Our focus is on the effect of the weighting scheme, characterized by the weighting parameter ␣, on c, the critical global coupling strength required for coherence in the network to set in. For networks of relatively large values of the average degree, the meanfield theory has been used, which is the standard theoretical tool for treating Kuramoto networks. We have also studied the onset of synchronization in terms of the eigenvalue approach. Our findings can be summarized as follows. Regardless of the coupling scheme, for weighted scale-free networks there exist two regimes with opposite synchronization behaviors. In the normal regime, the network’s ability to synchronize can be enhanced by increasing the number of links in the network, while the opposite behavior occurs in the abnormal regime. A striking phenomenon is that, in the vicinity of the point separating the two regimes, network synchronization has little dependence on the average degree. That is, no matter how the number of links is changed in the network, insofar as the network is connected, the synchronization dynamics are invariant in the sense that the value of c
L. F. Lago-Fernández, R. Huerta, F. Corbacho, and J. A. Sigüenza, Phys. Rev. Lett. 84, 2758 共2000兲; M. B. Barahona and L. M. Pecora, ibid. 89, 054101 共2002兲; X. F. Wang and G. Chen, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 187 共2002兲; IEEE Trans. Circuits Syst., I: Regul. Pap. 49, 54 共2002兲; X. F. Wang, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 885 共2002兲; M. Timme, F. Wolf, and T. Geisel, Phys. Rev. Lett. 89, 258701 共2002兲; T. Nishikawa, A. E. Motter, Y.-C. Lai, and F. C. Hoppensteadt, ibid. 91, 014101 共2003兲; J. Ito and K. Kaneko, Phys. Rev. E 67, 046226 共2003兲; F. M. Atay, J. Jost, and A. Wende, Phys. Rev. Lett. 92, 144101 共2004兲; Y. Moreno and A. F. Pacheco, Europhys. Lett. 68, 603 共2004兲; P. G. Lind, J. A. C. Gallas, and H. J. Herrmann, Phys. Rev. E 70, 056207 共2004兲; L. Huang, K. Park, Y.-C. Lai, and K. Yang, Phys. Rev. Lett. 97, 164101 共2006兲; C.-Y. Yin, W.-X. Wang, G. Chen, and B.-H. Wang, Phys. Rev. E 74, 047102 共2006兲; C.-Y. Yin, B.-H. Wang, W.-X. Wang, and G.-R. Chen, ibid. 77, 027102 共2008兲. 2 A. E. Motter, C. Zhou, and J. Kurths, Phys. Rev. E 71, 016116 共2005兲; M. Chavez, D.-U. Hwang, A. Amann, H. G. E. Hentschel, and S. Boccaletti, Phys. Rev. Lett. 94, 218701 共2005兲; C. Zhou, A. E. Motter, and J. Kurths, ibid. 96, 034101 共2006兲; T. Nishikawa and A. E. Motter, Phys. Rev. E 73, 065106共R兲 共2006兲. 3 Y. Kuramoto, Chemical Oscillations, Waves and Turbulence 共SpringerVerlag, Berlin, 1984兲. 4 S. H. Strogatz, Physica D 143, 1 共2000兲. 5 J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, Rev. Mod. Phys. 77, 137 共2005兲. 6 J. G. Restrepo, E. Ott, and B. R. Hunt, Phys. Rev. E 71, 036151 共2005兲; Chaos 16, 015107 共2006兲. 7 S. Guan, X. Wang, Y.-C. Lai, and C.-H. Lai, Phys. Rev. E 77, 046211 共2008兲. 8 A.-L. Barabási and R. Albert, Science 286, 509 共1999兲. 9 A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani, Proc. Natl. Acad. Sci. U.S.A. 101, 3747 共2004兲; P. J. Macdonald, E. Almaas, and A.-L. Barabási, Europhys. Lett. 72, 308 共2005兲. 10 T. Ichinomiya, Phys. Rev. E 70, 026116 共2004兲. 11 J. Gómez-Gardeñes, Y. Moreno, and A. Arenas, Phys. Rev. Lett. 98, 034101 共2007兲. 12 T. Kato, Perturbation Theory for Linear Operators 共Springer-Verlag, Berlin, 1995兲. 13 D.-H. Kim and A. E. Motter, Phys. Rev. Lett. 98, 248701 共2007兲. 14 E. P. Wigner, Ann. Math. 53, 36 共1951兲; T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Rev. Mod. Phys. 53, 385 共1981兲.
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Arm splitting and backfiring of spiral waves in media displaying local mixed-mode oscillations Qingyu Gao,1,2 Lu Zhang,1 Qun Wang,1 and I. R. Epstein2 1
College of Chemical Engineering, China University of Mining and Technology, Xuzhou 221008, People’s Republic of China 2 Department of Chemistry and Volen Center for Complex Systems, MS 015, Brandeis University, Waltham, Massachusetts 02454-9110, USA
共Received 16 December 2008; accepted 10 February 2009; published online 17 March 2009兲 The behavior of spiral waves is investigated in a model of reaction-diffusion media supporting local mixed-mode oscillations for a range of values of a control parameter. This local behavior is accompanied by the formation of nodes, at which the arms of the simple spiral waves begin to split. With further parameter changes, this nodal structure loses stability, becoming quite irregular, eventually evolving into turbulence, while the local dynamics increases in complexity. The breakup of the spiral waves arises from a backfiring instability of the nodes induced by the arm splitting. This process of spiral breakup in the presence of mixed-mode oscillations represents an alternative to previously described scenarios of instability of line defects and superspirals in media with perioddoubling and quasiperiodic oscillations, respectively. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3093047兴 Spiral waves are ubiquitous in nonlinear science, appearing in a wide range of biological, chemical, and physical systems. In many systems, simple spirals evolve into more complex structures, and several scenarios have been identified by which this process can take place. Often, formation of complex spiral structures leads to spiral breakup and spiral turbulence. Again, generic scenarios by which this behavior emerges have been studied. In this paper, we identify through the numerical investigation of a model of two coupled allosteric enzymes a novel mechanism, characterized by the splitting of spiral arms, that produces complex spiral waves, which can subsequently evolve into spiral turbulence via a distinctive breakup process involving collision of waves with backfired segments expelled by their neighbors. I. INTRODUCTION
Spiral waves are perhaps the most common dissipative structures1–5 and have been observed in diverse systems such as galaxies,6 populations of a unicellular slime mold,7 premixed flames,8 the Belousov–Zhabotinsky chemical reaction,9 intracellular Ca2+ release from Xenopus laevis oocytes,10 and chicken retina.11 Because of their ubiquity in nature and significance in human health,12 spiral waves have attracted attention from both experimental and theoretical researchers seeking generic principles that govern the various spiral-generating systems. Earlier studies focused mainly on simple spiral waves, but these shed little light on the more complex structures in natural spiral patterns such as galaxies, typhoons, seashells, and lichens. Recently, attention has shifted to investigations of complex spiral waves, and four classes of mechanisms responsible for their generation have been identified: 共1兲 tip meandering, which induces the formation of superspiral structures;13 共2兲 period-doubling bifurcation of the local dynamics, which leads to the formation of 1054-1500/2009/19共1兲/013135/6/$25.00
line defects on the period-2n spiral waves;14,15 共3兲 transverse wave instability, which results in the formation of rippling spiral arms16 or segmentation of spiral segments in the presence of a fast-diffusing inhibitor;17 共4兲 interaction of two steady states, one excitable and the other pseudo-Turing unstable, which also induces the formation of segmented spiral waves.18 The formation of complex spiral structures usually portends the onset of spiral turbulence, which occurs via spiral breakup. To date, two principal mechanisms for the breakup of spirals have been described: 共1兲 approach of two neighboring spiral fronts until the local wavelength is below the critical value allowed by the dispersion relation;19–21 共2兲 transverse instability of line defects in period-2 spiral waves.22 In this paper, using a three-variable reaction-diffusion 共RD兲 model, we elucidate a new mechanism that generates complex spiral waves, which can evolve to spiral turbulence via a distinctive breakup process, in which waves break on collision with backfiring wave segments of nearby waves.
II. MODEL
The Decroly–Goldbeter model is based on a homogeneous model23 of two coupled allosteric enzymes, which was developed to investigate systems in which two instabilitygenerating mechanisms are coupled. In the original system, the instabilities arise from the positive feedback of reaction products on the two enzymes phosphofructokinase and adenylate cyclase.24 The RD version of the model has been applied to the study of spiral waves with superimposed target waves.25 The model equations are
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FIG. 1. 共Color兲 Spiral waves and bifurcation diagram. 共a兲 Snapshot of the spatial distribution of species  at ks = 1.0 s−1 showing a simple spiral wave. 共b兲 Magnified three-dimensional concentration profile of the region in the rectangle in 共a兲. 共c兲 Plot of the oscillation maxima of  vs distance r from the spiral tip in 共a兲. 共d兲 Snapshot of the spatial distribution of species  at ks = 0.84 s−1. 共e兲 Magnified threedimensional concentration profile of the rectangle in 共d兲 showing a spiral arm with a one-hump oscillatory tail. 共f兲 Plot of the oscillation maxima of  vs distance r from the spiral tip in 共d兲. Other parameters: / Km1 = 0.1 s−1, 1 = 2 = 10.0 s−1, q1 = 50.0, q2 = 0.02, L1 = 5.0⫻ 108, L2 = 100.0, d = 1.0⫻ 10−6, and D = 1.0⫻ 10−6 cm2 s−1. Size of the reactive medium is 1000⫻ 1000 grid points.
␣ = 共/Km1兲 − 1⌽ + Dⵜ2␣ , t  = q11⌽ − 2 + Dⵜ2 , t
共1兲
␥ = q22 − ks␥ + Dⵜ2␥ , t with ⌽ = ␣共1 + ␣兲共1 + 兲2/关L1 + 共1 + ␣兲2共1 + 兲2兴 and
= 共1 + d兲共1 + ␥兲2/关L2 + 共1 + d兲2共1 + ␥兲2兴,
共2兲
where the variables ␣, , and ␥ represent three biochemical species. The parameters , Km1, 1, 2, q1, q2, L1, L2, d, and ks are determined by the reaction conditions. Detailed discussions of the model can be found in Ref. 23. We set the diffusion coefficients of all three species, denoted by D, equal to 1.0⫻ 10−6 cm2 s−1 in this study. The simulations were carried out with the Euler integration method using a spatial grid dx = dy = 0.002 cm and an integration time step dt = 0.02 s. With smaller spatial grids and time steps, the results were essentially unchanged. As in earlier studies,23 we took ks as the control parameter and fixed the other parameters. Zero-flux boundary conditions were used. III. RESULTS
The model supports simple spiral waves 关see Figs. 1共a兲 and 1共b兲兴 over a wide range of parameters. When we decrease ks from 1.0 to 0.84 s−1 as shown in Figs. 1共d兲 and 1共e兲, the uniform arm 关Fig. 1共b兲兴 spawns a second, adjunct arm 关Figs. 1共e兲兴. In Figs. 1共c兲 and 1共f兲 we plot the oscillation maxima of the species  against the distance r from the spiral core 共r = 0兲 at ks = 1.0 and 0.84 s−1, respectively. We define the symbol M N to represent an oscillation with M large and N small peaks per cycle. We see that the local
dynamics in the radial direction consists of 共simple兲 10 oscillations at ks = 1.0 s−1, but for ks = 0.84 s−1, beyond r ⬇ 66,26 the local dynamics bifurcates to 11 oscillations. This transformation of the local behavior coincides with the minimum distance at which splitting of a spiral arm takes place. If we continue to decrease the control parameter ks, the spiral arms begin to split beyond a critical distance from the spiral core 关Figs. 2共a兲 and 2共b兲兴. The split waves rejoin a small distance away from the splitting point, producing a nodal structure within the spiral arm, and a new split can occur near the new node 关see Fig. 2共b兲兴, where we use the term node to refer to a point at which an arm begins to divide. Accompanying the formation of the nodal structure within the spiral arm is a series of bifurcations of the local dynamics as we move out from the spiral core. As shown in Fig. 2共c兲, the local dynamics transforms from 10 to 11 oscillations at r = 15 and then to a regime of 12 关Fig. 2共d兲兴 and 13 关Fig. 2共e兲兴 mixed-mode oscillations at r = 142. The spatial bifurcation points where the local dynamics bifurcates from 1S to 1S+1 mixed-mode oscillations move inward as ks is lowered. At ks = 0.6 s−1, the nodal structure some distance away from the spiral core loses stability and becomes more irregular 关see Figs. 3共a兲 and 3共b兲兴. As this complex spatial structure emerges with decreasing ks, the bifurcation radii continue to move inward, so that at ks = 0.6 s−1, 10 oscillations transform to 11 at r = 10, and the regime of 11 oscillations gives way to 12, 13, and newly emerging 14 oscillations at r = 77 关Fig. 3共c兲兴. Figures 3共d兲 and 3共e兲, respectively, show 14 and 13 mixed-mode oscillations at two representative spatial points. When ks reaches 0.57 s−1, the irregular nodes dominate the entire spiral wave, except very near the spiral core 关Fig. 4共a兲兴, and the radial dynamics 关see Fig. 4共b兲兴 displays not only periodic 1N mixed-mode oscillations 关see Fig. 4共c兲, showing 13 oscillations兴, as found at higher kS, but more complex mixed-mode oscillations 关Figs. 4共d兲–4共f兲兴 as well. At ks = 0.56 s−1, the spiral wave begins to break up, as shown in Fig. 5共a兲. The breakup first occurs near the spiral core, resulting in a central small spiral fragment surrounded
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FIG. 2. 共Color兲 Spiral waves at ks = 0.7 s−1. 共a兲 Snapshot of the spatial distribution of  共scale at right兲. 共b兲 Magnification of the region in 共a兲 indicated by the white rectangle. 共c兲 Plot of the oscillation maxima of  vs r. 关共d兲 and 共e兲兴 Time series at points with r = 176 and 446, respectively. Other parameters are as in Fig. 1.
by a much larger spiral fragment. Later, the large fragment starts to split at the lower left corner of the medium. Further decreases in the control parameter cause the central and surrounding spiral fragments to rupture into still more fragments 关Figs. 5共b兲–5共d兲兴. The central spiral fragment disappears when ks is decreased to 0.5 s−1, as shown in Fig. 5共c兲. As long as ks ⬎ 0.5 s−1, each spiral fragment retains the arm splitting and nodal structures of the spiral arms 关see Figs. 5共a兲 and 5共b兲兴 as well as a sense of rotation, and the medium still supports local 10, 11, and more complex mixedmode oscillations in different regions. At ks = 0.50 s−1, the density of nodes significantly increases, and each spiral fragment contains multiple nodes 关Fig. 5共c兲兴. Although the spiral waves have dissociated in Figs. 5共a兲–5共c兲, the fragments maintain their integrity, and the patterns are not yet turbulent. At still lower ks, e.g., 0.46 s−1 as in Fig. 5共d兲, the pattern is dominated by continuously generated and annihilated frag-
FIG. 3. 共Color兲 Spiral waves at ks = 0.6 s−1. 共a兲 Snapshot of the spatial distribution of . 共b兲 Magnification of the region in 共a兲 indicated by the white rectangle. 共c兲 Plot of the oscillation maxima of  vs r. 关共d兲 and 共e兲兴 Time series at points with r = 346 and 446, respectively. Other parameters are as in Fig. 1.
ments and evolves into turbulence, while the local dynamics becomes aperiodic. A detailed view of the onset of spiral breakup is shown in Fig. 6, which shows the evolution of the system when ks is switched from 0.57 to 0.56 s−1. After ks is decreased, a section near the spiral core soon becomes quite complex 关Fig. 6共a兲兴, and 40 s later three wave segments are backfired from this section of the wave 关white arrows in Fig. 6共b兲兴. These segments propagate inward and collide with the neighboring outwardly propagating wave, as shown in Figs. 6共c兲 and 6共d兲, causing the wave to rupture at the collision sites. The spiral fragments formed due to these breaks propagate outward and continue rotating, growing, and colliding, as shown in Fig. 6共e兲, where a new backfired wave segment, indicated by the white arrow, can be observed. This segment collides with the central spiral fragment to create a crack, from which the spiral arm recovers without rupturing 关see Fig. 6共f兲兴.
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(c) FIG. 5. 共Color兲 Snapshots of the spatial distribution of . ks = 共a兲 0.56 s−1, 共b兲 0.54 s−1, 共c兲 0.5 s−1, and 共d兲 0.46 s−1. Other parameters are as in Fig. 1.
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FIG. 4. 共Color兲 Spiral waves at ks = 0.57 s−1. 共a兲 Snapshot of the spatial distribution of . 共b兲 Plot of the oscillation maxima of  vs r. 关共c兲–共f兲兴 Time series at points with r = 96, 176, 296, and 446, respectively. Other parameters are as in Fig. 1.
Note that the wave segments indicated by white arrows in Fig. 6共f兲 are generated by collision of spiral fragments rather than by backfiring, which cannot increase the number of spiral fragments 关see Fig. 6共g兲兴. About 800 s later, a new wave segment 关indicated by the white arrow in Fig. 6共h兲兴 is backfired from an arm near the lower left corner of the medium and leads to the wave fracture seen in Fig. 6共i兲. Figure 6共j兲 presents a time series 共t兲 at the point marked by the white square in Fig. 6共f兲. We note a long quiescent period, denoted by the black arrow, which results from the annihilation of the wave in that region by a large backfired wave segment.
IV. DISCUSSION
Our investigation of the mixed-mode spiral waves has been performed in a parameter region that supports only simple 10 oscillations in the homogeneous system. At all parameters where a definable core exists, the trajectory of the spiral tip is a small regular circle. How then does the instability of the simple spiral waves occur? To answer this question, we studied a one-dimensional 共1D兲 analog of the spiral waves: a 1D system with Dirichlet boundary conditions at one end and the usual zero-flux boundary conditions at the opposite end. The values of the variables at the fixed Dirichlet source were set at the unstable steady state, corresponding to the values in the spiral tip. This “1D spiral” approximates the radial dynamics of the two-dimensional 共2D兲 spiral waves. Figure 7 shows the variation of the maxima of the  oscillations as a function of ks at spatial point 301 in the 1D spiral. As ks decreases, the temporal dynamics at this point undergoes a series of bifurcations: the oscillations transform from 10 to 11, 12, 13, and more complex mixed-mode oscillations, mirroring the evolution of the temporal dynamics in the 2D spiral waves. Hence we conclude that the formation of spiral waves with mixed-mode local dynamics is due to the bifurcations of the radial dynamics of the simple spiral waves. Quantitative differences in the values of the bifurcation parameters and, more importantly, the generation of complex spatial behavior, result from 2D, notably curvature, effects. We have described here a new scenario of spiral breakup, which proceeds from spiral arm splitting, nodes, adjunct arms, and backfiring through turbulence. Although the full scenario has not yet been clearly demonstrated experimentally, some reported experiments and simulations
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FIG. 6. 共Color兲 Evolution of spiral breakup. ks is changed from 0.57 to 0.56 s−1 at t = 0, with Fig. 5共a兲 as the initial state. t = 共a兲 620 s, 共b兲 660 s, 共c兲 700 s, 共d兲 800 s, 共e兲 900 s, 共f兲 1000 s, 共g兲 1100 s, 共h兲 1900 s, and 共i兲 2000 s. 共j兲 Time series at point 共580, 570兲, denoted by the white square in 共f兲. Other parameters are as in Fig. 1.
show indications of the phenomena described here. Spiral waves with adjunct arms have been observed in aggregating populations of the amoeba Dictyostelium discoideum27 and in simulations of a FitzHugh–Nagumo model,28 but the further evolution of these structures has not been described. Spontaneous backfiring of traveling waves has been seen in a numerical simulation of an excitable Bär–Eiswirth model,29 in another two-variable model with three fixed points in the
local dynamics,30 and in experiments on an excitable, lightsensitive Belousov–Zhabotinsky reaction.31 We note that, while a minimum of three concentration variables is required to generate mixed-mode oscillations in a homogeneous system, RD systems are essentially infinite dimensional, so that local mixed-mode behavior can arise even with only two chemical species. Here we have demonstrated the local mixed-mode oscillations that give rise to arm splitting, nodes, and even backfiring breakup of a spiral. In the Decroly–Goldbeter RD model, the mixed-mode oscillations result from a homoclinic connection via collision of a saddle point and a limit cycle to form the spiral-type attractor. The local mixed-mode dynamics is critical in determining the fine structure of the spiral arm. It is well known that the properties of the spiral tip, which is the source 共or in less common antispirals,32 the sink兲 of the spiral wave plays a key role in the spiral dynamics. In most models studied to date, the tip of the spiral is an unstable focus, but here the spiral tip is a saddle point, which may be the underlying cause of the bifurcations found in the radial dynamics. Studies of complex chemical oscillations in homogeneous systems have revealed several bifurcation scenarios starting from simple oscillations: quasiperiodicity, period doubling, and mixed-mode oscillations, all of which can ultimately lose stability and give way to chaos as a control parameter is varied.33 Bifurcations of spiral dynamics in RD systems are found to mirror the behavior of the corresponding homogeneous systems: simple spiral waves with simple local dynamics can evolve to complex spiral waves with quasiperiodic local dynamics and then to spiral turbulence13,25 or can proceed via period-doubling local dynamics to complex spiral waves and then to spiral turbulence.14 Here we have studied a new route from simple to complex spiral waves containing such features as adjunct arms and nodes via bifurcation of the local dynamics into mixed-mode oscillations and then to spiral breakup through arm splitting and backfiring. The radial distance from the spiral core and the model parameters decide the complexity of the local oscillations. ACKNOWLEDGMENTS
This work was supported by Grant No. CHE-0615507 from the U.S. National Science Foundation, Grant No. 20573134 from the National Science Foundation of China, and Grant No. 05-0477 from the Plan of Supporting the New Talents of the New Century. Q.G. is grateful for the financial support of the visiting program to Brandeis University from the Chinese Scholarship Council. The authors are indebted to Jun Li for checking the calculations and for his graphic work. A. T. Winfree, When Time Breaks Down 共Princeton University Press, Princeton, 1987兲. 2 A. S. Mikhailov and K. Showalter, Phys. Rep. 425, 79 共2006兲. 3 M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 共1993兲. 4 L. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99 共2002兲. 5 Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence 共SpringerVerlag, Berlin, 1984兲. 6 R. Irion, Science 307, 64 共2005兲. 7 J. J. Tyson and J. D. Murray, Development 106, 421 共1989兲. 8 S. K. Scott, J. C. Wang, and K. Showalter, J. Chem. Soc., Faraday Trans. 93, 1733 共1997兲. 1
FIG. 7. Bifurcation diagram obtained by the simulation of a 1D spiral showing the maxima of the oscillations of  as a function of ks at a representative spatial point. Inset shows bifurcation sequence of the 1D spiral in the parameter region in which the 2D spiral waves are studied. Size of the system is 501. Other parameters are as in Fig. 1.
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O. Decroly and A. Goldbeter, Proc. Natl. Acad. Sci. U.S.A. 79, 6917 共1982兲. 24 A. Goldbeter and S. R. Caplan, Annu. Rev. Biophys. Bioeng. 5, 449 共1976兲. 25 L. Zhang, Q. Y. Gao, Q. Wang, H. Wang and J. C. Wang, Phys. Rev. E 74, 046112 共2006兲. 26 All positions r and distances are measured in grid points, which are separated by 0.002 cm, with r = 0 located at the center of the spiral core. 27 K. J. Lee, E. C. Cox, and R. E. Goldstein, Phys. Rev. Lett. 76, 1174 共1996兲. 28 K. J. Lee, Phys. Rev. Lett. 79, 2907 共1997兲. 29 M. Bär and M. Eiswirth, Phys. Rev. E 48, R1635 共1993兲. 30 M. Or-Guil, J. Krishnan, I. G. Kevrikidis, and M. Bär, Phys. Rev. E 64, 046212 共2001兲. 31 T. Okano, Y. Matsuda, and K. Miyakawa, Phys. Rev. E 74, 066103 共2006兲. 32 V. K. Vanag and I. R. Epstein, Science 294, 835 共2001兲. 33 S. K. Scott, Chemical Chaos 共Oxford University Press, Oxford, 1991兲.
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A dynamical systems proof of Kraft–McMillan inequality and its converse for prefix-free codes Nithin Nagaraja兲 Department of Electronics and Communications Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Amritapuri Campus, Kerala 690525, India
共Received 10 October 2008; accepted 23 January 2009; published online 24 March 2009兲 Uniquely decodable codes are central to lossless data compression in both classical and quantum communication systems. The Kraft–McMillan inequality is a basic result in information theory which gives a necessary and sufficient condition for a code to be uniquely decodable and also has a quantum analogue. In this letter, we provide a novel dynamical systems proof of this inequality and its converse for prefix-free codes 共no codeword is a prefix of another—the popular Huffman codes are an example兲. For constrained sources, the problem is still open. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3080885兴 The connection between classical data communications and physics goes back to Shannon’s paper in 1949 where he defined “entropy” (term borrowed from statistical thermodynamics) as the amount of information content of a message source.1 He showed that the ultimate limit of lossless data compression in the absence of noise is bounded by the entropy of the source. Quantum mechanics has provided an alternate form of data communications by altering the physical states of quantum systems. The other branch of physics that is being increasingly applied to communications is nonlinear dynamics/chaos. Data compression is a fundamental block in both classical and quantum communication systems. Lossless data compression is often a necessity in scenarios requiring high accuracy such as medical imaging applications. Uniquely decodable codes are central to lossless data compression irrespective of the mode of communication (classical or quantum). A subset of this class is the prefix-free codes, where no codeword is a prefix of another and enable every encoded message to be uniquely decoded without the need for special markers for delimiting successive codewords. They provide instantaneous decoding which avoids delay in practical applications. Huffman codes,2 Shannon–Fano codes,1 and arithmetic codes3 are a few examples of prefix-free codes. Quantum versions of prefix-free codes also exist.4–6 The Kraft–McMillan inequality7 is a basic result in information theory which gives a necessary and sufficient condition for a code to be uniquely decodable and also has a quantum analogue.4 We provide a new proof of this inequality and its converse for (classical) prefix-free codes by a dynamical systems approach. I. INTRODUCTION
The central problem of classical and quantum communications is to transmit information from the sender to the receiver in an efficient, secure, and robust way. The message is compressed for efficiency of transmission, subsequently ena兲
Electronic mail: [email protected]. URL: nithin.nagaraj.googlepages.com.
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crypted using cryptographic algorithms to make it secure, and finally protected with error correction codes for robustness to noise induced errors. In 1948, Shannon formulated a mathematical theory of communication where he was the first to define entropy 共the term was borrowed from statistical thermodynamics兲 as a measure of information content and further showed that the fundamental limit of lossless data compression in the absence of noise is bounded by the entropy of the message source.1 Information and coding theory was thus born and its goal is to study the fundamental limits to efficient and robust transmission while providing practical solutions 共algorithms兲 for achieving these limits. Quantum mechanics provides an alternate way of communication by employing physical information in the form of states of a quantum system. Quantum information theory studies the fundamental principles behind such a “physical” form of communications.8 In particular, quantum data compression is a growing area of research.9 Ever since Shannon established this link between communications and physics, there has been a surge of research activity in employing the tools of physics, especially nonlinear dynamics/chaos theory and quantum mechanics to theoretical and computational aspects of communications. Nonlinear dynamics or chaos theory has been used to model and analyze information sources.10 Recently, we have discovered a connection between a popular lossless compression algorithm and chaotic dynamical systems.11 Lossless compression is normally achieved by assigning codewords to the alphabet in such a fashion that frequently occurring alphabets get shorter codewords 共e.g., the Morse code used in telegraphy兲, thereby enabling savings in storage and transmission. Uniquely decodable codes are an important class of lossless compression codes which allow for unambiguous decoding. Prefix-free codes 共no codeword is a prefix of another兲, a subset of uniquely decodable codes, provide instantaneous decoding which makes it very useful in practical applications as it avoids delay. The most popular example of a uniquely decodable code which is also prefixfree is the Huffman code2 which is used in JPEG,12 the older
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international standard for still image compression. Besides Huffman codes, there are a number of uniquely decodable prefix-free codes which are routinely used in lossless data compression applications 共e.g., Shannon–Fano codes and arithmetic codes兲. Quantum versions of prefix-free codes also exist. Schumacher and Westmoreland proposed prefixfree codes for their implementation of variable-length quantum coding.4 Braunstein et al. developed a quantum analogue of Huffman coding.5 Chuang and Modha developed a quantum version of arithmetic coding.13 In this letter, we are interested in a famous inequality due to Kraft7 for determining whether a given code is uniquely decodable or not. McMillan proved this inequality by employing arguments from complex analysis.14 Subsequently, Karush15 gave a simpler proof of the inequality by using a counting argument. The Quantum Kraft inequality was derived for the first time by Schumacher and Westmoreland.4 In this work, a new proof of the classical Kraft–McMillan inequality and its converse for prefix-free codes using a dynamical systems approach is derived. We also indicate an open problem related to the inequality for 共classical兲 constrained sources 共a quantum version of this open problem is not investigated in this paper兲. II. UNIQUE DECODABILITY AND PREFIX-FREE CODES
Let W+ denote the set of all strings of finite length composed of elements from the set W. Thus, 兵0 , 1其+ = 兵0 , 1 , 00, 01, 10, 11, 000, 001, . . .其. A binary symbol code C is a mapping from the alphabet set A = 兵a1 , a2 , . . . , aN其 to 兵0 , 1其+. c共x兲 will denote the codeword corresponding to x and l共x兲 will denote its length. The extended code C+ is a mapping from A+ to 兵0 , 1其+ obtained by concatenation, without punctuation, of the corresponding codewords: c+共x1x2 ¯ xk兲 = c共x1兲c共x2兲 ¯ c共xk兲.
共1兲 16
Definition: A code C is uniquely decodable if, under the extended code C+, no two distinct strings have the same encoding, i.e., ∀x,y 苸 A+,
x ⫽ y ⇒ c+共x兲 ⫽ c+共y兲.
共2兲
Prefix-free code: A symbol code C is called a prefix-free16 code 共sometimes also known as prefix code兲 if no codeword is a prefix of any other codeword. A prefix-free code is also known as a self-punctuating code or instantaneous code because, an encoded string can be decoded from left to right instantaneously, without looking ahead to subsequent codewords. The end of each codeword is immediately recognizable. A prefix-free code is uniquely decodable.16,17 Examples: The code C1 = 兵0 , 11011其 is prefix-free code since 0 is not a prefix of 11011. The code C2 = 兵0 , 01, 110, 0101其 is not a prefix-free code since 0 is a prefix of 01 and 0101 and further 01 is a prefix of 0101. Similarly C3 = 兵1 , 11011其 is not a prefix-free code since 1 is a prefix of 11011. While it is true that all prefix-free codes are uniquely decodable, non-prefix-free codes can also be uniquely decodable. For example, C3 as given above is uniquely decodable
FIG. 1. Kraft–McMillan inequality and converse are satisfied by the entire set of uniquely decodable codes for unconstrained sources. For constrained sources, the problem is still open.
although it is not prefix-free. Prefix-free codes play a very important role in communications. The popular Huffman codes2 which is used in JPEG,12 the older international standard for still image compression, is a prefix-free variablelength code 共different codewords need not necessarily have the same length兲. The Kraft–McMillan inequality gives a necessary and sufficient condition for unique decodability 共only for unconstrained sources, see Fig. 1兲. A new dynamical system proof of the inequality along with its converse 共for prefix-free codes兲 will be provided in the next section. III. A DYNAMICAL SYSTEM PROOF OF THE KRAFT–McMILLAN INEQUALITY FOR PREFIX-FREE CODES
Given a uniquely decodable binary prefix-free code C for an alphabet A, the codewords c1 , c2 , . . . , cN with lengths l1 , l2 , . . . , lN necessarily satisfy N
2−l ⱕ 1, 兺 i=1
共3兲
i
where N = 兩A兩, the cardinality of set A. We shall work with binary codes in this paper; however, it is easy to extend our arguments for nonbinary codes. A. The binary map
Consider the binary map 关Fig. 2共a兲兴 T : 关0 , 1兲 → 关0 , 1兲 defined as x 哫 2x,
0 ⱕ x ⬍ 21 ,
哫2x − 1,
1 2
ⱕ x ⬍ 1.
It is well known that the binary map is a nonlinear chaotic dynamical system, which preserves the Lebesgue measure 共ordinary length measure兲.18 Furthermore, every initial condition in 关0,1兲 has a unique symbolic sequence and every finite length 共⬎0兲 symbolic sequence corresponds to a subset of 关0,1兲 of nonzero measure. Since the binary map has the maximum topological entropy for two symbols 关=ln共2兲兴, all possible arrangements of 0 and 1 can occur in its space of symbolic sequences. We shall prove two simple lemmas re-
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FIG. 2. 共a兲 Binary map. 共b兲 Symbolic sequences of length 1. 共c兲 Symbolic sequences of length 2. 共d兲 Symbolic sequences of length 3.
garding the binary map which will be used to prove the Kraft–McMillan inequality. Lemma 1: Given any sequence 共or string兲 S of 0 and 1 of length m, there exists an unique interval on the binary map of length 2−m such that all initial conditions in that interval will have S as the binary symbolic sequence corresponding to the first m iterations. Proof: Consider the given string S of length m as a binary prefix in 关0,1兲 共i.e., think of S as the real number 0 S in ¯ , 0.S1 ¯ 兲, where the binary representation兲. The interval 关0.S0 overline indicates infinite repetition, consists of all possible binary numbers in 关0,1兲 which have S as the desired prefix. All these binary numbers when fed as initial conditions to the binary map will yield S as the symbolic sequence in m iterations 共this is because the binary map can be seen as a shift map which spits out the leading bits of the binary representation of the initial condition兲. The length of this inter¯ − 0 S0 ¯ which is 2−m. 䊐 val is 0.S1 Lemma 2: Two symbolic sequences S1 and S2 of lengths m1 and m2, respectively, which are not prefixes of each other, correspond to two disjoint intervals of lengths 2−m1 and 2−m2, respectively. Proof: From Lemma 1, we know that S1 and S2 correspond to two intervals I1 and I2 of lengths 2−m1 and 2−m2, respectively 共both are subsets of 关0,1兲兲. All that we need to show is that I1 and I2 are disjoint whenever S1 and S2 are prefix-free. Without loss of generality, assume that m1 ⬎ m2 which implies that I2 is larger than I1 共since 2−m2 ⬎ 2−m1兲. Suppose that the intervals are not disjoint. This implies that the intersection of the two intervals is not a null set. This means that there exists at least one initial condition which belongs to both I1 and I2. Call this initial condition as x. When x is iterated, since x 苸 I1 the symbolic sequence of x should begin with “S1¯.” Also, since x 苸 I2, its symbolic sequence should begin with “S2¯.” Since there is only one unique symbolic sequence for every point in the binary map, these two symbolic sequences should be the same: “S1 ¯ ” = “ S2 ¯ ”. Since m1 ⬎ m2, this implies that S2 is a prefix of S1
which is a contradiction. Thus, the two intervals I1 and I2 are disjoint. 䊐 Converse of Lemma 2: Two disjoint intervals I1 and I2 共subsets of 关0,1兲兲 have respective symbolic sequences S1 and S2 on the binary map with lengths m1 and m2, such that they are not prefixes of each other. Proof: Without loss of generality, assume that S2 is a prefix of S1 共m1 ⬎ m2兲, which implies S1 = S2Q where Q is a finite length string of 0 and 1. By a backward iteration of the symbolic sequence S1, we get an initial condition, say, x, which belongs to I1 共since I1 corresponds to all initial conditions whose symbolic sequence begins with S1兲. The point x also belongs to I2 since the symbolic sequence of x is S2Q. This implies that I1 and I2 are not disjoint, which is a contradiction. Thus, S2 is not a prefix of S1. 䊐 Proof of the Kraft–McMillan inequality: Since c1 , c2 , . . . , cN with lengths l1 , l2 , . . . , lN are prefix-free codes, using Lemmas 1 and 2, these can be seen as symbolic sequences of disjoint intervals of the binary map on 关0,1兲 with lengths 2−l1 , 2−l2 , . . . , 2−lN respectively. Any finite collection of disjoint intervals with lengths 2−l1 , 2−l2 , . . . , 2−lN and which are subsets of 关0,1兲 necessarily satisfy Eq. 共3兲. 䊐 IV. PROOF OF THE CONVERSE OF THE KRAFT–McMILLAN INEQUALITY
Given a set of codeword lengths that satisfy Eq. 共3兲, there exists a uniquely decodable binary prefix-free code with these codeword lengths. Proof: Let l1 , l2 , . . . , lM be the specified distinct codeword lengths such that they satisfy Eq. 共3兲. Without loss of generality, let us assume that l1 ⬍ l2 ⬍ ¯ ⬍ l M . Let there be a1 codewords of length l1, a2 codewords of length l2, and so on up to a M codewords of length l M . The Kraft–McMillan inequality can be rewritten as M
a i2 兺 i=1
M
−li
ⱕ 1,
ai = N, 兺 i=1
共4兲
where N = 兩A兩 as before. Let us determine the maximum number of codewords that can have a particular codeword length li while still satisfying Eq. 共4兲. If there are 2li + 1 or more codewords with length li, then 共2li + 1兲2−li = 1 + 2−li ⬎ 1, violating Eq. 共4兲. Thus there can at most be 2li codewords of length li. Let us begin with l1. We know that there are exactly 2l1 disjoint intervals with length 2−l1 on the binary map which have symbolic sequence of length l1. Since the intervals are disjoint, the symbolic sequences are necessarily prefix-free codewords 共by the converse of Lemma 2兲. We first assign the symbolic sequences of a1 of these disjoint intervals as codewords. Once a1 disjoint intervals of length 2−l1 are used up, we have lost a12l2−l1 intervals of length 2−l2. The number of available disjoint intervals of length 2−l2 is 2l2 − a12l2−l1. If a2 ⬍ 2l2 − a12l2−l1 then we can allocate disjoint intervals to a2 codewords of length l2. This requires a2 ⬍ 2l2共1 − a12−l1兲, which reduces to a12−l1 + a22−l2 ⬍ 1 which is necessarily true from Eq. 共4兲. Thus we can use the symbolic sequence of a2 disjoint intervals as prefix-free codewords 共of length l2兲. This argument is repeated for a3 and so on until we have allocated
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1/2
A
1/4 1/2 1/4 FIG. 3. Converse of the Kraft–McMillan inequality. 共a兲 Assigning intervals to codewords of lengths 兵3, 3, 2, 2其 which satisfies the Kraft–McMillan inequality. 共b兲 Assigning intervals to codewords of lengths 兵3, 3, 2, 1其 which satisfies the Kraft–McMillan inequality with equality. This is known as a complete code.
unique disjoint intervals to all codewords 共see example in Fig. 3兲. We have thus proved the converse of the Kraft– McMillan inequality by construction of prefix-free codewords using symbolic sequences of disjoint intervals on the binary map. 䊐 A. Comment on the proof
We could have chosen a different chaotic map 共e.g., the tent map or the logistic map兲 for the proof and not necessarily the binary map. The key aspect of dynamical systems that is used in the proof is the fact that the binary map has maximum topological entropy for two alphabets, i.e., all possible arrangements of 0 and 1 of all possible finite lengths can occur. This enabled us to associate codewords with symbolic sequences. Note that the binary map is not just a pedagogical tool but a Generalized Luröth Series 共GLS兲 which is chaotic, Lebesgue measure preserving, and ergodic 共in fact, Bernoulli兲.18 In another related work, we have recently shown that GLS can achieve Shannon’s entropy rate for independent and identically distributed sources and thus provide optimal lossless data compression.11 The Kraft–McMillan inequality is very important since it enables a simple proof of Shannon’s inequality 共here pi refers to the probability of the ith message alphabet and H is Shannon’s entropy of the message source measured in bits/ symbol兲: N
N
i=1
i=1
兺 pili ⱖ − 兺 pi log2共pi兲 = H.
共5兲
The above inequality is actually a version of the noiseless source coding theorem for symbol codes.16 It is important because it gives the lower bound on the average codeword length of uniquely decodable symbol codes. The proof of the above inequality is given in Ref. 14. The above inequality becomes an equality only if pi = 2−yi where all of the y i’s are nonzero integers. B. Extension to nonbinary codes
The arguments used in the proof of the Kraft–McMillan inequality and its converse can be extended in a straightforward manner for nonbinary codes 共base B ⬎ 2兲. In the case of codewords of base B, the B-ary dynamical system is used 共x 哫 Bx mod 1 for all x 苸 关0 , 1兲兲. The B-ary dynamical system is also a type of GLS.
C
1/2
1/4
B
1/2
1/4 FIG. 4. A constrained Markov source with AB as a forbidden word. The numbers indicate the transition probabilities between the alphabets.
V. THE KRAFT–McMILLAN INEQUALITY FOR CONSTRAINED SOURCES
An important fact regarding the Kraft–McMillan inequality that we did not mention is that it is a necessary and sufficient condition for those sources which produce all possible symbolic sequences as messages. We shall call such a source as an unconstrained source. The topological entropy of such an unconstrained source is the maximum possible 关e.g., a source with N ⬍ ⬁ alphabets, the maximum possible topological entropy is ln共N兲兴. On the other hand, constrained sources 共with finite alphabet size兲 are those that have less than the maximum allowed topological entropy. This implies that they have certain restrictions in the space of all symbolic sequences. Certain symbolic sequences are forbidden and these are known as forbidden words. Dalai and Leonardi19 showed that uniquely decodable codes for such constrained sources need not necessarily satisfy the Kraft–McMillan inequality. To illustrate this point, they considered a Markov source with three symbols A, B, and C, and suppose that symbol A can never be followed by B 共AB is a forbidden word兲 as shown in Fig. 4. Then, by choosing the codewords “0,” “1,” and “01” for the source symbols A, B, and C, respectively, any sequence of symbols can be uniquely decoded. Notice that in this case, the Kraft– McMillan inequality is not satisfied 共2−1 + 2−1 + 2−2 ⬎ 1兲. They generalize the Kraft–McMillan inequality for constrained sources 共they use the term constrained sequences兲. However, they report that their generalization is only necessary and not sufficient. A necessary and sufficient condition for unique decodability for constrained sources remains an open problem. The dynamical systems framework is able to capture both constrained and unconstrained sources by means of the topological entropy. We are working on extending our proof to the case where the topological entropy is not maximum, but a proof for constrained sources is still elusive. VI. CONCLUSIONS
The Kraft–McMillan inequality is an important result in information theory which gives a necessary and sufficient condition for a code to be uniquely decodable. It also has a quantum analogue. We have provided a new proof of this inequality and its converse for 共classical兲 prefix-free codes using a dynamical systems approach. We have used the binary map and its special dynamical system properties for the proof. The binary map is a type of GLS, a chaotic dynamical
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system. Furthermore, the proof is readily generalizable for higher alphabet sizes by constructing the appropriate GLS and studying its symbolic sequences. The problem of deriving a necessary and sufficient condition for unique decodability for constrained sequences remains open. It will also be interesting to investigate a quantum version of this open problem. ACKNOWLEDGMENTS
Nithin Nagaraj would like to express sincere gratitude to his Ph.D. thesis advisor Professor Prabhakar G. Vaidya for useful discussions, constant support, and encouragement. He is grateful to the School of Natural Sciences and Engineering, National Institute of Advanced Studies, Indian Institute of Science Campus, Bangalore where much of this work was carried out. He would also like to thank the reviewers and the editor for useful comments. C. E. Shannon, Bell Syst. Tech. J. 27, 379 共1948兲. D. A. Huffman, Proc. IRE 40, 1098 共1952兲. 3 J. J. Rissanen, IBM J. Res. Develop. 23, 146 共1979兲. 1 2
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B. Schumacher and M. P. Westmoreland, Phys. Rev. A 64, 042304 共2001兲. S. L. Braunstein, C. A. Fuchs, and D. Gottesman, IEEE Trans. Inf. Theory 46, 1644 共2000兲. 6 K. Bostroem and T. Felbinger, Phys. Rev. A 65 032313 共2002兲. 7 L. G. Kraft, “A device for quantizing, grouping and coding amplitude modulated pulses,” M.S. thesis, MIT, 1949. 8 M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information Cambridge 共Cambridge University Press, Cambridge, 2000兲. 9 R. Jozsa and B. Schumacher, J. Mod. Opt. 41, 2343 共1994兲; B. Schumacher, Phys. Rev. A 51, 2738 共1995兲; P. Hayden, R. Josza, and A. Winter, J. Math. Phys. 43, 4404 共2002兲. 10 T. Kohda, Proc. IEEE 90, 641 共2002兲. 11 N. Nagaraj, P. G. Vaidya, and K. G. Bhat, Commun. Nonlinear Sci. Numer. Simul. 14, 1013 共2009兲. 12 G. K. Wallace, Commun. ACM 34, 30 共1991兲. 13 I. L. Chuang and D. S. Modha, IEEE Trans. Inf. Theory 46, 1104 共2000兲. 14 B. McMillan, IEEE Trans. Inf. Theory 2, 115 共1956兲. 15 J. Karush, IRE Trans. Inf. Theory 7, 118 共1961兲. 16 D. J. C. MacKay, Information Theory, Inference and Learning Algorithms 共Cambridge University Press, Cambridge, 2003兲. 17 G. A. Jones and J. Mary Jones, Information and Coding Theory 共Springer, New Delhi, India, 2004兲. 18 K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers 共29 Mathematical Association of America, Washington, DC, 2002兲. 19 M. Dalai and R. Leonardi, Proceedings of the International Symposium on Information Theory, 2005 共unpublished兲, p. 1534. 4 5
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A new criterion to distinguish stochastic and deterministic time series with the Poincaré section and fractal dimension Abbas Golestani,1 M. R. Jahed Motlagh,1 K. Ahmadian,1 Amir H. Omidvarnia,2 and Nasser Mozayani1 1
Computer Engineering Department, Iran University of Science and Technology, 164885311 Narmak, Tehran, Iran 2 Electrical and Computer Engineering Department, University of Tehran, 141746619 Tehran, Iran
共Received 9 August 2008; accepted 17 February 2009; published online 24 March 2009兲 In this paper, we propose a new method for detecting regular behavior of time series: this method is based on the Poincaré section and the Higuchi fractal dimension. The new method aims to distinguish random signals from deterministic signals. In fact, our method provides a pattern for decision making about whether a signal is random or deterministic. We apply this method to different time series, such as chaotic signals, random signals, and periodic signals. We apply this method to examples from all types of route to chaotic signals. This method has also been applied to data about iris tissues. The results show that the new method can distinguish different types of signals. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3096413兴 The main contribution of this paper is a new method to discriminate random and deterministic signals. This new method is based on two specific signal features. The test cases include random signals, deterministic signals, and combinations thereof. These test cases were sufficient to test the method’s performance. This method was also applied to a real biological signal, yielding results that are consistent with medical experiments. I. INTRODUCTION
Nonlinear signal processing is an important research area with many applications. Specifications and identifications of nonlinear signals can help us to detect nonlinear behavior of dynamical systems. One specification, the discrimination of stochastic and deterministic behaviors of nonlinear time series, is a basic topic in nonlinear dynamic fields.1 This specification has attracted researchers for a long time.2 Some studies have shown that when fractal dimension is a function of the state-space dimension, deterministic properties of a system can be determined by low values of fractal dimension, and stochastic properties of a system can be determined by high values of fractal dimension.3–5 Although this idea has seen wide use, some researchers have cast doubt on the usefulness of the fractal dimension for identifying the behavior of time series.6–8 The fractal dimension indicates the amount of complexity and repeatability in different scales.9 Among different versions of fractal dimension that are used for complexity estimation, the Higuchi fractal dimension is a precise and applicable mechanism to estimate self-similarity that also gives a stable value for the fractal dimension.10–13 The Higuchi fractal dimension is a stable method to compute the fractal dimension of irregular time series of natural phenomena. In contrast with the conventional methods to estimate fractal dimension in the state space, the Higuchi fractal dimension can be calculated di1054-1500/2009/19共1兲/013137/13/$25.00
rectly in the time domain and is therefore simple and fast. It has also been found to estimate samples as short as 150–500 data points reliably.14 In the past few years, fractal analysis techniques have gained increasing attention in medical signal and image processing. For example, analyses of encephalographic data and other biosignals are among its applications.15 Fractal complexity of the signal in the time domain 共calculated using Higuchi’s algorithm兲 seems to be the simplest method. The same method may also be used in other biomedical applications.16–18 Because of the similarity between chaotic and stochastic signals, distinguishing these two types is difficult. To overcome this problem, specification of chaotic signals can be used. Out of the different properties of chaotic signals, we focus on the Poincaré section: a method to detect periodic signals. In this paper, we propose a new, efficient method to distinguish random signals from deterministic signals based on the Poincaré section and the modified version of the Higuchi fractal dimension: the P&H method. The P&H method has several steps. The first step is to intersect the time series trajectory with the Poincaré section. This intersection induces a set of points that indicate dynamic flow. Now there is a series 共S兲 specified by these points. Applying the Higuchi fractal dimension to the S series yields a vector L共k兲. This vector is a basis for decision making about series specification: it suggests typical criteria to distinguish stochastic signals from deterministic signals. We also apply this method to examples from all types of route to chaotic signals. All chaotic systems can be mapped into their examples based on their route to chaos 共period doubling, quasiperiodic, or intermittent兲, and that examples of each of these routes produce a peak in L共k兲 which shows that these time series are deterministic. This paper is organized as follows. In Sec. II, we explain Poincaré section. In Sec. III, we describe the Higuchi fractal
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val k, we get k sets of new time series. The length of the curve Lm共k兲 is defined as follows: Lm共k兲 =
兺i=1兩x共m + ik兲 − x共m + 共i − 1兲k兲兩共N − 1兲 , N−m k k
共3兲
where N is the number of samples and N−1 N−m k k
FIG. 1. Order of obtained points in the intersection between the flow 共⌫兲 and the Poincaré section共s兲.
dimension. In Sec. IV, we present the P&H method. Finally, in Sec. V, we present the results of applying the new method to different time series.
is the normalization factor. We define the length of the curve for the time interval k , L共k兲, as the average value over k sets of Lm共k兲. If L共k兲 ⬀ k−D, then the curve is a fractal with dimension D. In the modified version of the Higuchi fractal dimension that is used in this paper, the normalization coefficient changes from N−1 N−m kk k
冋 册
II. THE POINCARÉ SECTION
Analyzing the dynamics of a high dimensional flow in the corresponding phase space is a nontrivial task. Conventionally, rather than analyzing the continuous flow in the 共d兲th dimension, we observe the dynamics induced by the flow on a particular section of the phase space. The chosen section is called the Poincaré section.19 The intersection induces a set of points in 共d − 1兲th-dimension space. For instance, suppose that we have a three-dimensional 共3D兲 flow 共⌫兲. Rather than directly examining the 3D flow, we consider the intersection of the flow with the section. In fact, this set of points induces a time series in which the points come from a specific side of the section 共see Fig. 1兲. P0, P1, P2, and so on come from intersections of the flow with the Poincaré section. We can define the Poincaré map as a discrete function T : S → S that contacts successive points of intersection. S is the Poincaré section, Pk+1 = T共Pk兲 = T关T共Pk−1兲兴 = T2共Pk−1兲 = ¯ , 共1兲
Pk+1, Pk, . . . , 苸 P,
where P is a mapping of flow points 共⌫兲 toward Poincaré section 共S兲.20 Flow intersects the Poincaré section in q共q ⬎ 1兲 points. The resulting point inherits some important features from the flow. Many of the flow features such as periodicity and quasiperiodicity could follow from the corresponding Poincaré map of the flow attractor. III. THE HIGUCHI FRACTAL DIMENSION
In Higuchi’s method, which is used for fractal dimension k calculation, we must construct a new time series, xm , from the input time series, x共1兲 , x共2兲 , . . . , x共N兲 as follows:
再
冉 冊冎
k = x共m兲,x共m + k兲,x共m + 2k兲, . . . ,x m + xm
m = 1,2, . . . ,k,
N−m k k
,
共2兲
where both m and k are integers and is Gauss’ notation. m is the initial time and k is the interval time. For a time inter-
to N−1 . N−m k k
冋 册
IV. THE P&H METHOD
This method is based on the Poincaré section and the modified version of the Higuchi fractal dimension that distinguishes random signals from deterministic signals. The details of the P&H method are as follows. Suppose that X is a time series. The Poincaré section is shown by the P notation. The new time series is obtained by intersecting these flows, and it can be indicated as P 艚 X = 兵P0, P1, P2, . . . , Pn其. These points have a time series structure and the orders of these points are as follows: Pk+1 = T共Pk兲 = T关T共Pk−1兲兴 = T2共Pk−1兲 = ¯ , Pk+1, Pk, . . . , 苸 P.
共4兲
The obtained time series is shown by the S notation. All in all, intersection occurs in two or more dimensions. One-dimensional signals must be embedded according to the embedding theorem.21 The result of computing the Poincaré section of an embedded time series is a set of vector points with dimension D − 1, if the input data set of vectors has dimension D. The projection is done orthogonally to the tangent vector of the vector with the index. Now we calculate the Higuchi fractal dimension of each variable in the S series. Next, we compute the average of these results. For example, if the initial trajectory has three dimensions, then the obtained points from the intersection have two dimensions, such as 共x1 , y1兲 , 共x2 , y2兲 , . . .. Therefore, we preserve the x values and y values of points sepa-
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FIG. 3. The horseshoe map after a single iteration. FIG. 2. 共Color online兲 The decreasing trend of the P&H method’s outputs for random time series.
rately and introduce each of them as inputs to the Higuchi processes to L1共k兲, L2共k兲 obtained, then average of L1共k兲, L2共k兲 assumes as final L共k兲 vector,
再
冉 冊冎
k = S共m兲,S共m + k兲,S共m + 2k兲, . . . ,S m + Sm
N−m k k
,
共5兲
m = 1,2, . . . ,k. k For each Sm , Lm共k兲 is
Lm共k兲 =
兺i=1兩S共m + ik兲 − S共m + 共i − 1兲k兲兩共N − 1兲 , N−m k k
共6兲
where N is the number of samples and N−1 N−m k k
is the normalization factor. Thus, we obtain k sets of Lm共k兲 that indicate the series of S specification. So L共k兲 is obtained from the average value over k sets of Lm共k兲. Now consider the value of log共L共k兲兲. For random time series, this value decreases according to Fig. 2. As shown in Sec. III, Lm共k兲 is obtained by summing approximately 共N / k兲 terms. If we ignore the normalization factor, then as k increases, the number of summed terms k time series were random series, then the decreases. If the xm positive subtraction of consequent terms yields a value that is also random. So in the value of Lm共k兲, 共N / k兲 random terms are summed, and in the value of Lm共k + 1兲, N / k + 1 random terms are summed. Therefore,
In deterministic time series, because of the deterministic relationship between series’ terms, at least for a specific value of k, the value of Lm共k + 1兲 is greater than the value of Lm共k兲 even though k increases. For chaotic and periodic time series, we will show that there is at least one k value by which the value of Lm共k + 1兲 is greater than Lm共k兲. First of all, for chaotic time series, because of the stretching and folding property, the L共k兲 vector indicates a zigzag pattern. We illustrate this with a simple example. Assume the Smale horseshoe22 example as in Fig. 3. The horseshoe map takes a rectangular region and stretches it horizontally by a factor a ⬎ 1 and then folds it vertically by a factor b ⬍ 1.23 So the stretching and folding process is repeated over and over, as illustrated in Fig. 4. Then, we intersect the second transformed Smale horseshoe with the Poincaré section 共line兲. According to Fig. 4, the first point that comes from the intersection between the flow and the Poincaré section belongs to the upper part. Consequently, the second point comes from the lower part. This process continues, and successive points coming from the intersection have a zigzag order relative to each other. So, stretching and folding generate a series whose points are in a zigzag relation with each other. If the Higuchi process is applied to this series, then a zigzag pattern also arises. So, for chaotic time series, the P&H method can yield a zigzag pattern, x1x2 ¯ x1+k ¯ x共k+1兲+k ¯ xN , Dk = x1+k − x1 = x共k+1兲+k − x1+k = ¯ = and x1x2 ¯ x1+共k+1兲 ¯ x1+共k+1兲+共k+1兲 ¯ xN , 共9兲
共N/k兲X → Lm共k兲, 共N/k + 1兲X → Lm共k + 1兲.
共7兲
As k increases, the number of summands decreases. Therefore, the value of Lm共k兲 decreases. Based on this specification, we propose an irregularity criterion to distinguish between random time series and deterministic time series. For random time series, we observe a decreasing pattern 共Fig. 2兲, and for deterministic time series, we observe a nondecreasing pattern 共such as in Fig. 7兲.
共8兲
FIG. 4. 共Color online兲 The horseshoe map after two iterations.
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FIG. 5. 共Color online兲 The intersection of time series trajectory and the Poincaré section. Intersection indicates the stretching and folding specification.
Dk+1 = x1+共k+1兲 − x1 = x1+共k+1兲+共k+1兲 − x1+共k+1兲 = ¯ = M . For even values of k, Lm共k + 1兲 ⬎ Lm共k兲. Lm共k兲 is obtained from the number of N / k terms in the summation: that is, N / k ⫻ Dk. In the value of Lm共k + 1兲, there are 共N / k + 1兲 terms that we write as 共N / k + 1兲Dk+1. As the stretching and folding process progresses, approaches zero, but the M interval is fixed, Lm共k + 1兲 ⬎ Lm共k兲 ⇒ ⬎
N N N Dk+1 ⬎ Dk ⇒ M k+1 k k+1
N M ⇒ ⬎ k k+1 k
共 → 0兲.
共10兲
Therefore, for chaotic time series, the L共k兲 values have the stretching and folding property.24 For instance, the Lorenz time series attractor has a butterfly motion. The stretching and folding property makes the Lorenz attractor move the left and right sides of the attractor periodically, without repeating its trajectory. As shown in Fig. 5, the intersection of the trajectory and Poincaré section yields points in the left and right sides of the attractor, alternately. This process continues, producing the Poincaré map. This yields points that alternate between being far away and close together. Figure 6 illustrates points in the intersection between the time series trajectory and the Poincaré section. As can be seen in Fig. 6, points alternate between different distances. These vibrations suggest the stretching and folding property.
FIG. 6. 共Color online兲 Time series obtained by intersecting the Poincaré section with the Lorenz attractor. It shows vibrations corresponding to the stretching and folding property 共zoomed兲.
FIG. 7. 共Color online兲 Obtained pattern from the P&H method’s output for all chaotic time series.
According to this property, L共k兲 is greater for odd values of k than for previous even values of k. So as shown in Fig. 7, for all chaotic time series, a specific pattern is obtained. In periodic signals, because of the periodicity property of signals 共T兲, the points from the Poincaré section have periodicity. So when the Higuchi dimension is applied, in the L共k兲 vector, there is a peak value every T points. As can be seen in Fig. 14, the sine describes this. We could propose a corresponding numerical criterion decision making with the P&H method, but for the sake of better understanding and avoiding ambiguity, we chose not to.
V. EXPERIMENTAL RESULTS
We present the results of applying the P&H method to different stochastic and deterministic signals. Besides the P&H method, we also show the power spectrum of each signal, so we can compare the results. We will see that the P&H method is more efficient. For example, the power spectrum of the Hénon time series is similar to the power spectrum of white noise. First, we examine stochastic time series.
A. White noise
As mentioned in Sec. IV, the P&H method has a decreasing pattern for random time series. So when k increases, the value of L共k兲 decreases and has a decreasing pattern. As Fig. 8共a兲 shows, the power spectral density of uniform white noise is uniformly distributed over all frequencies. The uniform white noise signal contains uniformly distributed pseudorandom scalar values drawn from a uniform distribution on the unit interval, generated by a modified version of Marsaglia’s subtract with borrow algorithm.25 We embedded white noise series in 3D space because we were able to use the P&H process. Figure 8共b兲 shows white noise signals with a uniformly distributed function 共left兲 points resulting from intersecting the Poincaré section with a stochastic signal 共middle兲 and the output of the P&H method over the resulting time series 共right兲. According to Fig. 8共b兲, the outputs of the P&H method for all stochastic signals are equal. In Fig. 8共b兲, the pattern is
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FIG. 8. 共Color online兲 共a兲 Power spectrum of uniform white noise. 共b兲 Applying the P&H method over uniform white noise.
specific to the stochastic signals: for deterministic signals, this kind of pattern is not produced. Now, a white noise signal has been tested with a Gaussian-distributed function. The Gaussian white noise signal contains normally distributed pseudorandom scalar values drawn from a normal distribution with mean of 0 and standard deviation of 1 using Marsaglia’s ziggurat algorithm.26 The power spectral density of this signal is also uniformly distributed over all frequencies, similar to Fig. 8共a兲. As is apparent in Fig. 9, the P&H method generates a
decreasing monotonous pattern for all stochastic signals. This specification can be a criterion for distinguishing random signals from deterministic signals. B. Linearly filtered noise class
The proposed approach has been applied to a colored noise that was produced by applying a bandpass filter over a white noise. Uniform white noise was assumed to be sampled at a 50 Hz sampling rate and filtered by a tenth-
FIG. 9. 共Color online兲 Applying of P&H method over Gaussian white noise.
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FIG. 10. The linear filtering process for white noise.
order Chebyshev type I filter with normalized bandpass edge frequency of 10–40 Hz and 0.1 dB of peak-to-peak ripple in the bandpass 共see Fig. 10兲. When colored noise was produced, we embedded it in 3D space. As Fig. 11共b兲 shows, the P&H method determines that the input signal is a colored noise. C. Monotonic nonlinear transformation of a linearly filtered noise class
By applying the P&H method on the extracted irregularity criteria of a colored noise that has been passed through a monotonic nonlinear transform 共see Fig. 12兲, we observe that the method can determine the noise behavior of the input signal, as illustrated in Fig. 13共b兲. We also examined some deterministic time series. Deterministic time series contain periodic deterministic series and nonperiodic deterministic series. First we examine periodical deterministic series.
FIG. 12. Nonlinear transformation process of linearly filtered noise.
D. Periodic signals
The Poincaré section is proposed to detect periodicity. Because of the periodicity property of signals 共T兲, the points resulting from the Poincaré section show periodicity. So when we apply the Higuchi dimension, in the L共k兲 vector, there is a peak value every T points, as can be seen in Fig. 14. The sine function is embedded in three dimensions. In Fig. 14, the Poincaré map has only two repeated points due to the periodic behavior of the signal. So, as we show next, frequent points in the Poincaré map of signals are attainable only for periodic signals. In this case, every second output point of the P&H method is a peak. Another example is a signal with a five-period motion. This motion repeats itself exactly every five periods of the external drive. In contrast with the previous example, the output of the P&H method for this case decreases in k = 4 , 5 , 6 , 7, and in k = 8, it increases. By other means, the zigzag pattern has not been obtained, and every five points a peak value occurs.
FIG. 11. 共Color online兲 共a兲 Power spectrum of colored noise. 共b兲 Applying the P&H method to colored noise.
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FIG. 13. 共Color online兲 共a兲 Power spectrum of nonlinear transformation of colored noise. 共b兲 Applying the P&H method to a nonlinear transformation of colored noise.
FIG. 14. 共Color online兲 共a兲 Power spectrum of sine time series. 共b兲 Applying the P&H method to the embedded sine function.
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FIG. 15. 共Color online兲 共a兲 Power spectrum of the pendulum time series. 共b兲 Applying the P&H method to the periodic pendulum time series.
Another example is a pendulum time series. This series was periodic with a five-period motion with the following parameters: dy 1 = a sin共bt兲 − cby 2 − d sin y 1, dt
the type of flow or map is very different. Thus, we examined the P&H method over Hénon, Rössler, and Lorenz time series. E. Lorenz time series
dy 2 = y1 , dt 共11兲
a = 1.5, b = 0.666 66, c = 0.1, d = 1.36. As mentioned above, the period of the signal 共Fig. 15兲 is five, and this leads to the output of the P&H method: every five points, it has a peak. Next, we consider the chaotic signals that are known to be a part of a nonperiodical deterministic series. Chaotic signals have different behaviors27 in some respects, which are specific to discrete-time or continuous-time chaotic systems, the dimension of chaotic signals, the case study application, etc. It is likely that some routes by which stable equilibrium becomes chaotic are yet to be identified. These can be categorized into three basic types: period doubling, quasiperiodic, and intermittent.24 For the sake of testing the P&H method, we apply this method to all types of route to chaotic signals. We examined the logistic map as an example of the intermittent type, and we examined the van der Pol flow as an example of the quasiperiodic type. The other flows fall into to the period-doubling type. In the period-doubling case,
As an example of a chaotic signal, we examine Lorenz time series. The attractor and its derived equations were introduced by Lorenz in 1963. The system arises in lasers, dynamos, and specific waterwheels.28 For the following set of parameters, these PDEs produce chaotic behavior: dy 1 = a共y 1 − y 2兲, dt
dy 2 = by 1 − y 2 − y 1y 3 , dt
dy 3 = y 1y 2 + cy 3 , dt
共12兲
A = − 10, b = 28, c = − 2.666 666. According to Fig. 16, applying the P&H method to a Lorenz series yields a specific and unusual pattern. F. Rössler time series
Rössler designed the Rössler attractor in 1976. These originally theoretical later equations turned out to be useful for modeling equilibrium in chemical reactions. The original Rössler paper says that the Rössler attractor was intended to
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FIG. 16. 共Color online兲 共a兲 Power spectrum of Lorenz time series. 共b兲 Applying the P&H method over Lorenz time series.
behave similar to the Lorenz attractor, but is easier to analyze qualitatively.29 This series has a chaotic behavior with the following parameters. For the following set of parameters, these PDEs produce chaotic behavior: dy 1 = − y 2 − y 3, dt
dy 2 = y 1 + ay 2 , dt
dy 1 = y 2, dt
dy 2 = a共1 − y 21兲y 2 − y 1 + b sin共ct兲, dt 共15兲
a = 1, b = 0.61, c = 1.1. According to Figs. 17–20, the P&H method for chaotic series produces an equal zigzag pattern.
共13兲 dy 3 = b + y 3共y 1 − c兲, dt
I. Logistic map time series
a = 0.45, b = 2.0, c = 4.0
G. Hénon time series
The Hénon map is a discrete-time dynamical system. The map was introduced by Hénon as a simplified model of the Poincaré section of the Lorenz model.30 This series has a chaotic behavior with the following parameters: Xn+1 = 1 + aX2n + bY n,
Y n+1 = Xn , 共14兲
For r = 3.8284, we observe that the logistic map is intermittent.24 As mentioned above, an intermittent series can yield chaotic behavior. We embed the logistic map series in three dimensions and then apply the P&H method. Xn+1 = rXn共1 − Xn兲, r = 3.8284.
共16兲
We emphasize that the points obtained from the Poincaré map of chaotic signals are not repeated, in contrast with the points obtained from periodic signals. This feature can be used to distinguish periodic signals from chaotic signals.
a = − 1.4, b = 0.3. J. The X-coordinate of Lorenz system with interfering noise H. van der Pol time series
The van der Pol oscillator was originally discovered by the Dutch electrical engineer and physicist van der Pol. The van der Pol equation is used in both the physical and the biological sciences.31,32 This series has a chaotic behavior with the following parameters:
In this example, we evaluate the performance of the irregularity criterion in the presence of noise. So, the first variable of the Lorenz system is accompanied by an additive white noise in three different amplitudes based on the standard deviation of the original time samples. For the time series, the standard deviation of the samples is 7.9130.
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FIG. 17. 共Color online兲 共a兲 Power spectrum of Rössler time series. 共b兲 Applying the P&H method over the Rössler time series.
FIG. 18. 共Color online兲 共a兲 Power spectrum of Hénon time series. 共b兲 Applying the P&H method to the Hénon time series.
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FIG. 19. 共Color online兲 共a兲 Power spectrum of the van der Pol time series. 共b兲 Applying the P&H method over the van der Pol time series.
FIG. 20. 共Color online兲 共a兲 Power spectrum of logistic map time series. 共b兲 Applying the P&H method over the logistic map time series.
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FIG. 21. 共Color online兲 Depicts the influence of additional noise on the performance of our proposed method. As the noise level increases, the irregularity criterion approaches the decreasing pattern. 共a兲 The amplitude of additional noise is 0.1x. 共b兲 The amplitude of additional noise is 0.3x. 共c兲 The amplitude of additional noise is 0.5x.
Figure 21 shows embedding space reconstructions for three noisy time series with 0.1x, 0.3x, and 0.5x noise amplitudes. As Fig. 21 illustrates, reconstructed trajectories become less smooth in the presence of noise. So, we expect the value of the irregularity criterion to approach the decreasing pattern. However, as additional noise increases, detecting regularity will become more difficult. Nonetheless, in the presence of high levels of noise, the P&H method provides a reasonable answer.
K. Iris tissue
The high degree of irregularity of the iris tissue is the main reason why it is useful in identification systems. In some literature, this irregularity has been shown to be the result of a stochastic genetic process.33 Medical reports have suggested that the genetic formation of iris tissue follows a stochastic process,34 so we examine whether the iris structure has a stochastic behavior or a specific order. So, the iris
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less, the P&H method can be used as a criterion to distinguish between stochastic and deterministic signals. 1
FIG. 22. 共Color online兲 共a兲 The half iris image. 共b兲 Applying the P&H method.
information should be projected from two dimensions into one dimension. Converting two-dimensional time series to one-dimensional one is trivial and can be done in various ways, so there are no any differences in the result of our method. According to Fig. 22共a兲, the important point in this case is the existence of some redundant factors, such as eyelash, eyelid, and reflections. The size of the iris images is 128 ⫻ 512 pixels. As shown in Fig. 22共b兲, the deterministic assumption of the iris is rejected, and it has been characterized as a stochastic signal. The results are obtained from applying this analysis to the “CASIA I” and “CASIA III 共Interval兲” databases,35,36 and they indicate that the iris patterns are stochastic. VI. CONCLUSION
In this paper, we introduced a new method 共P&H method兲 to distinguish between stochastic and deterministic signals. This method is based on the Poincaré section and the Higuchi fractal dimension. We tested the P&H method on different signals such as examples from all types of route to chaotic signals, the obtained results showed the efficacy of this method. In addition, there is no limitation on the dimensionality of input series in the P&H method. In contrast with the classic methods that are based on numerical results, the P&H method has a graphically based decision-making process, which may limit its use in some applications. Neverthe-
J. Hubbard and B. West, Differential Equations: A Dynamical Systems Approach: Ordinary Differential Equations 共Springer, Berlin, 1991兲, Vol. 5. 2 P. Grassberger, T. Schreiber, and C. Schaffrath, Int. J. Bifurcation Chaos Appl. Sci. Eng. 1, 521 共1991兲. 3 A. H. Omidvarnia and A. Nasrabadi, Fractals 16, 129 共2008兲. 4 A. Brandstater, J. Swift, and H. L. Swinney, Phys. Rev. Lett. 51, 1442 共1983兲. 5 P. Berge, Y. Pomeau, and C. Vidal, “Order within Chaos: Towards a Deterministic Approach to Turbulence 共Wiley, New York, 1984兲. 6 A. R. Osborne and A. Provenzale, Physica 35D, 357 共1989兲. 7 J. Theiler, Phys. Lett. A 155, 480 共1991兲. 8 N. P. Greis and H. S. Greenside, Phys. Rev. A 44, 2324 共1991兲. 9 K. Najarian, K. Gopalakrishnan, and R. H. Zadeh, Int. J. Bioinformatics Research and Applications 共IJBRA兲 1, 102 共2005兲. 10 T. Higuchi, Physica D 31, 277 共1988兲. 11 T. Higuchi, Physica D 46, 254 共1990兲. 12 L. Telesca, Chaos, Solitons Fractals 18, 385 共2003兲. 13 W. Klonowski, E. Olejarczyk, and R. Stepien, Signal processing for functional analysis of protein mutants, Attractors, Signals, and Synergetics, Frontiers of Non-linear Dynamics 共Pabst Science, Lengerich, 2002兲, Vol. 1, pp. 553–560. 14 A. Anier, T. Lipping, S. Melto, and S. Hovilehto, Proceedings of the 26th Annual International Conference of the IEEE IEMBS APOS’04, 2004 共unpublished兲, Vol. 1, pp. 526–529. 15 A. Accardo, M. Affinito, and M. Carrozzi, Biol. Cybern. 77, 339 共1997兲. 16 W. Klonowski, Chaos, Solitons Fractals 14, 1379 共2002兲. 17 W. Klonowski, Nonlinear Biomed. Phys. 1, 5 共2007兲. 18 M. Moradi, P. Abolmaesumi, and A. Phillip, Proceedings of the 28th Annual International Conference of the IEEE EMBS APOS’06, 2006 共unpublished兲, pp. 2400–2403. 19 A. Basu, Project in School of Electrical and Computer Engineering Georgia Institute of Technology, Atlanta, GA, 2007. 20 N. B. Tufillaro, Poincare Map, 1997. 21 F. Takens, Dynamical Systems and Turbulence, Lecture Notes in Mathematics, Vol. 898 共Springer, Berlin, 1981兲 p. 366. 22 S. Smale, Bull. Am. Math. Soc. 73, 747 共1967兲. 23 M. Branson, Proceedings of the AIAA Space, 2006 共unpublished兲. 24 J. C. Sprott, Chaos and Time Series Analysis 共Oxford University Press, Oxford, 2003兲. 25 G. Marsaglia and A. Zaman, Ann. Appl. Probab. 1, 462 共1991兲. 26 G. Marsaglia and W. W. Tsang, J Stat. Software 5, 1 共2000兲. 27 E. Ott, Chaos in Dynamical System 共Cambridge University Press, New York, 2002兲. 28 E. N. Lorenz, J. Atmos. Sci. 20, 130 共1963兲. 29 O. E. Rössler, Phys. Lett. 57A, 397 共1976兲. 30 P. Grassberger and I. Procaccia, Physica 9D, 189 共1983兲. 31 B. van der Pol and J. van der Mark, Nature 共London兲 120, 363 共1927兲. 32 R. Fitzhugh, Biophys. J. 1, 445 共1961兲. 33 R. P. Wildes, Proc. IEEE 85, 9 共1997兲. 34 F. H. Adler, Physiology of the Eye 共Mosby, St. Louis, MO, 1965兲. 35 National Laboratory of Pattern Recognition, Chinese Academy of Sciences, available at http://www.sinobiometrics.com. 36 C. Merkwirth, U. Parlitz, and I. Wedekind, W. Lauterborn, TSTOOL, Version, 1.11, 2002, downloadable from http:// www.physik3.gwdg.de/tstool/index.html.
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Capture and release of traveling intrinsic localized mode in coupled cantilever array Masayuki Kimura1,a兲 and Takashi Hikihara1,2,b兲 1
Department of Electrical Engineering, Kyoto University, Kyoto 615-8510, Japan Photonics and Electronics Science and Engineering Center, Kyoto University, Kyoto 615-8510, Japan
2
共Received 19 September 2008; accepted 18 February 2009; published online 24 March 2009兲 A method to manipulate intrinsic localized mode 共ILM兲 is numerically discussed in a nonlinear coupled oscillator array, which is obtained by modeling a microcantilever array. Prior to the manipulation, coexistence and dynamical stability of standing ILMs are first investigated. The stability of coexisting ILMs is determined by a nonlinear coupling coefficient of the array. In addition, the global phase structure, which dominates traveling ILMs, is also changed with the stability. It makes possible to manipulate a traveling ILM by adjusting the nonlinear coupling coefficient. The capture and release manipulation of the traveling ILM is shown numerically. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3097068兴 Energy localization is very often observed in spatially extended system. Even in a homogeneous coupled oscillator, a localized excitation is caused by discreteness and nonlinearity. The localized excitation is intrinsic localized mode, which is also called discrete breather, first found by Sievers and Takeno. By various theoretical and numerical studies, properties of intrinsic localized mode have been revealed. In addition, experimental studies recently appear to confirm the properties. Intrinsic localized mode is observed in electronic circuit ladders, optical wave guides, and micromechanical oscillator arrays. It has been suggested that intrinsic localized mode can be utilized for applications to practical engineering. A new application using localized oscillations should include appropriate control methods. The basis of such control methods is the manipulation of the intrinsic localized mode. We discuss how to manipulate a localized excitation. I. INTRODUCTION
Intrinsic localized mode 共ILM兲 is a spatially localized and temporally periodic solution in nonlinear discrete systems. ILM was analytically discovered by Sievers and Takeno in 1988.1 They identified the ILM in the Fermi–Pasta– Ulam lattice.2 After the discovery, ILM in nonlinear discrete systems has attracted many researchers. That is, the existence, the stability, the movability, and other properties of ILM have been investigated theoretically and numerically for a variety of physical systems.3 Experimental studies have recently appeared.4 ILM is experimentally generated or observed in various systems, for instance, Josephson-junction array,5,6 optic wave guides,7,8 photonic crystals,9 micromechanical oscillators,10 mixed-valence transition metal complexes,11,12 antiferromagnets,13 and electronic circuits.14 These experiments suggest the phenomenological universala兲
Electronic mail: kimura@dove kuee kyoto-u.ac.jp. Electronic mail: hikihara@kuee kyoto-u.ac.jp.
b兲
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ity of ILM and the possible application phase. Studies have indeed appeared toward potential applications in both fundamental science and practical engineering.4 Sato and co-workers10,15 showed the existence of intrinsic localized modes in microcantilever arrays. By externally exciting the array, standing ILMs fixed at a site of the array were observed. It was also identified that a localized excitation wandered in the array. The wandering excitation is called traveling ILM in this paper. In the experiments, the traveling ILM was finally captured at a site and survived as a standing ILM. Therefore, standing ILMs can be generated by capturing a traveling ILM at a site. Recently, Sato and co-workers16,17 manipulated the position of ILM using a localized impurity. A standing ILM was attracted or repulsed by the impurity. The attractive and repulsive manipulations allow shifting the position of standing ILM without decaying the concentrated energy of ILM. The observations and the manipulations show that the stable standing ILM exists and it can be relocated even in a micrometer scale device such as the microcantilever array. That is, ILM can be applicable to both micro- and nanoengineering. A microcantilever array can be modeled by a simple coupled nonlinear differential equation17 with nonlinearity in both on-site and intersite terms. It has been shown that several ILMs coexist in the array, and the dynamical stability of these coexisting ILMs depends on the ratio of nonlinearity in on-site and intersite terms.18 In addition to that, the bifurcation structure concerning the stability change has been investigated. To study the mechanism of the transition of traveling ILMs, it is also necessary to analyze on how a traveling ILM behaves in phase space. The relationship between traveling waves and the global phase structure was discussed in a coupled magnetoelastic beam system, which is modeled as a simple coupled nonlinear differential equation having the same form as of a microcantilever array.19–21 They suggested that the transition of traveling wave is governed by the phase structure in the vicinity of coexisting standing waves. Standing ILMs in the
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Chaos 19, 013138 共2009兲
M. Kimura and T. Hikihara Overhang Support
tion of the overhang region. The restoring force caused by the deformation of the overhang has cubic nonlinearity.17 Therefore the microcantilever array is expressed by u¨n = − ␣1un − ␣2兵共un − un−1兲 + 共un − un+1兲其 − 1u3n
Edge
Cantilevers
FIG. 1. Schematic configuration of coupled cantilever array. The array has eight cantilevers arranged in one dimension. Both ends of the array are fixed by support.
microcantilever array are assumed to have similar dynamics as the standing wave solutions. That is, it is conjectured that a traveling dynamics of ILM is governed by the phase structure around standing ILMs. The purpose of this study is to propose a new method for manipulation of ILMs in the coupled oscillators which has nonlinearities in both on-site and intersite terms. Both the traveling ILMs and the corresponding global phase structure are numerically investigated. Then a manipulation method is discussed based on the phase structure. The manipulation can be achieved by adjusting the coefficient of the nonlinear intersite term. For the research presented in this paper, a microcantilever array that exhibited ILMs was chosen as coupled oscillators. Parameter settings for the coupled oscillators are discussed in Sec. II. In Sec. III, the coexistence and the stability are investigated with respect to various values of the nonlinear coupling coefficient. The relationship between a traveling ILM and the global phase structure is discussed in Sec. IV. Finally manipulations of an ILM are shown in Sec. V. II. COUPLED CANTILEVER ARRAY
A microcantilever array fabricated by Sato et al. is treated as a coupled oscillator array in this paper. A schematic configuration of the cantilever array is shown in Fig. 1. Eight cantilevers are arranged with equal intervals in one dimension. Adjacent cantilevers are coupled by the overhang. The size of the array is determined as in Ref. 15. A cantilever has a length of 50 m, a width of 15 m, and a thickness of 300 nm. All cantilevers are arranged with a pitch of 40 m. Overhang region has a length of 60 m and a thickness of 300 nm. The vibration of a single cantilever is described by partial differential equation. Since the cantilever is thin, Euler– Bernoulli beam theory can be applied to the cantilever. The theory gives us resonant frequencies of the cantilever. The lowest frequency corresponds to the first mode oscillation of the cantilever. Because the microcantilever array was excited near the lowest frequency,15 the motion of the tip of cantilever is modeled by a simple spring-mass system, which has the same resonant frequency as the original cantilever. On the basis of the theoretical22–24 and the experimental analyses,17 the spring has cubic nonlinearity in the restoring force. Therefore the single cantilever is represented by a spring-mass system having cubic nonlinearity in the spring. The overhang is modeled the same way.17 The difference in displacement of neighboring cantilevers causes the deforma-
− 2兵共un − un−1兲3 + 共un − un+1兲3其,
共1兲
where the displacement of the tip of nth cantilever is depicted by un. The first and third terms represent the restoring force caused by bending each cantilever. The coupling force is depicted by the second and forth terms. Equation 共1兲 is nondimensional and is obtained by scaling time and length. The time scaling is applied to the original differential equation so that ␣1 is unity. The coefficient of nonlinear on-site restoring force is set to 0.01 by the scaling in length. Values of 5.38 s and 2.67 m are chosen as units of time and length. The linear coupling force was experimentally estimated to be one-tenth of the on-site restoring force.15 Thus we assumed that the linear and nonlinear coupling coefficients are one-tenth of the corresponding coefficient to on-site force, i.e., ␣2 = 0.1 and 2 = 0.001. The nondimensional form shows that a microcantilever array is characterized by coupling coefficients in the nondimensional differential equation. These coupling coefficients can be thought of as the ratio of on-site and intersite coefficients. Because the coefficients in Eq. 共1兲 depend on the length and the thickness of the cantilever and the overhang, the pitch of arranged cantilevers, Young’s modulus, and the second moment of area,17 the ratio can be selected in the design of a microcantilever array. In particular, it has been reported that the ratio in nonlinear on-site and intersite coefficients governs the stability of ILM.18 The behavior of traveling ILM can be changed with the nonlinearity ratio. Thus, properties of standing ILMs and traveling ILMs are investigated numerically for various values of the nonlinearity ratio in this paper. III. STANDING ILMs AND ITS STABILITY A. Numerical techniques
In order to obtain a time-periodic solution, the shooting method using Newton–Raphson method is applied.25 Since ILMs for Eq. 共1兲 are time-periodic solutions, they can be found by the shooting method. Equation 共1兲 is integrated with an initial guess over a given period T. The initial guess is estimated by a computational technique using anticontinuous limit.26 If the initial guess is close enough to an ILM solution for Eq. 共1兲, the ILM is obtained with the given period T. The total energy of the ILM, H, is a function of the period of the ILM, H = H共T兲. The function, H共T兲, is a single valued function when the ILM is sufficiently localized. The total energy monotonically increases with decreasing the period T of ILM because Eq. 共1兲 has hard nonlinearity for the on-site and the intersite restoring forces. Therefore, an ILM on certain energy surface, which is defined by a given total energy, can easily be found by Newton–Raphson method. The total energy is set at 250.
5 0 -5
15
(b)
10 5 0 -5
5 0 -5
-10
-10
-10
-15
-15
-15
0 1 2 3 4 5 6 7 8 9
Amp tude
5 0 -5
5 0 -5
5 0 -5
-10
-10
-15
-15
-15
0 -5 -15
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
共2兲
where p is the index of cantilever and is set at 4 without loss of generality. An ILM solution corresponds to a fixed point xⴱ 苸 ⌺ p for the Poincaré map P : ⌺ p → ⌺ p. Then the stability of the ILM is equivalent to the stability of the fixed point. The linearization of the map P leads to the linear map at the fixed point xⴱ: 共3兲 ⴱ
where depicts a variation. The matrix DP共x 兲 has 15 different eigenvalues which correspond to Floquet multipliers. An ILM is determined to be unstable if one of the 15 eigenvalues is outside unit circle on complex plane. Because Eq. 共1兲 is a Hamiltonian system, an ILM is not determined to be unstable if and only if the all of eigenvalues are on a unit circle. B. Coexistence of ILM
Equation 共1兲 has several coexisting ILMs. In this paper, these ILMs are roughly classified into ST and P modes only by the symmetry of amplitude distribution. ST modes have odd-symmetric amplitude distributions. Since the locus of ST mode is at a site, its value becomes an integer. P modes have even-symmetric amplitude distribution. Then XILM takes half-integer. Coexisting ILMs in the cantilever array can be distinguished by the index number of oscillators having large amplitude because the translational symmetry is broken by the fixed ends of array. In this paper, a ST mode standing at mth site is called STm. A P mode is depicted Pm − m⬘, where m⬘ = m + 1, because the P mode has even symmetric in amplitude distribution. The locus of the P mode is found at m + 1 / 2. Coexisting ILMs at 2 = 0.001 are shown in Fig. 2. The ends of array correspond to n = 0 and 9. Figure 2共a兲 shows a
Site n
FIG. 3. Coexisting ILMs at 2 = 0.01, H = 250: 共a兲 ST2, 共b兲 ST4, 共c兲 P1-2, and 共d兲 P4-5.
ST mode standing at n = 1. The ST mode is labeled as ST1. Other ILMs shown in Figs. 2共b兲–2共d兲 are labeled as ST4, P1-2, and P4-5, respectively. Eight ST modes and seven P modes are found at 2 = 0.001. Figure 3 shows coexisting ILMs at 2 = 0.01, which is ten times larger than the former case. In this array, six ST modes and seven P modes coexist. ST2, ST4, P1-2, and P4-5 are shown in Figs. 3共a兲–3共d兲, respectively. ST1 was not found for this array by using our method. In addition, the symmetry of P1-2 is obviously broken, that is, the amplitude of the first and second oscillators are different. The disappearance of ST1 and the symmetry breaking of P1-2 are due to an increase in the influence of the fixed boundary with 2. It implies that the effect of the fixed boundary for ILMs depends on the nonlinear coupling coefficient. This dependency is discussed in more detail with the stability of ILMs in Sec. III C. C. Stability of coexisting ILMs
Eigenvalues of ST4 and P4-5 at 2 = 0.001 are shown in Fig. 4. All of the eigenvalues for ST4 are on a unit circle, as shown Fig. 4共a兲. Then ST4 is not determined to be unstable. A solution stays around ST4 for a long period when an initial condition of the solution is chosen near ST4. Thus, the stability of ST4 is called “marginally stable.”3 In fact, if Eq. 共1兲 has some damping terms, then the absolute values of all eigenvalues are less than unity. Figure 4共b兲 shows eigenvalues of P4-5. One of the eigenvalues is outside the unit circle. 1.5
1.5
(a)
1
1
0.5
0.5 Im(λ)
A periodic solution can be treated as a fixed point on a hyper surface. The dynamics around the fixed point is represented by the Poincaré map defined on the surface. Here we introduce a hyperplane
k+1 = DP共xⴱ兲k ,
5
Site n
FIG. 2. Coexisting ILMs at 2 = 0.001, H = 250: 共a兲 ST1, 共b兲 ST4, 共c兲 P1-2, and 共d兲 P4-5. The sixth order symplectic integrator is applied for the integration.
⌺ p = 兵共u,u˙ 兲 苸 R16兩u p典0, u˙ p = 0其,
(d)
10
-10
Site n
Im(λ)
Site n
0 1 2 3 4 5 6 7 8 9
15
(c)
10
-10 0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
Site n
15
(d)
10
0 -5 -15
0 1 2 3 4 5 6 7 8 9
Amp tude
15
(c)
10
5
Site n
Amp tude
15
0 1 2 3 4 5 6 7 8 9
(b)
10
-10
Site n
Site n
15
(a)
10
Amp tude
15
(a)
10
Amp tude
Amp tude
15
Amp tude
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Capture and release of traveling ILM
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0
-0.5
(b)
0
-0.5
-1
-1
-1.5
-1.5
-1 -0.5
0
0.5
1 1.5 Re(λ)
2
2.5
3
-1 -0.5
0
0.5
1 1.5 Re(λ)
2
2.5
3
FIG. 4. Eigenvalues of the linearized map DP共xⴱ兲 at 2 = 0.001: 共a兲 ST4 and 共b兲 P4-5. The circle drawn by dashed curve indicates unit circle in the complex plane.
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(a)
1
0.5
0.5 Im(λ)
1
Im(λ)
Chaos 19, 013138 共2009兲
M. Kimura and T. Hikihara
0
-0.5
rameter gap can also be obtained by adding an impurity. Thus, Fig. 6 also shows the magnitude of the effect of an impurity with respect to the coupling nonlinearity. The influence of an impurity has a peak near 2 ⯝ 0.005 45 that corresponds to the stability change in ILMs standing near the center of array.
(b)
0
-0.5
-1
-1
-1.5
-1.5
-1 -0.5
0
0.5
1 1.5 Re(λ)
2
2.5
3
-1 -0.5
0
0.5
1 1.5 Re(λ)
2
2.5
3
IV. TRAVELING ILM AND PHASE STRUCTURE
FIG. 5. Eigenvalues of the linearized map DP共xⴱ兲 at 2 = 0.001: 共a兲 ST4 and 共b兲 P4-5.
Then P4-5 is unstable. All coexisting P modes are unstable and ST modes are stable at 2 = 0.001. However the stability of coexisting ILMs are flipped at 2 = 0.01. ST4 at 2 = 0.01 is unstable, as shown in Fig. 5共a兲. Then P4-5 is stable. It implies that the stability change occurs between 2 = 0.001 and 0.01. Figure 6 shows the stability and the locus of each coexisting ILM with respect to the nonlinear coupling coefficient, 2. The locus of ILM is obtained by XILM =
8 兺n=1 n ⫻ 兩un兩 8 兺n=1 兩un兩
共4兲
,
where 兩un兩 is the absolute value of nth oscillator’s displacement on the hyper surface ⌺4. The set of stable ILMs is represented by the solid curve. The dashed curves correspond to unstable ILMs. The figure clearly shows that the stability of coexisting ILMs is flipped with the nonlinear coupling coefficient, 2. In particular, ILMs standing around the center of array almost simultaneously gain or lose stability at 2 ⯝ 0.005 45.18 On the other hand, ST1 and P1-2 coincide at 2 ⯝ 0.002 38 and disappear with increment of 2. The P1-2 appears again with ST2 at 2 ⯝ 0.007 16. Such appearance and disappearance are classified as saddle-node bifurcation.18 Figure 6 shows that a bifurcation point tends to depart from 2 ⯝ 0.005 45 as the locus of ILM approaches the end of array. This parameter gap seems to be affected by the fixed ends. In fact, the gap is vanished in the ringed array.18 Since the fixed end can be thought as an impurity, the pa-
4.5 4 3.5
XILM
3 2.5 2 1.5
P4-5 (stable)
P4-5 (unstable)
ST4 (unstable)
ST4 (stable)
P3-4 (stable)
P3-4 (unstable)
ST3 (unstable)
ST3 (stable) P2-3 (unstable)
P2-3 (stable)
ST2 (stable)
ST2 (unstable)
P1-2 (unstable)
P1-2 (stable)
1 ST1 (stable) 0.5 0
0
0.002
0.004
¯2
0.006
0.008
0.01
FIG. 6. Locus and stability of coexisting ILMs. The solid curve corresponds to stable ILMs. Unstable ILMs are represented by the dashed curve.
A. Relationship between traveling ILM and invariant manifold
Perturbation against an unstable ILM generates a traveling ILM. Trajectory of the traveling ILM is determined by phase structure around the unstable ILM.21 The phase structure is characterized by invariant manifolds of the unstable ILM. Each unstable ILM in coexisting ILMs has only one eigenvalue outside the unit circle. Then the unstable manifold of each unstable ILM is one-dimensional. In addition, the stable manifold is also one dimensional since Eq. 共1兲 can be reversed with respect to time. We applied a projection G : ⌺ p → C for drawing invariant manifolds, where C means the set of all complex numbers. The projection is defined as27 N
hk = G共xk兲 = 兺
n=0
再
1 2 i共2/N兲n u˙ e + UOn共un兲ei共2/N兲n 2 n
冎
+ UIn共un − un−1兲ei共2/N兲共n+共1/2兲兲 ,
共5兲
where N = 9. The fixed boundaries are denoted by u0 and u9. The values of u0 and u9 are kept at zero. On-site and intersite potentials of the nth oscillator are represented by UOn and UIn, respectively. The on-site and the intersite potentials are given by UOn共un兲 =
␣1 2 1 4 u + u , 2 n 4 n
2 ␣2 UIn共un − un−1兲 = 共un − un−1兲2 + 共un − un−1兲4 . 2 4
共6兲
The locus of an ILM is given by k = arg hk. The velocity of an traveling ILM can be estimated from the difference between k and k+1.27 The structure of unstable manifolds is schematically drawn by projection. Figure 7共a兲 shows the structure at 2 = 0.001. In Fig. 7共a兲, stable and unstable ILMs are represented by open circles and squares, respectively. Unstable manifolds are drawn by solid curves. The unstable manifold of P3-4 has a cyclic structure, which is centered between P3-4 and ST4 in Fig. 7共a兲. The structure implies that a traveling ILM initially excited near P3-4 wanders between P3-4 and ST4. In Fig. 7共b兲, the behavior of the traveling ILM is shown with energy distribution given by Eq. 共6兲. The dark region corresponds to high energy state. At first, the energy is mainly distributed on the third and fourth sites and between them. The locus of the energy distribution is at 3.5. In other words, the traveling ILM stays near P3-4. The traveling ILM suddenly moves to ST4 at t ⯝ 50. The energy concentrates on
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Capture and release of traveling ILM 0.05
ST4 and ST5. The reciprocal behavior corresponds to the structure of the unstable manifolds. In addition, the traveling ILM maintains its localized energy distribution, while it wanders in the array. Therefore, the structure of unstable manifolds implies the behavior of a traveling ILM, which is generated near an unstable ILM.
0.04 0.03
Velocity
0.02 0.01 0
ST4
P3-4
-0.01
P4-5
-0.02 -0.03
B. Invariant manifold and nonlinear coupling coefficient
-0.04 3.5
Site
4 Site
4.5
6
200
5
160
4
120
3
80
2
40
1 0
0
50
(b)
100
150 200 Time
250
300
350
Energy
-0.05
(a)
FIG. 7. 共a兲 Unstable manifold of P3-4 and P4-5. Coexisting ILMs are represented by circles and squares. The arrow in the upper panel implies the direction of a perturbation against P3-4. The dots lying on the upper left are caused by definition of the hyper plane ⌺4. 共b兲 Temporal development of a traveling ILM. Darkness corresponds to the energy.
the third oscillator at t = 60. Then the traveling ILM immediately returns to P3-4. Finally the traveling ILM reciprocally moves for a long period. Figures 7共a兲 and 7共b兲 show that the behavior of the traveling ILM is predictable by the structure of unstable manifolds in the phase space. Unstable manifolds of unstable ST modes at 2 = 0.01 are shown in Fig. 8共a兲. The unstable manifolds have cyclic structures as well as the unstable manifolds of P modes at 2 = 0.001. However, the cyclic structures of the unstable manifolds of ST modes are centered at a stable P mode. The unstable manifold of ST4 is located nearby ST5 and vice versa. Thus, a traveling ILM will move from the vicinity of ST4 toward ST5 if the traveling ILM is excited near the ST4. The behavior of the traveling ILM is shown in Fig. 8共b兲. The traveling ILM that was initially at n = 4 begins to move at t ⯝ 50. Finally the traveling ILM reciprocally moves between 0.05 0.04 0.03
Velocity
0.02 0.01
P4-5
0 -0.01
ST4
ST5
-0.02 -0.03 -0.04 4
Site
4.5 Site
5
7
140
6
120 100
5
80 60
4
40
3 2 0
(b)
Energy
-0.05
(a)
20
50
100
150 200 Time
250
300
350
0
FIG. 8. 共a兲 Unstable manifold of ST3, ST4, and ST5. 共b兲 Temporal development of a traveling ILM.
The structure of unstable manifolds in phase space also depends on the nonlinear coupling coefficient because the stability of coexisting ILMs is flipped by varying 2. Figure 8 shows the structure of unstable manifolds at 2 = 0.003, 0.005, and 0.006. The stability change occurs between 2 = 0.005 and 0.006. Unstable manifolds of P modes at 2 = 0.003 have cyclic structures. The phase structure shown in Fig. 8共a兲 asymptotically changes to Fig. 8共b兲 with increasing 2. Figure 8共b兲 shows a simple structure similar to the phase space of a pendulum system. The inset in Fig. 8共b兲 shows the structure around P3-4. There is no hetero- or homoclinic connection. The phase structure at 2 = 0.006 is also a simple structure. In Fig. 8共c兲, the unstable manifold of ST4 reaches both the vicinity of ST5 and ST4. If 2 increases from 0.006 to 0.01, the structure changes to Fig. 8共a兲. Therefore, the structure of unstable manifolds in phase space is changed with its cyclic structures maintained with respect to the nonlinear coupling coefficient. However, the structure is drastically changed when the stability change occurs. In Sec. V, the drastic change is applied to a manipulation of a traveling ILM. V. CAPTURE AND RELEASE
The behavior of a traveling ILM is determined by the structure of unstable manifolds in phase space. Thus, the traveling ILM can be manipulated by changing the phase structure. In the microcantilever array, the nonlinear coupling coefficient flips the stability of coexisting ILMs and changes the phase structure. In this section, capture and release of a traveling ILM is numerically discussed. We assume that a traveling ILM is initially excited near a coexisting ILM, which is stable. The traveling ILM stays around the stable ILM. If the stability change is caused by rapid shift of the nonlinear coupling coefficient, the traveling ILM begins to move along the unstable manifold of the destabilized ILM. That is to say, the traveling ILM is released. The released ILM will wander in the array. The wandering ILM can be captured by the stability change if the wandering ILM approaches to the vicinity of an unstable ILM. The captured ILM stays around the stabilized ILM. Capture and release of a traveling ILM is numerically shown in Fig. 9共a兲. An ILM initially stands at P3-4. In the initial state, the nonlinear coupling coefficient is set at 2 = 0.006. Thus, the P3-4 is stable. The nonlinear coupling coefficient is discontinuously changed from 0.006 to 0.005 at k = 69, where k denotes the map number corresponding to the time evolution. As a result, the first stability change is caused. The ILM is released. The released ILM leaves from the destabilized P3-4 at k ⯝ 110. Then it approaches to the
Chaos 19, 013138 共2009兲
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0
ST4
P4-5
ST5
8 (a)
120
6
80
4 2
k = 69
0
0
80
4 2
0 1e 5 3 9998
120
6
1e 5
3.5
k = 118
(c)
8
(c) ¯2 = 0.006
0
-0.05
k = 69
0
3 502
S te
Velocity
-0.05 0.03
4 2
1e 4
35
0 200 160 120 80 40 0
6
0
1e 4 3 498
40
k = 134
(b)
8
(b) ¯2 = 0.005
S te
Velocity
-0.05 0.03
4
4
0
4 0002
Site
4.5
5
Energy
(a) ¯2 = 0.003
Energy
Velocity
0.05
k = 69 0
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40
k = 115 100
150
200
k
250
300
0
FIG. 9. Schematic relationship between coexisting ILMs and unstable manifolds for 共a兲 2 = 0.003, 共b兲 0.005, and 共c兲 0.006. The structure of the vicinity of an unstable ILM is shown in the insets. Unstable manifolds are located very close to the unstable ILM. There is no homo- or heteroclinic connection.
FIG. 10. Capture and release of traveling ILM. A stable ILM is initially excited. The stability of coexisting ILMs is changed at k = 69, where k depicts the number of the map and corresponds to the time development. The nonlinear coupling coefficient is instantaneously changed here. The stability is flipped again at 共a兲 k = 134, 共b兲 118, and 共c兲 115.
vicinity of P4-5 because the unstable manifold of P3-4 reaches near the P4-5 关see Fig. 8共b兲兴. After the released ILM reaches the P4-5, 2 is set at 0.006 again at k = 134. The P4-5 gains stability by the second stability change. The released ILM is captured around the stabilized P4-5. Consequently, the ILM travels from P3-4 to P4-5. However, the released ILM is not captured if the second stability change is caused at k = 118, as shown in Fig. 9共b兲. The ILM travels in the whole of array. The direction of the traveling ILM is turned by the end of array. On the other hand, Fig. 9共c兲 shows that the released ILM is captured around the ILM where the traveling ILM initially stands. The second stability change is caused at k = 115. The traveling ILM reciprocally moves around P3-4. The difference in the behavior is due to the phase structures. A released ILM travels along unstable manifolds from unstable P3-4, as shown in Fig. 10共a兲. After the second stability change, the phase structure is drastically changed. Figure 10共b兲 shows the phase structure and trajectories of traveling ILMs after the second stability change. For the trajectory corresponding to Fig. 9共b兲, the traveling ILM is located outside all cyclic structures of unstable manifolds. On the other hand, the traveling ILM is inside the cyclic structure for the trajectory in Figs. 9共a兲 and 9共c兲. It implies that if a traveling ILM released by the first stability change is inside the cyclic structure when the second stability change occurs, the traveling ILM is captured around a stabilized ILM. In addition, a traveling ILM wandering the whole of array is generated if the traveling ILM is located outside all cyclic structures 共see Fig. 11兲. The manipulation using the stability change requires that the nonlinear coupling coefficient 2 of Eq. 共1兲 is adjustable. The nonlinear on-site coefficient 1 can also flip the stability because the ratio 2 / 1 determines the stability of coexisting ILMs.18 It has been reported that an on-site nonlinearity can be adjustable by applying a static electric field to a micro-
cantilever array.17 The electric field is applied between each cantilever and a substrate facing the array, and an electric force is thus induced in each cantilever. The on-site potential is modified as UOn共un兲 = ␣1u2n / 2 + 1u4n / 4 − ␦1V2 / 共d⬘0 + un兲,17,28 where d0⬘ and V depict a nondimensionalized distance and the voltage between the cantilever array and the substrate, respectively. The relative magnitude of the static electric potential is determined by a coefficient ␦1 which depends on the size of cantilever. According to Maclaurin’s expansion, an applied electric field changes the on-site nonlinear coefficient as 1⬘ = 1 − 4␦1V2 / d⬘05. Therefore the on-site nonlinearity can be varied as a function of the voltage. If a microcan-
Ve oc ty
0.03
0.005
k = 134 k = 118 k = 115
k = 118
0.02
k = 115 0.01
0 0.03
Ve oc ty
(a): ¯2
ST3
ST4 P3 4
(b): ¯2
k = 134
ST5 P4 5
0.006 k = 118
0.02
0.01
k = 115
k = 134
P3 4
0 3
ST3
3.5
4
ST4
Site
4.5
P4 5
5
ST5
FIG. 11. Trajectories of traveling ILMs and unstable manifolds of unstable ILMs. The trajectories are projected by G. Open circles and boxes correspond to coexisting ILMs. Gray curves represent unstable manifolds of unstable ILMs. The trajectories are drawn by sequences of points. 共a兲 Trajectories during the first and the second stability changes, and 共b兲 trajectories after the second stability change.
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Chaos 19, 013138 共2009兲
Capture and release of traveling ILM
tilever array is fabricated to have the stable ST modes, the manipulation is possible because an applied voltage decreases the nonlinear on-site coefficient. In addition, a macromechanical cantilever array with tunable on-site nonlinearity is also proposed to study the manipulations of ILM.29 The dynamics of the macrocantilever array is similar to the coupled oscillators discussed in this paper. On-site potential of each cantilever is adjusted by an static magnetic field caused by an electromagnet facing to the tip of the cantilever. Because each cantilever has a permanent magnet at the free end, the magnetic field causes a nonlinearity in the on-site potential. The nonlinearity is easily varied by adjusting the current flowing in the electromagnets. Consequently, the manipulation method using parameter adjustment can be confirmed experimentally for both micro- and macrocantilever arrays. VI. CONCLUSION
In this paper, it has been shown how a traveling ILM behaves in the phase space. Unstable manifolds of unstable coexisting ILMs strongly affect the behavior of traveling ILM. That is, it is suggested that the structure of unstable manifolds in the phase space governs the traveling ILM. On the basis of the fact that the global phase structure is changed by a nonlinear coefficient in the coupled oscillators, we have proposed a new method to manipulate ILM. Since the equation of motion of the coupled oscillators is generalized by nondimensionalization, the results in this paper can be applied to various systems without loss of generality. In addition, the possibility to realize the manipulation in experiments is discussed for micro- and macrocantilever arrays. In micrometer scale, a static electric field can vary a nonlinear on-site coefficient as a function of the voltage between the cantilever array and the substrate. On the other hand, a static magnetic field can be used to adjust the nonlinear on-site coefficient in macroscale. Therefore the manipulation using stability change can be possibly applied to experimental systems. ACKNOWLEDGMENTS
The authors would like to thank Professor Masayuki Sato, Kanazawa University, Japan, for his helpful comments and discussions. One of the authors 共M.K.兲 would like to
thank Dr. Kazuyuki Yoshimura, NTT corporation, Japan, for fruitful discussions regarding the stability of ILMs. This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology in Japan, The 21st Century COE Program No. 14213201, and the Global COE program. A. J. Sievers and S. Takeno, Phys. Rev. Lett. 61, 970 共1988兲. E. Fermi, J. Pasta, and S. Ulam, The Collected Papers of Enrico Fermi 共University of Chicago Press, Chicago, 1955兲, Vol. 2, pp. 978–988. 3 S. Flach and C. R. Willis, Phys. Rep. 295, 181 共1998兲. 4 D. K. Campbell, S. Flach, and Y. S. Kivshar, Phys. Today 57共1兲, 43 共2004兲. 5 E. Trías, J. J. Mazo, and T. P. Orlando, Phys. Rev. Lett. 84, 741 共2000兲. 6 P. Binder, D. Abraimov, A. V. Ustinov, S. Flach, and Y. Zolotaryuk, Phys. Rev. Lett. 84, 745 共2000兲. 7 H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 共1998兲. 8 R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 2726 共1999兲. 9 J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 共London兲 422, 147 共2003兲. 10 M. Sato, B. E. Hubbard, A. J. Sievers, B. Ilic, D. A. Czaplewski, and H. G. Craighead, Phys. Rev. Lett. 90, 044102 共2003兲. 11 B. I. Swanson, J. A. Brozik, S. P. Love, G. F. Strouse, A. P. Shreve, A. R. Bishop, W.-Z. Wang, and M. I. Salkola, Phys. Rev. Lett. 82, 3288 共1999兲. 12 K. Kisoda, N. Kimura, H. Harima, K. Takenouchi, and M. Nakajima, J. Lumin. 94–95, 743 共2001兲. 13 M. Sato and A. J. Sievers, Nature 共London兲 432, 486 共2004兲. 14 M. Sato, S. Yasui, M. Kimura, T. Hikihara, and A. J. Sievers, Europhys. Lett. 80, 30002 共2007兲. 15 M. Sato, B. E. Hubbard, L. Q. English, A. J. Sievers, B. Ilic, D. A. Czaplewski, and H. G. Craighead, Chaos 13, 702 共2003兲. 16 M. Sato, B. E. Hubbard, A. J. Sievers, B. Ilic, and H. G. Craighead, Europhys. Lett. 66, 318 共2004兲. 17 M. Sato, B. E. Hubbard, and A. J. Sievers, Rev. Mod. Phys. 78, 137 共2006兲. 18 M. Kimura and T. Hikihara, Phys. Lett. A 372, 4592 共2008兲. 19 T. Hikihara, Y. Okamoto, and Y. Ueda, Chaos 7, 810 共1997兲. 20 T. Hikihara, K. Torii, and Y. Ueda, Phys. Lett. A 281, 155 共2001兲. 21 T. Hikihara, K. Torii, and Y. Ueda, Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, 999 共2001兲. 22 M. R. M. Crespo da Silva and C. C. Glynn, J. Struct. Mech. 6, 437 共1978兲. 23 M. R. M. Crespo da Silva, Int. J. Solids Struct. 24, 1225 共1988兲. 24 P. Malatkar and A. H. Nayfeh, Nonlinear Dyn. 31, 225 共2003兲. 25 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. 共Cambridge University Press, New York, 1992兲. 26 L. M. Marín and S. Aubry, Nonlinearity 9, 1501 共1996兲. 27 P. A. Houle, Phys. Rev. E 56, 3657 共1997兲. 28 M. I. Younis and A. H. Nayfeh, Nonlinear Dyn. 31, 91 共2003兲. 29 M. Kimura and T. Hikihara, Phys. Lett. A 373, 1257 共2009兲. 1 2
CHAOS 19, 013139 共2009兲
Cantori of the dissipative sawtooth map Alessandra Celletti1,a兲 and Massimiliano Guzzo2,b兲 1
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy 2 Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, I-35121 Padova, Italy
共Received 5 September 2008; accepted 12 February 2009; published online 31 March 2009兲 We investigate the existence of cantori for a dissipative version of the sawtooth map. Making use of an explicit parametric representation of the solution, we prove that cantori exist for any irrational value of the frequency. We also perform a numerical study aimed to determine some dynamical properties of the dissipative sawtooth map. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3094217兴 A relevant role in the dynamics of (discrete or continuous) nearly integrable dynamical systems is played by invariant tori, which are characterized by an irrational winding number. Nearly integrable systems are ruled by a perturbing parameter, which measures the nonlinearity of the dynamics. As this parameter is increased, invariant tori become distorted and displaced until a critical threshold is reached. At this point the invariant tori break down and leave place to the so-called Aubry– Mather sets, which are still invariant, but they are graphs of Cantor sets. For this reason one usually refers to the remnant tori as cantori; despite their gaps, the cantori can still provide a barrier to the diffusion in the phase space.1,2 The proof of the existence of cantori in conservative models is based on the application of a variational principle to the action functional (see Refs. 3 and 4). Here we address the question of the existence of cantori in dissipative nearly integrable systems. I. INTRODUCTION
We concentrate on a discrete sample known as the sawtooth map, which is described by a piecewise linear mapping arising from the linearization of the standard map about a fixed point. The conservative version is described by the equations y n+1 = y n + f共xn兲,
xn+1 = xn + y n+1 ,
共1.1兲
where xn 苸 S1 = R / Z, y n 苸 R, 苸 R denotes the perturbing parameter and the analytic expression of the perturbation f on the covering R of S1 is f共x兲 ⬅ mod共x,1兲 − f共x兲 ⬅ 0
1 2
if 0 ⬍ mod共x,1兲 ⬍ 1,
The mapping 共1.1兲 is area preserving, since its Jacobian J is equal to Electronic mail: [email protected]. b兲 Electronic mail: [email protected]. 1054-1500/2009/19共1兲/013139/6/$25.00
冉
1
1 1+
冊
;
it undergoes an elliptic regime as −4 ⬍ ⬍ 0, a hyperbolic regime whenever ⬍ −4 or ⬎ 0, and a parabolic regime for = −4 or = 0 共see Ref. 5兲. For ⬎ 0 there are no invariant circles and the phase space is populated by cantori and by periodic orbits. An explicit analytic solution for the cantori associated with Eq. 共1.1兲 was provided in Ref. 6 共see also Refs. 7–10兲 through a suitable parametric representation. In Ref. 11 it was proven that all cantori have zero fractal dimension 共see also Ref. 12兲. In the dissipative setting the sawtooth map is described by the equations y n+1 = by n + c + f共xn兲,
xn+1 = xn + y n+1 ,
共1.2兲
where b 苸 R is the dissipative parameter and c 苸 R is the net flux. Notice that we consider ⬎ 0,b in 共0,1兲, while one reduces to the conservative case setting b = 1 and c = 0. Mimicking the conservative case, we parametrize the solutions as x共兲 = + u共兲, where the parametric coordinate 苸 S1 is a linear function of the frequency of the motion and where u = u共兲 is a suitable function for which we provide an analytical expression 共see Sec. II兲. Using this representation of the solution we prove the existence of cantori associated with irrational frequencies 共see Sec. III兲. We conclude with some numerical experiments to investigate the phase space structure, the relation between cantori and periodic orbits, and the hyperbolic character of the dynamics 共see Sec. IV兲. II. A PARAMETRIC REPRESENTATION OF THE SOLUTION
if x 苸 Z.
a兲
J=
From the definition 共1.2兲 of the dissipative sawtooth map, one easily obtains that the following equation holds: xn+1 − 共1 + b兲xn + bxn−1 = c + f共xn兲.
共2.1兲
Let us parametrize the orbits with real frequency with a function x共兲, where 苸 S1 is the parametric coordinate such that, if xn = x共兲, then xn+1 = x共 + 兲. We require that in 19, 013139-1
© 2009 American Institute of Physics
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Chaos 19, 013139 共2009兲
A. Celletti and M. Guzzo
关0,1兴 x共兲 is a monotonically increasing function of and that x共 + 1兲 = x共兲 + 1. It will be convenient to look for parametrizations x共兲 of the form x共兲 = + u共兲,
共2.2兲
where u = u共兲 is an unknown function periodic with respect to . Equation 共2.1兲 becomes u共 + 兲 − 共1 + b兲u共兲 + bu共 − 兲 = c + 共b − 1兲 + f共 + u共兲兲.
The equations for j ⱕ −2 and j ⱖ 2 provide, respectively, − and +, i.e., 1+b+ + 2b
冑冉
1+b+ 2b
1+b+ + + = 2
冑冉
1+b+ 2
− =
␣− =
b␣2− ⬅ A −␣ , 共1 + b + 兲− − 1
␣+ =
␣2+ ⬅ A +␣ , 共1 + b + 兲+ − b
⬁
兺 a j f共 + j兲 j=−⬁
共2.3兲
for suitable coefficients u0, a j 共j 苸 Z兲. We remark that the representation 共2.3兲 is valid both for invariant tori and periodic orbits. By setting 1 u0 ⬅ 关共1 − b兲 − c兴,
共2.4兲
one obtains
兺 关a j−1 − 共1 + b兲a j + ba j+1兴f共 + j兲 = c + 共b − 1兲 + f共 + u共兲兲.
We proceed to look for an explicit solution u共兲 which satisfies the above equation in the domain 苸 共0 , 1兲. We will show that, having fixed a frequency for given values of the parameters and b, there exists a constant c and a function u = u共兲 such that Eq. 共2.2兲 is a solution of Eq. 共2.1兲. We remark that f共 + u共兲兲 = f共兲 + u共兲 whenever + u共兲 苸 共0 , 1兲 for 苸 共0 , 1兲 关indeed we will show that x共兲 苸 共0 , 1兲 for any 苸 共0 , 1兲; see Lemma 3 below兴; then, one finds ⬁
兺
␣=
关a j−1 − 共1 + b兲a j + ba j+1兴f共 + j 兲
j=−⬁
⬁
= f共兲 + 兺 a j f共 + j 兲. j=−⬁
The coefficients a j can be found by comparison of terms of the same order. Indeed, for j = 0 one has a−1 − 共1 + b兲a0 + ba1 = 共1 + a0兲, while for j ⫽ 0:
a0 ⬅ − ␣
for j = 0,
a j ⬅ − ␣+−兩j兩 +
for j ⬎ 0
for some real numbers ␣−, ␣, ␣+, −, + to be defined as follows.
共2.7兲
Combining the above formulas, one easily obtains the relations
␣− = ␣+ = ␣,
b− = + .
Finally we obtain that u = u共兲 is given by Eq. 共2.3兲 with a⫾j = − ␣−兩j兩 ⫾ , where ␣ is defined in Eq. 共2.7兲 and ⫾ in Eq. 共2.5兲. The coefficient u0 is determined imposing that the solution satisfies x共0兲 = 0, as described by the following lemma. Lemma 1: If u0 in Eq. 共2.3兲 satisfies ⬁
−j u0 ⬅ ␣ 兺 −j − 兵f共− j 兲 + b f共j 兲其,
共2.8兲
j=1
then the function x = x共兲 given by Eq. 共2.2兲 satisfies x共0兲 = 0. Proof: It suffices to show that u共0兲 = 0. Let us expand u共兲 as ⬁
−1
u共兲 = u0 +
兺
j=−⬁
a j f共 + j 兲 + a0 f共兲 + 兺 a j f共 + j 兲; j=1
using that f共0兲 = 0, one computes the value in = 0 as u共0兲 = u0 +
⬁
兺
Let us write the coefficients a j as for j ⬍ 0,
共2.6兲
−1 . 1 + b + − A−−1 − − bA++
再
a j f共j兲 + 兺 a j f共j兲
j=−⬁
aj ⬅ −
− b.
−1 + ␣−−1 − + b ␣ + + , 1+b+
−1
a j−1 − 共1 + b兲a j + ba j+1 = a j .
␣−−兩j兩 −
共2.5兲 2
where A− and A+ are auxiliary quantities defined by the first two equations of Eq. 共2.6兲. Then one has
␣=
⬁
j=−⬁
1 − , b
For j = −1, 1, 0, one obtains
We make the following ansatz on u by assuming that it can be expanded in the form u共兲 = u0 +
冊 冊
2
= u0 − ␣
j=1
−1
兺
j=−⬁
⬁
−兩j兩 −兩j兩 − f共j 兲 + 兺 + f共j 兲 j=1
⬁
−j = u0 − ␣ 兺 兵−j − f共− j 兲 + + f共j 兲其 j=1 ⬁
−j = u0 − ␣ 兺 −j − 兵f共− j 兲 + b f共j 兲其; j=1
冎
013139-3
Chaos 19, 013139 共2009兲
Cantori of the dissipative sawtooth map
due to Eq. 共2.8兲 one finds u共0兲 = 0, which implies x共0兲 = 0. 䊐
u共兲 −兩j兩 = − ␣ 兺 −兩j兩 − − ␣ − ␣ 兺 + j⬍0 j⬎0
III. EXISTENCE OF DISSIPATIVE CANTORI
=−
In this section we prove the existence of cantori for the dissipative sawtooth mapping. Let us denote by D the discontinuity set defined as
䊐 Lemma 3: For any 苸 共0 , 1兲 the function x = x共兲 satisfies x共兲 苸 共0 , 1兲. Proof: Due to Lemma 2 the variation of u = u共兲 outside the gaps is equal to ⫺1 共recall that 1 + u / = 0兲. If we prove that the variation of u on the gaps is 1, we obtain that the total variation is zero; therefore x = + u共兲 spans between 0 and 1, being x共0兲 = 0. To measure the variation of u on the gaps, let us start from
D = 兵 苸 关0,1兴 : = mod共j ,1兲, j 苸 Z其. We state the following. Proposition: For an irrational frequency , let ˜ = 兵共x,y兲 苸 R2:共x,y兲 − 共x共兲,y共兲兲 苸 Z ⫻ 兵0其 M for some 苸 关0,1兴 \ D其 ˜ / 共Z ⫻ 兵0其兲. Then, M is a Cantor set for the and let M = M mapping 共1.2兲 with c as in Eq. 共2.4兲. We consider the function x共兲 constructed in Sec. II, and we remark that the sequence 共xn , y n兲 = 共x共 + n兲 , + x共 + n兲 − x共 + 共n − 1兲兲兲 defines an orbit of the map if mod共 + n , 1兲 苸 共0 , 1兲 for any n, namely, if mod共 , 1兲 苸 关0 , 1兴 \ D. As a consequence, the set M is a nonempty invariant set whose properties can be obtained by the following lemmas. Lemma 2: The derivative of x共兲 with respect to is zero for any 苸 共0 , 1兲 \ D. Proof: Since x共兲 / = 1 + u共兲 / , let us show that u共兲 / = −1. In fact, starting from
−1
u共兲 = u0 − ␣ f共兲 − ␣ 兺 −兩j兩 − f共 + j 兲 j=−⬁
⬁
− ␣ 兺 −兩j兩 + f共 + j 兲, j=1
where the discontinuity points = mod共k , 1兲. Next we remark that
located
at
k
→0−
is equal to 0 if k ⫽ −j and it is equal to ⫺1 if k = −j. In fact, the function f共兲 is, by definition, discontinuous over the integers where the variation amounts to ⫺1. If k = −j, then k = mod共共k + j兲 , 1兲 = 0 and the variation of f共兲 is ⫺1. If k ⫽ −j, being 共k + j兲 irrational, then k = mod共共k + j兲 , 1兲 is irrational; therefore f共兲 is continuous in k and has zero variation. Finally, we obtain
兺 a j f共 + j兲
j=−⬁
−兩j兩 = u0 − ␣ 兺 −兩j兩 − f共 + j 兲 − ␣ f共兲 − ␣ 兺 + f共 + j 兲, j⬍0
are
lim f共mod共共k + j兲,1兲 + 兲 − lim f共mod共共k + j兲,1兲 + 兲
→0+
⬁
u共兲 = u0 +
␣ ␣ −␣− = − 1. − − 1 + − 1
j⬎0
one has
lim u共k + 兲 − lim u共k + 兲
→0+
冋
→0−
= lim − ␣ f共mod共k,1兲 + 兲 − ␣ 兺 →0+
⬁
−1
冋
−兩j兩 − f共mod共共k
j=−⬁
j=1
⬁
−1
− lim − ␣ f共mod共k,1兲 + 兲 − ␣ 兺 →0−
+ j兲,1兲 + 兲 − ␣ 兺 −兩j兩 + f共mod共共k + j兲,1兲 + 兲
−兩j兩 − f共mod共共k
j=−⬁
册
+ j兲,1兲 + 兲 − ␣ 兺 −兩j兩 + f共mod共共k + j兲,1兲 + 兲 j=1
册
−兩k兩 = ␣共␦k0 + −兩k兩 − + + 兲.
Notice that the latter equality implies that the jumps are positive at the discontinuities; summing over the jumps one concludes that ⬁
−兩k兩 ␣ 兺 共−兩k兩 − + + 兲 + ␣ = k=1
冉
1 1 − −1 −
−1+
1 1 − −1 +
冊
− 1 + 1 ␣ = 1.
䊐 Proof of the Proposition: By Lemmas 2 and 3 we have proven that the set M is closed, bounded, nonempty, without isolated points, and totally disconnected. 䊐
013139-4
Chaos 19, 013139 共2009兲
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b=1,lambda=0.0001
IV. NUMERICAL REPRESENTATION OF THE DISSIPATIVE CANTORI
0.678 0.676 0.674 0.672 0.67 0.668
y
We report some numerical experiments obtained using the parametric representation of the mapping 共1.2兲 for different values of the parameters, with c varying according to Eq. 共2.4兲. The sum in Eq. 共2.3兲 has been truncated up to a suitable value, say, j max, and the following approximation has been used:
0.666 0.664 0.662
jmax
兺
a j f共 + j 兲;
0.66
共4.1兲
0.658
j=−jmax
0
0.2
0.4
0.6
0.8
1
0.8
1
0.8
1
x b=1,lambda=0.001 0.68
0.675
0.67 y
in our experiments we set jmax = 2000. The coordinate runs on an equally spaced grid of 2000 points in the interval 共0,1兲; the axes provide the parametric representation x = + u共兲 and y = + u共兲 − u共 − 兲, where u is given in Eq. 共2.3兲. We start from the conservative case setting b = 1 and we vary the parameters and as shown in Fig. 1. In each panel ten different values of have been considered, starting from 0.658 to 0.676 with step size of 2 ⫻ 10−3. Results on the dissipative case are reported in Fig. 2; here has been set equal to 10−2, while the dissipative parameter b varies between 0.95 and 0.5. Again, ten different values of have been considered as in Fig. 1; for each set of parameters the value of c varies, thus yielding cantori belonging to different mappings, though plotted on the same phase space. Next we fix ⬅ g = 共冑5 − 1兲 / 2 and we set = 10−2; using the parametrization 共2.2兲 and 共2.3兲, we compute the cantorus with rotation number g as well as some periodic orbits with frequency approximating the golden ratio. We stress that the parametric solution 共2.3兲 is valid also for periodic orbits since no zero divisors appear for the sawtooth map, contrary to what happens for different mappings, for example, in the case of the standard map 共see, e.g., Ref. 13兲. In Fig. 3, the golden curve and the periodic orbits with frequencies 2/3, 8/13, 34/55, and 144/233 are shown within the conservative case, while Fig. 4 provides the same objects in the dissipative setting with b = 0.8. A qualitative analysis of the stability of the cantori can be performed by comparing the parametric solution 共2.3兲 to the direct iteration of the mapping 共1.2兲 and 共2.4兲. Precisely, by selecting a point 共x0 , y 0兲 of the cantorus obtained with the parametric approximation 共4.1兲 computed at = 1 / 2, we compare the sequence 共xn , y n兲 defined by
0.656
(a)
0.665
0.66
0.655
(b)
0
0.2
0.4
0.6 x
b=1,lambda=0.01 0.69 0.685 0.68 0.675 0.67 y
u jmax共兲 = u0 +
0.665 0.66 0.655 0.65 0.645
(c)
0
0.2
0.4
0.6 x
FIG. 1. The conservative sawtooth map for ten values of ranging between 0.658 and 0.676, while takes the values 10−4 共a兲, 10−3 共b兲, and 10−2 共c兲.
xn = n + u jmax共n兲,
N = 10−N. For these computations the parameter jmax was chosen depending on the numerical precision as follows:
y n = + u jmax共n兲 − u jmax共n − 兲, ˜ n , ˜y n兲 of the where n = mod共1 / 2 + n , 1兲, with the points 共x orbit of 共x0 , y 0兲 computed numerically iterating the map 共1.2兲. Figure 5 reports the distance dn = 兩xn − ˜xn兩 + 兩y n − ˜y n兩
共4.2兲
versus n in semilogarithmic scale, obtained with two computations characterized by different numerical precision
jmax =
冋
册
N . log10 +
From Fig. 5 it is clear that the distance among the points of the two sequences increases exponentially with respect to the iteration parameter n, providing a typical behavior of hyperbolic dynamics both in the conservative and in the dissipative sawtooth map.
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Cantori of the dissipative sawtooth map b=0.95,lambda=0.01
b=0.8,lambda=0.01
0.69
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y
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0
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1
(c)
0
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0
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b=0.9,lambda=0.01 0.685
0.685
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y
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1
y 0
0.1
0.2
0.3
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0.7
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1
(d)
V. CONCLUSIONS
Several physical situations are described by a nearly integrable dynamical system subject to a 共small兲 dissipation. Indeed, many interesting cases are provided by solar system dynamics, where the nearly integrable N-body problem can be assumed to be perturbed by small dissipations, such as the solar wind, the Yarkovski effect, or a tidal torque due to internal nonrigidity. In this context, the existence of attractive invariant tori has been established in Refs. 14 and 15 under general assumptions. As we know from the conservative setting, once the perturbing parameter increases above a critical threshold, the invariant tori break down and leave b=1,lambda=0.01
0.4
0.5 x
0.6
0.7
place to cantori. In this work we have addressed the question of the existence of cantori in the dissipative case. A positive answer comes from the investigation of a peculiar example, the dissipative sawtooth map. In this case we have been able to provide an analytic construction of the parametric solution and we have proved the existence of cantori provided the frequency of motion is irrational. The parametric representation of the solution is valid also for periodic orbits. In particular, having fixed an irrational winding number, the periodic orbits with frequency given by its rational approximants tend to the cantorus. This statement is corroborated by some numerical experiments. b=1,lambda=0.01
0.67
0.622 0.621
0.66
0.62
0.65
0.619 0.618
0.64
y
y
0.8
FIG. 2. The dissipative sawtooth map for ten values of ranging between 0.658 and 0.676; here = 10−2, while b = 0.95 共a兲, b = 0.9 共b兲, b = 0.8 共c兲, and b = 0.5 共d兲.
0.655
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x b=1,lambda=0.01
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(c)
b=1,lambda=0.01
y
1
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(b)
0.9
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0.61
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(b)
0.5 0.6 0.7 x b=0.5,lambda=0.01
0.675
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0.4
0
0.2
0.4
0.6 x
0.8
0.613
1
(d)
0
0.2
0.4
0.6 x
FIG. 3. The conservative sawtooth map with = 10−2; top left: the cantorus 共denoted by +兲 with frequency g is displayed together with the periodic orbit 共denoted by ⫻兲 with frequency 2/3; bottom left: the periodic orbit 共⫻兲 has frequency 8/13; top right: the periodic orbit has frequency 34/55; bottom right: the periodic orbit 共⫻兲 has frequency 144/233.
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b=0.8,lambda=0.01
0.67
0.623 0.622
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y
y
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0
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b=0.8,lambda=0.01
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1
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0.8
0.9
1
b=0.8,lambda=0.01
0.624
0.623 0.622
0.622
0.621 0.62
0.62
FIG. 4. The dissipative sawtooth map with = 10−2 and b = 0.8; top left: the cantorus 共denoted by +兲 with frequency g is displayed together with the periodic orbit with frequency 2/3; bottom left: the periodic orbit 共denoted by ⫻兲 has frequency 8/13; top right: the periodic orbit 共⫻兲 has frequency 34/55; bottom right: the periodic orbit 共⫻兲 has frequency 144/233.
0.618
y
y
0.619 0.618 0.617 0.616
0.616 0.615
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0
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The generalization of the proof to the standard map as well as to different systems requires advanced techniques due to the fact that the analytic form of the cantori is not known except for the peculiar case of the sawtooth map. We plan to face such problem in a future work.
1000
2000
0.3
0.4
(d)
3000
4000
5000
20
40
60
80
FIG. 5. 共Color online兲 Stability analysis of a cantorus of the dissipative sawtooth map with = 10−2 and b = 0.9; the figure reports the logarithm log10 dn 共in ordinate兲 vs n 共in abscissa兲, where dn was defined in Eq. 共4.2兲 as the distance between the parametric solution of the cantorus and the iterates of the dissipative sawtooth mapping. The top curve is obtained by means of a numerical computation characterized by a numerical precision of 10−40, while the bottom curve is characterized by a numerical precision of 10−80.
0.5 x
0.6
ACKNOWLEDGMENTS
We thank G. Benettin for useful discussions. M.G. has been supported by Project No. CPDA063945/06 of the University of Padova. A.C. and M.G. acknowledge PRIN 2007B3RBEY Dynamical Systems and applications of MIUR. 1
C. Efthymiopoulos, G. Contopoulos, and N. Voglis, Celest. Mech. Dyn. Astron. 73, 221 共1999兲. 2 R. S. MacKay, J. D. Meiss, and I. C. Percival, Physica D 13, 55 共1984兲. 3 J. Mather, Topology 21, 457 共1982兲. 4 S. Aubry and P. Y. Le Daëron, Physica D 8, 381 共1983兲. 5 P. Ashwin, Phys. Lett. A 232, 409 共1997兲. 6 I. C. Percival, AIP Conf. Proc. 57, 302 共1980兲. 7 Q. Chen, R. S. MacKay, and J. D. Meiss, J. Phys. A 23, L1093 共1990兲. 8 Q. Chen and J. D. Meiss, Nonlinearity 2, 347 共1989兲. 9 R. Coutinho, R. Lima, R. Vile la Mendes, and S. Vaienti, Ann. Inst. Henri Poincare, Sect. A 56, 415 共1992兲. 10 R. S. MacKay and J. D. Meiss, Nonlinearity 5, 149 共1992兲. 11 H.J. Schellnhuber, H. Urbschat, and A. Block, Phys. Rev. A 33, 2856 共1986兲. 12 H. J. Schellnhuber and H. Urbschat, Phys. Rev. A 38, 5888 共1988兲. 13 A. Celletti and C. Falcolini, Physica D 170, 87 共2002兲. 14 H. W. Broer, G. B. Huitema, and M. B. Sevryuk, Quasi-periodic motions in families of dynamical systems, Lecture Notes in Mathematics, Vol. 1645 共Springer–Verlag, Berlin, 1996兲. 15 A. Celletti and L. Chierchia, Arch. Ration. Mech. Anal. 191, 311 共2009兲.
CHAOS 19, 013140 共2009兲
Chaos of elementary cellular automata rule 42 of Wolfram’s class II Fang-Yue Chen,1 Wei-Feng Jin,2 Guan-Rong Chen,3 Fang-Fang Chen,4 and Lin Chen4 1
School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, People’s Republic of China 2 College of Pharmaceutical Sciences, Zhejiang Chinese Medical University, Hangzhou, Zhejiang 310018, People’s Republic of China 3 Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, People’s Republic of China 4 Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People’s Republic of China
共Received 30 October 2008; accepted 24 February 2009; published online 31 March 2009兲 In this paper, the dynamics of elementary cellular automata rule 42 is investigated in the bi-infinite symbolic sequence space. Rule 42, a member of Wolfram’s class II which was said to be simply as periodic before, actually defines a chaotic global attractor; that is, rule 42 is topologically mixing on its global attractor and possesses the positive topological entropy. Therefore, rule 42 is chaotic in the sense of both Li-Yorke and Devaney. Meanwhile, the characteristic function and the basin tree diagram of rule 42 are explored for some finite length of binary strings, which reveal its Bernoulli characteristics. The method presented in this work is also applicable to studying the dynamics of other rules, especially the 112 Bernoulli-shift rules of the elementary cellular automata. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3099610兴 In sample patterns of spatiotemporal evolution of one dimensional cellular automata, the human eye readily discerns between simple and complex structures. The generic behavior displayed by cellular automata was conjectured by Wolfram to fit into four classes, numbered from I to IV. Loosely speaking, the first two classes exhibit the simple space-time patterns as fixed points and periodic orbits, and the latter two contain the chaotic behaviors. Over the past few years much has been discussed concerning the relationship between the direct “visual” complexity and the notions of chaotic dynamical systems. It seems hard to argue the intrinsic complexity of classes I and II to be very high, according to any of the usual measures of complexity, particularly that organized around the symbolic dynamics of stationary symbol sequences. Recently, Chua et al. provided a nonlinear dynamics perspective to Wolfram’s empirical observations. Based on this work, the dynamics of rule 42, which is a member of class II, is fully characterized in the bi-infinite symbolic sequence space below. Surprisingly, it is found that rule 42 defines a chaotic global attractor and possesses very rich and complicated dynamical behaviors. This interesting result suggests that “visual” simply rules may be endowed with highly temporal chaotic properties. I. INTRODUCTION
Cellular automata 共CAs兲 are a class of spatially and temporally discrete, deterministic mathematical systems characterized by local interactions and an inherently parallel form of evolution. CAs, formally introduced by von Neumann in the early 1950s, are able to produce complex dynamical phenomena by means of designing simple local rules.1 Due to their simple mathematical constructions and distinguishing features, CAs have been widely used to model a variety of 1054-1500/2009/19共1兲/013140/6/$25.00
dynamical systems. The study of topological dynamics of CAs began with Hedlund in 1969, who viewed onedimensional CA 共1D CAs兲 in the context of symbolic dynamics as endomorphisms of the shift dynamical system,2 where the main results are the characterizations of surjective and open CAs. Based on the theoretical concept of universality, researchers have tried to develop even simpler and more practical architectures of CAs which can be used for widely diverse applications. In 1970, Conway proposed his now-famous game of life,3 which received widespread interests among researchers in different fields. In the early 1980s, Wolfram introduced space-time representations of 1D CAs and informally classified them into four classes by using dynamical concepts such as periodicity, stability, and chaos.4–6 In 2002, he introduced his monumental work A New Kind of Science.7 Based on this work, Chua et al. provided a nonlinear dynamics perspective to Wolfram’s empirical observations via the concepts such as characteristic function, forward time- map, basin tree diagram, and isle-of-Eden digraph.8–11 Although the dynamical behaviors of CAs can be analyzed—as for other dynamical systems—under different frameworks, the exact determination of their temporal evolutions is in general very hard if not impossible. In particular, many topological properties such as topological entropy, sensitivity, and topological mixing of CAs are undecidable.12 However, if one focuses on particular subclasses of CAs, many topological properties become tractable. For example, there are two well-understood subclasses of CAs, equicontinuous CAs and positively expansive CAs, with remarkably different stability properties.13,14 The chaotic properties of CAs are investigated in Refs. 15–18. As for nonlinear CAs belonging to class IV, many of them can produce gliders in their space-time evolutions.19 In addition, the dynamical be-
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haviors of surjective CAs, permutive and closing CAs, and linear CAs have been studied in Refs. 20–22. However, there are many distinct CAs whose dynamical behaviors are much less understood, especially those capable of universal computation. A special type of 1D CAs is the Bernoulli -shift CAs, which show Bernoulli-shift dynamics.8 Each of the 112 Bernoulli -shift rules has a ID code BN关␣ ,  , 兴, where ␣ denotes the number of attractors of rule N,  denotes the slope of the Bernoulli -shift map, and denotes the relevant forward time- map. Using finite-length bit strings with periodic boundary conditions, the space-time evolution of any 1 of the 112 Bernoulli -shift rules on their attractors can be uniquely predicted by two parameters:  = ⫾ 2 and . For example, the parameters of rule 42 are  = 2, = 1, and = 1, so the space-time evolution of rule 42 on this attractor can be predicted as follows: shift the bit strings to the left by 1 bit in every iteration. Note that rule 42 is neither linear nor permutive, so the methods of analyzing the dynamics of linear and permutive CAs are not applicable to it. The objective of this paper is to thoroughly characterize the dynamical behaviors of rule 42 under the framework of bi-infinite symbolic sequence space. It is surprising to find that rule 42, a member of Wolfram’s class II which was considered to be simple, actually possesses very rich and complex dynamics. The rest of the paper is organized as follows: Section II presents the basic concepts of 1D CAs and symbolic dynamics. Section III explores the qualitative properties of rule 42 via using the concepts of characteristic function and basin tree diagram on a special subset of the bi-infinite sequence space. Section IV identifies the global attractor of rule 42 which is simply a subshift of finite type. Section V demonstrates the chaotic dynamics of rule 42; that is, rule 42 is topologically mixing on the global attractor and possesses the positive topological entropy. Finally, Sec. VI highlights the main results in the analysis of Chua’s Bernoulli -shift rules and prospects for future studies.
II. PRELIMINARIES
For a finite symbol S, a word over S is finite sequence a = 兵a0 , . . . , an−1其 of elements of S. The length of a is denoted by 兩a兩 = n and the word of zero length is denoted by . Denote the set of all words of length n by Sn, the set of words over S by Sⴱ = 艛nⱖ0Sn, and the set of nonzero words by S+ = 艛nⱖ0Sn. If a is a finite or infinite word and I = 关i , j兴 is an interval of integers on which a is defined, then denote a关i,j兴 = 共ai , . . . , a j兲 and a关i,j兲 = 共ai , . . . , a j−1兲. b is a subword of a, denoted by b Ɱ a, if b = aI for some interval I 債 Z; otherwise, write b 葓 a. The set of bi-infinite configurations is denoted by SZ, and a metric d on SZ is defined as ⬁
¯兲 = d共x,x
1
¯兲 d 共x ,x
i i i , 兺 兩i兩 ¯兲 2 1 + d 共x ,x i=−⬁ i
i
i
where x, ¯x 苸 SZ, and di共· , ·兲 is the metric on S defined as
共1兲
TABLE I. The truth table of Boolean functions of rules 42, 112, 171, and 241. xi−1
xi
xi+1
N = 42
N = 112
N = 171
N = 241
0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 0
0 0 0 0 1 1 1 0
1 1 0 1 0 1 0 1
1 0 0 0 1 1 1 1
¯ i兲 = di共xi,x
再
1 if xi = ¯xi 0 if xi ⫽ ¯xi .
冎
共2兲
In SZ, the cylinder set of a word a 苸 Sn is 关a兴k = 兵x 苸 SZ 兩 x关k,k+n兲 = a其, where k 苸 Z. Obviously, such a set is both open and closed 共called clopen兲. The cylinder sets generate a topology on SZ and form a countable basis for this topology. Therefore, every open set is a countable union of cylinder sets. In addition, SZ is a Cantor space. The classical left-shift map is defined by 关共x兲兴i = xi+1 for any x 苸 SZ, i 苸 Z. By a theorem of Hedlund,2 a map T : SZ → SZ is a CA if and only if it is continuous and commutes with , i.e., ⴰ T = T ⴰ . For any CA, there exists a radius r ⱖ 0 and a local rule N : S2r+1 → S such that 关T共x兲兴i = N共x关i−r,i+r兴兲. Moreover, 共SZ , T兲 is a compact dynamical system. A set X 債 SZ is T invariant if T共X兲 債 X and strongly T invariant if T共X兲 = X. If X is closed and T invariant, then 共X , T兲 or simply X is called a subsystem of T. A set X 債 SZ is an attractor23 if there exists a nonempty clopen T-invariant set Y such that 艚nⱖ0Tn共Y兲 = X. Thus, there always exists a global attractor, denoted by ⌳ = 艚nⱖ0Tn共SZ兲, which is also called the limit set of T. For instance, let A denote a set of some finite words over S, and ⌳ = ⌳A is the set which consists of the bi-infinite configurations made up of all the words in A. Then, ⌳A is a subsystem of 共SZ , 兲, where A is said to be the determinative block system of ⌳. For a closed invariant subset ⌳ 債 SZ, the subsystem 共⌳ , 兲 or simply ⌳ is called a subshift of . For bi-infinite, 1D elementary cellular automata 共ECA兲, r = 1 and S is denoted by 兵0,1其. Each local rule can be expressed by a Boolean function. For example, the Boolean function of rule 42 is expressed as N共x关i−1,i+1兴兲 = ¯xi · xi+1 丣 ¯xi−1 · xi · xi+1,
∀ i 苸 Z,
共3兲
where xi 苸 S, “·,” “ 丣 ,” and “−” denote “AND,” “XOR” and “NOT” logical operations, respectively.24 The truth table of this Boolean function is shown in Table I, where rules 112, 171, and 241, which belong to the equivalence class of rule 42, are also listed. Then, the global map of rule 42 can be induced as follows: for any x 苸 SZ, 关T42共x兲兴i = ¯xi · xi+1 丣 ¯xi−1 · xi · xi+1,
∀ i 苸 Z,
where 关T42共x兲兴i denotes the ith symbol of T42共x兲.
共4兲
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Chaos of rule 42 of Wolfram’s class II
III. QUALITATIVE PROPERTIES OF RULE 42
In this section, the qualitative properties of rule 42 are examined on a special subset ⌺共I兲 傺 SZ, where 兺共I兲 兵x 苸 SZ兩x关kI,共1+k兲I−1兴 = x关0,I−1兴,
∀ k 苸 Z其.
共5兲
Thus, one can uniquely associate each x 苸 兺共I兲 with a binary string 共x0 , . . . , xI−1兲 with a finite length I and periodic boundary conditions. Therefore, all information needed to understand the dynamical evolutions of rule 42 on 兺共I兲 can be obtained via using a finite length I of binary strings with periodic boundary conditions. On one hand, the CA characteristic function 42 of rule 42, defined as I−1
42:RI → RI,
哫 42共兲 = 兺 y i · 2−共i+1兲 ,
共6兲
i=0
can be used to understand its dynamical evolutions, where is the decimal form of 共x0 , . . . , xI−1兲 defined by −共i+1兲 = 兺I−1 , and 共y 0 , . . . , y I−1兲 is the output binary string i=0 xi · 2 of 共x0 , . . . , xI−1兲 determined by rule 42. Here, the characteristic functions 42 and 242 are displayed in Fig. 1 showing only 500 points 共colored in black兲 to avoid clutter. In other words, each graph in Fig. 1 shows only 500 uniformly distributed values of 42, each calculated from a binary string with the length I = 90. On the other hand, the concept of basin tree diagram of rule 42 is also exploited to completely characterize its longterm time-asymptotic behaviors. It is worth pointing out that a period-n orbit is a period-n isle of Eden of rule N in the sense that there are no binary strings different from the points in this orbit which converge to it, i.e., it has an empty basin of attraction. The basin tree diagrams of rule 42 are studied for some finite length I. Because of lack of space, only representative attractors are exhibited in Fig. 2 for I = 9, where each binary string is displayed in black and white 共i.e., white pixel stands for 0 and black for 1兲 along with its decimal identification number, calculated from the decimal equivalent of the binary string as in Ref. 11. These numbers are enclosed by small circles and are represented as nodes of a digraph where a directed edge pointing from node s1 to node s2 means that binary string s1 maps to binary string s2 after one iteration under rule 42. Additionally, the qualitative properties of rule 42 extracted from its basin tree diagrams for some relatively small I are summarized in Table II. A careful examination of Figs. 1 and 2 and Table II reveals that the dynamics of rule 42 on each attractor obeys the same Bernoulli-shift law with the parameters = 1 and = 1 independent of I. Therefore, rule 42 is endowed with a global attractor, which is simply a left Bernoulli-shift. In addition, Fig. 2共b兲 provides valuable and insightful information to argue that different random initial bit strings could converge to this global attractor after one iteration under rule 42. These empirical characteristics will be proved to be true for the bi-infinite case in the following section. Observe that all possible periods of rule 42 are only factors of I, which implies that the largest period of attractors equals the length I. This qualitative property shows additional significant information to uncover the characteristic features of its space-
FIG. 1. Characteristic functions of rule 42. Although only 500 points are shown, the abscissa of each point is calculated with a 90 bit string resolution. All 500 points fall onto the left Bernoulli-shift map after one iteration.
time evolutions. Meanwhile, it is inferred that the periods of the periodic orbits of rule 42, especially its isles of Eden, increase at a linear rate as a function of I. Such linear growth of period-n as a function of I, compared with the exponential growth, is a signature of all Bernoulli-shift rules. It is remarked that the exponential growth is a signature of all complex-Bernoulli-shift rules and hyper-Bernoulli-shift rules.10,11 IV. GLOBAL ATTRACTOR OF RULE 42
An empirical glimpse at Figs. 1 and 2 reveals that the global attractor of rule 42 is a subshift. In this section, this
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TABLE II. Summary of qualitative properties of rule 42.
I 8 9 10 11 12 15 17
Number of attractors/number Bernoulli parameters of isles of Eden Period of attractors 共 , 兲 19/11 29/16 48/25 75/36 132/61 627/235 1857/606
1, 2, 4, 8 1, 3, 9 1, 2, 5, 10 1, 11 1, 2, 3, 4, 6, 12 1, 3, 5, 15 1, 17
共1,1兲 共1,1兲 共1,1兲 共1,1兲 共1,1兲 共1,1兲 共1,1兲
*
such that T42 兩⌳ = 兩⌳ if and only if ∀x = 共. . . , x−1 , x0 , x1 , . . .兲 苸 ⌳, xi−1, xi, and xi+1 cannot be 1 simultaneously, ∀i 苸 Z. Proof: (Necessity) Suppose that there exists a subset *
⌳ 傺 SZ such that T42 兩⌳ = 兩⌳. Then, ∀x = 共. . . , x−1 , x0 , x1 , . . .兲 苸 ⌳, we have 关T42共x兲兴i = 关共x兲兴i = xi+1, ∀i 苸 Z. It follows from Eq. 共4兲 that ¯xi · xi+1 丣 ¯xi−1 · xi · xi+1 = xi+1, which implies xi−1 · xi · xi+1 = 0, ∀i 苸 Z. Thus, if xi = 0, then xi−1 · xi · xi+1 = 0 regardless of the value of xi−1 and xi+1, ∀i 苸 Z. If xi = 1, then xi−1 · xi+1 = 0. This implies x关i−1,i+1兴 苸 兵共0 , 1 , 0兲 , 共0 , 1 , 1兲 , 共1 , 1 , 0兲其, ∀i 苸 Z. Hence, it is evident that ∀x 苸 ⌳, xi−1, xi, and xi+1 cannot be 1 simultaneously, ∀i 苸 Z. (Sufficiency) Assume that there exists a subset ⌳ 傺 SZ, and ∀x 苸 ⌳, xi−1, xi, and xi+1 cannot be 1 simultaneously, ∀i 苸 Z. This yields that xi−1 · xi · xi+1 = 0, ∀i 苸 Z. Thus, xi−1 · xi · xi+1 丣 xi+1 = xi+1; namely, ¯xi · xi+1 丣 ¯xi−1 · xi · xi+1 = xi+1, ∀i 苸 Z. Therefore, it is proved that T42共x兲 = 共x兲, ∀x 苸 ⌳. The entire proof of the theorem is therefore complete. In the following discussion, the set A is always denoted by A⫽兵共0,0,0兲,共0,0,1兲,共0,1,0兲,共0,1,1兲,共1,0,0兲,共1,0,1兲,共1,1,0兲其. Proposition 2: ⌳ = ⌳A is a subshift of finite type. Proof: The proof follows directly from the definition of subshift of . Proposition 3: T42共x兲 苸 ⌳A, ∀x 苸 SZ; that is, ⌳A is the global attractor of T42. Proof: To prove that ⌳A is the global attractor of T42, we must show that ⌳A = 艚nⱖ0Tn42共SZ兲. We now consider separately two situations. 共a兲 共b兲 FIG. 2. Attractors choosing from the basin tree diagram of rule 42 for I = 9. 共b兲 and 共d兲 are also isles of Eden. There are a total of 29 attractors with the possible periods n = 1 , 3 , 9. The evolution dynamics on each attractor, especially 共a兲–共d兲, is a left Bernoulli-shift with the same parameters.
qualitative property is proved to be true in the bi-infinite symbolic sequence space. Proposition 1: For rule 42, there exists a subset ⌳ 傺 SZ
x 苸 ⌳A. Since ⌳A is invariant under T42, we have T42共x兲 苸 ⌳A. x 苸 SZ − ⌳A; we claim that T42共x兲 苸 ⌳A. Next, the proof is by contradiction.
Suppose that there exists y 苸 SZ − ⌳A such that T42共x兲 = y. Then, we have 共1 , 1 , 1兲 Ɱ y by Proposition 1. Without loss of generality, let y 关−1,1兴 = 共1 , 1 , 1兲. Since y i = 关T42共x兲兴i, ∀i 苸 Z, we especially have 关T42共x兲兴−1 = ¯x−1 · x0 丣 ¯x−2 · x−1 · x0 = 1,
共7兲
关T42共x兲兴0 = ¯x0 · x1 丣 ¯x−1 · x0 · x1 = 1,
共8兲
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关T42共x兲兴1 = ¯x1 · x2 丣 ¯x0 · x1 · x2 = 1.
共9兲
It therefore follows from Table I that x关−1,1兴 苸 兵共1 , 0 , 0兲 , 共1 , 0 , 1兲 , 共1 , 1 , 0兲其 by Eq. 共8兲. Furthermore, if x关−1,1兴 = 共1 , 0 , 0兲 or 共1,0,1兲, then ¯x1 · x2 丣 ¯x0 · x1 · x2 = 0 which is in contradiction to Eq. 共9兲. If x关−1,1兴 = 共1 , 1 , 0兲, then ¯x−1 · x0 丣¯ x−2 · x−1 · x0 = 0 which is contradictory to Eq. 共7兲. This completes the proof. Remark 1: It follows from Propositions 2 and 3 that rule 42 exhibits the same dynamics as the shift map on its global attractor. V. DYNAMICAL BEHAVIORS OF T42
The objective of this section is to investigate the complexity and chaotic dynamics of T42 on its global attractor ⌳A obtained in Sec. IV. It follows from Proposition 2 that dynamical behaviors of T42 on ⌳A can be characterized via a subshift ⌳A, which is the subshift of finite type. As the topological dynamics of a subshift of finite type is largely determined by the properties of its transition matrix, it is helpful to briefly review some definitions from Ref. 25. A matrix A is positive if all of its entries are non-negative, irreducible if ∀i , j, there exists n such that Anij ⬎ 0, aperiodic if there exists N, such that Anij ⬎ 0, n ⬎ N, ∀i , j. If ⌳A is a two-order subshift of finite type, then it is topologically mixing if and only if A is irreducible and aperiodic, where A is its associated transition matrix with Aij = 1, if 共i , j兲 Ɱ ⌳; otherwise Aij = 0. First, the topologically conjugate relationship between 共⌳A , 兲 and a two-order subshift of finite type is established, and the dynamical behaviors of T42 on ⌳A are discussed based on existing results. Let Sˆ = 兵r0 , r1 , r2 , r3 , r4 , r5 , r6其 be a new symbolic set, where ri, i = 0 , . . . , 6, stand for 共000兲, 共001兲, 共010兲, 共011兲, 共100兲, 共101兲, 共110兲, respectively. Then, one can construct a new symbolic space SˆZ on Sˆ. Denote by B = 兵共rr⬘兲 兩 r = 共b0b1b2兲 , r⬘ = 共b0⬘b1⬘b2⬘兲 苸 Sˆ , ∀ 1 ⱕ j ⱕ 2 such that b j = b⬘j−1其. Further, the two-order subshift ⌳B of is defined by ⌳B
ⴱ
where ri = 共xixi+1xi+2兲, ∀i 苸 Z. Then, it follows from the definition of ⌳B that for any x 苸 ⌳A, one has 共x兲 苸 ⌳B; namely, 共⌳A兲 債 ⌳B. One can easily check that is a homeomorphism and ⴰ = ⴰ . Therefore, 共⌳A , 兲 and 共⌳B , 兲 are topologically conjugate. Proposition 5. T42 is topologically mixing on ⌳A. Proof: It follows from Ref. 25 that a two-order subshift of finite type is topologically mixing if and only if its transition matrix is irreducible and aperiodic. Meanwhile, it is easy to verify that An is positive for n ⱖ 4, where A is the transition matrix of the two-order subshift ⌳B. This implies that A is irreducible and aperiodic. The proof is thus completed. Proposition 6. The topological entropy of T42 on ⌳A 3 equals log共a + b + 31 兲 ⬇ 0.609, where a = 31 冑 19+ 3冑33 and b 1冑 3 = 3 19− 3冑33. Proof: Recall that two topologically conjugate systems have the same topological entropy and the topological entropy of on ⌳B equals log 共A兲, where 共A兲 is the spectral radius of the transition matrix A of the subshift ⌳B. It is easy to calculate the spectral radius of the transition matrix A, obtaining the topological entropy of T42 on ⌳A as log 共A兲 3 3 = log关 31 冑 19+ 3冑33+ 31 冑 19− 3冑33+ 31 兴 ⬇ 0.609. This completes the proof. It follows from Ref. 26 that the positive topological entropy implies chaos in the sense of Li-Yorke. Meanwhile, the topological mixing is also a very complex property of dynamical systems. A system with topologically mixing property has many chaotic properties in different senses. For example, positive topological
冢 冣 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0
A=
0 0 0 0 0 0 1
.
1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0
Proposition 4. 共⌳A , 兲 and 共⌳B , 兲 are topologically conjugate; namely, 共⌳A , T42兲 and 共⌳B , 兲 are topologically conjugate. Proof: Define a map from ⌳A to ⌳B as follows:
:⌳A → ⌳B ,
chaos in the sense of ⇒
entropy
Li-Yorke ⇑
.
chaos in the sense of ⇐ topologically mixing
ⴱ
= 兵r = 共. . . , r−1 , r0 , r1 , . . .兲 苸 SˆZ 兩 ri 苸 Sˆ , 共ri , ri+1兲 葓 B , ∀ i 苸 Z其. Therefore, it is easy to calculate the transition matrix A of the subshift ⌳B as
ⴱ
x = 共. . .,x−1,x0,x1, . . .兲 哫 共. . .,r−1,r0,r1, . . .兲,
Devaney In conclusion, the mathematical analysis presented above provides the rigorous foundation for the following theorem. Theorem 1: T42 is chaotic in the sense of both Li-Yorke and Devaney on the global attractor ⌳A. Remark 2: Through this section an interesting and important observation is obtained: rule 42, a member of Wolfram’s class II which was said to be simply as periodic before, actually identifies a chaotic global attractor and possesses very rich and complex dynamical properties in the bi-infinite sequence space. This result verifies that the “visual” simply rules can also be endowed with highly temporal chaotic properties. VI. CONCLUSION AND DISCUSSION
How complex is rule 42? It is known that Wolfram divided the 256 ECA rules informally into four classes by
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adopting dynamical concepts such as periodicity, stability, and chaos through large amount of computer simulations and empirical observations. Thus, rule 42 is classified into class II which was said to be simple as periodic before. It seems hard to argue the intrinsic complexity of rule 42 to be very high, according to any of the usual measures of complexity, particularly that organized around the symbolic dynamics of stationary symbol sequences. Through this work, it is shown that rule 42 actually identifies a chaotic global attractor in the bi-infinite sequence space. More precisely, T42 is topologically mixing on ⌳A and possesses the positive topological entropy. Therefore, it is inferred that T42 is chaotic in the sense of both Li-Yorke and Devaney on ⌳A. This interesting result suggests that CA rules, which were considered to be simple before, may have much more complicated dynamical behaviors from the viewpoint of symbolic dynamics. Also, it suggests that spatial complexity may be higher than temporal complexity. Noticeably, rule 42 is not positively expansive CA rule, which is conjugated 共isomorphic兲 to one-sided shift having dense orbits, dense periodic configurations, positive topological entropy, and sensitive dependence on the initial condition. Meanwhile, rule 42 is also nonlinear and nonpermutive, so the methods presented in Refs. 20–22 and 27 are not applicable to it. It seems worth mentioning that the analysis method developed in this paper is also applicable to studying the dynamical behaviors of all 112 Bernoulli-shift rules therein. Among these Bernoulli-shift rules, there are 11 equivalence classes which have only one robust Bernoulli-shift attractor with the Bernoulli-shift parameters:  = 2, = 1 共or ⫺1兲, = 1; namely, rules 2, 10, 24, 34, 42, 46, 130, 138, 152, 162, and 170. Therefore, all these rules are possible candidate ECA rules for the chaotic properties of rule 42. Meanwhile, the dynamics of the Bernoulli-shift rules with all possible distinct parameters has been studied.28–30 The method also gives some support for investigating the dynamics of subsystems of other rules, especially the hyper-Bernoulli-shift rules. For example, several chaotic subsystems of rule 110, which is capable of universal computation, have been found so far. Additionally, it is important to further find the nonrobust Bernoulli-shift rules with Bernoulli-shift attractors with relatively small basins of attraction, which are not listed in Chua’s Bernoulli-shift rule table. However, the usefulness of the characteristic function and the basin tree diagram cannot be overemphasized. Since there are 2I distinct states for 1D CAs with I cells, plotting the characteristic function and the basin tree diagram of rule N becomes impractical in terms of computer time as the length I increase beyond modest values. Last but not least, it is necessary to uncover the relation-
ship between the universality of CAs and the CA characteristic functions in future research. ACKNOWLEDGMENTS
This research was jointly supported by NSFC 共Grant Nos. 60872093 and 10832006兲 and the City University of Hong Kong 共Grant No. SRG 700274兲. 1
J. von-Neumann, in Theory of Self-reproducing Automata, edited by A. W. Burks 共University of Illinois Press, Urbana, 1966兲. G. A. Hedlund, Theory Comput. Syst. 3, 320 共1969兲. 3 M. Gardner, Sci. Am. 223共10兲, 120 共1970兲. 4 S. Wolfram, Rev. Mod. Phys. 55, 601 共1983兲. 5 S. Wolfram, Physica D 10, 1 共1984兲. 6 S. Wolfram, Theory and Applications of Cellular Automata 共World Scientifc, Singapore, 1986兲. 7 S. Wolfram, A New Kind of Science 共Wolfram Media, Champaign, 2002兲. 8 L. O. Chua, V. I. Sbitnev, and S. Yoon, Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, 1045 共2005兲. 9 L. O. Chua, V. I. Sbitnev, and S. Yoon, Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 1097 共2006兲. 10 L. O. Chua, J. B. Guan, I. S. Valery, and J. Shin, Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2839 共2007兲. 11 L. O. Chua, K. Karacs, V. I. Sbitnev, J. B. Guan, and J. Shin, Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 3741 共2007兲. 12 L. P. Hurd, J. Kari, and K. Culi, Ergod. Theory Dyn. Syst. 82, 255 共1992兲. 13 M. D’amico, G. Manzini, and L. Margara, Theor. Comput. Sci. 290, 1629 共2003兲. 14 K. Culikii, L. P. Hurd, and S. Yu, Physica D 45, 357 共1990兲. 15 J. Urias, R. Rachtman, and A. Enciso, Chaos 7, 688 共1997兲. 16 M. Courbage, D. Mercier, and S. Yasmineh, Chaos 9, 893 共1999兲. 17 F. Ohi, Complex Systems 17, 295 共2007兲. 18 F. Ohi, AUTOMATA-2008: Theory and Applications of Cellular Automata, 2008 共unpublished兲. 19 J. G. Freire, O. J. Brison, and J. A. C. Gallas, Chaos 17, 026113 共2007兲. 20 J. P. Allouche and G. Skordev, J. Comput. Syst. Sci. 67, 174 共2003兲. 21 P. Kurka, Ergod. Theory Dyn. Syst. 17, 417 共1997兲. 22 P. Favati, G. Lotti, and L. Margara, Theor. Comput. Sci. 174, 157 共1997兲. 23 Using a finite length I with periodic boundary conditions, the concept of an attractor of rule N was introduced in Ref. 8. In this paper, the concept of an attractor for both finite and bi-infinite cases has no distinction, which can be easily distinguished from the context. 24 J. B. Guan, S. W. Shen, C. B. Tang, and F. Y. Chen, Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 4245 共2007兲. 25 B. Kitchens, Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts 共Springer-Verlag, Berlin, 1998兲. 26 F. Blanchard, E. Glasner, and S. Kolyada, J. Reine Angew. Math. 547, 51 共2002兲. 27 G. Cattaneo, M. Finellt, and L. Margara, Theor. Comput. Sci. 244, 219 共2000兲. 28 W. F. Jin, F. Y. Chen, G. R. Chen, L. Chen, and F. F. Chen, J. Cell. Automata 共unpublished兲. 29 F. F. Chen, F. Y. Chen, W. F. Jin, and L. Chen, International Workshop on Chaos-Fractals Theories and Applocations, 2008 共unpublished兲, p. 2863. 30 L. Chen, F. Y. Chen, F. F. Chen, and W. F. Jin, International Workshop on Chaos-Fractals Theories and Applocations, 2008 共unpublished兲, p. 2866. 2
CHAOS 19, 015101 共2009兲
Introduction to Focus Issue: Nonlinear Dynamics in Cognitive and Neural Systems F. Tito Arecchi1,a兲 and Jürgen Kurths2,b兲 1
C.N.R.-Istituto Nazionale di Ottica Applicata, Largo E. Fermi 6, 50125 Firenze, Italy and Dipartimento di Fisica, Universitá di Firenze, 50019 Sesto Fiorentino, Italy 2 Potsdam Institute for Climate Impact Research, 14412 Potsdam, Germany and Institute of Physics, Humboldt University Berlin, 12489 Berlin, Germany
共Received 5 March 2009; accepted 5 March 2009; published online 31 March 2009兲 In this Focus Issue, two interrelated concepts, namely, deterministic chaos and cognitive abilities, are discussed. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3106111兴 Chaotic dynamics as loss of initial information affects, in general, any nonlinear system with three or more degrees of freedom. This finding by Poincaré was disregarded for some decades and then reconsidered by Lorenz in 1963.1 Since then, deterministic chaos has been observed and analyzed in many physical systems, starting from simple laboratory situations as in, e.g., coupled pendula, nonequilibrium fluids, and lasers. In the case of large systems made of many components, it is crucial to discriminate chaotic motion from statistical effects due to the linear sum of uncorrelated causes; many indicators have been suggested and tested in this regard. The distinctive feature of deterministic chaos is a positive rate of information loss called Kolmogorov entropy. On the other hand, cognition means extracting from the environment some features on the basis of which a living being reacts. This occurs at any level, starting from unicellular organisms that build decisions by processing the information provided by chemical and thermal gradients. In pluricellular organisms, a processing speed up is, in several cases, obtained by electrical rather than chemical mutual communication. This is not limited to animals but is now being observed in plants as well.2 As animals get more complex, the distributed processing networks specialize into a dedicated organ—the brain. A brain is made of huge networks of coupled units—the neurons—each one sufficiently rich in dynamics to have its own complex behavior if studied in isolation from other partners. It is, however, known that small neuron networks are stabilized against chaos by a combination of inhibitory mutual feedbacks. They can be considered as stable modules with a specific function, such as a stereotyped reaction to a stimulus as it occurs in central pattern generator or encoding mechanisms as explored in the olfactory system of insects.3,4 Encoding is just the first step, then how to read information and make good use of it. An attractive paradigm is that of fixed point attractors,5,6 which allows a sound processing model. Any input is classified according to its resemblance to a template previously stored in the system; this fact endows an attractor network of a capacity scalable with its size. However, a stable attractor network has a strong limitation in a兲
Electronic mail: [email protected]. Electronic mail: juergen [email protected].
b兲
1054-1500/2009/19共1兲/015101/3/$25.00
its limited capacity. Recurring to the dynamical attractors associated with chaotic dynamics, one can build richer scenarios, flexibly adaptable to many situations rather than restricted to a fixed repertoire. Here, the central cognitive issue emerges. In the presence of chaos, the information loss rate can be high enough to preclude a convenient reaction. In fact, the only operational way to attribute cognitive ability to an agent is to look at its reactions. A smart cognitive agent can compensate for chaotic information loss by recurring to memory resources that add extra signals perturbing the original input and, hence, recoding the original dynamical space. This provides a slowdown of the information loss rate. The strategy is called “control of chaos;” its introduction7 signaled a breakthrough in chaotic scenarios. In the case of many coupled chaotic units, a way of displaying a coherent behavior, that is, holding some collective information for a time much longer than the chaotic decay rate of a single unit, is mutual synchronization.8 A complete approach to synchronization strategies is provided in Refs. 9 and 10 and, more generally, for synchronization in complex networks in Ref. 11. However, there remain several basic open problems in this topic, especially the following. 共1兲 How does chaos affect some brain areas? Do these areas behave as separate domains which maintain their individuality by holding inner correlations and yet elaborate global messages that they exchange at their boundary? 共2兲 Low resolution measurement techniques, such as EEG 共electro-encephalography兲 and fMRI 共functional magnetic resonance imaging兲, are unable to detect the single neuron electrical activity; they rather average out the behavior of a large number of neurons. On the other hand, sampling single neuron signals requires invasive methods as microelectrodes or inspection of calcium ion release by two-photon fluorescence. These microanalyses can be carried on sparse samples, and their connection toward the global behavior of a whole network remains an open problem. 共3兲 Consider a set of coupled neurons in an array acting globally as a feature detector; it is hypothesized that the interplay between bottom-up stimuli arriving from sensory areas and top-down signals fed from memory stores11 yields collective synchronization.12 How well do
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these attractive hypotheses match the observed types of behavior? 共4兲 The main cognitive problem is how a given sensorial input elicits a decision 共motor response兲, which affects the same environment from where the input originated兲. Once some early neuron groups encode the sensory signals into specific sequences of neuron spikes,3,4 is that code already driving an appropriate action, so that an external observer can establish that a cognitive act has occurred or is there a successive further recoding? Calling cognition the loop perception action, this can be fast 共around 100 ms兲 and explained by the Bayes procedures13 or slower 共around, say, 800 ms兲 and mediated—in the case of humans—by processing in the prefrontal cortex.14 共5兲 In this second case, the cognitive agent makes use of its own resources, as modeled, e.g., by adaptive resonance theory11 in order to reduce the original Kolmogorov entropy and be able to build an appropriate reaction on its environment. The amount of this reduction can change from an individual to another. Let us call creativity15 the “best” recoding that lengthens the information loss 共not too short, otherwise it would preclude an appropriate reaction, nor too long, otherwise it would make the agent blind to successive inputs兲. How do we measure creativity? How is it related to symbolic language? Is there creativity in nonhuman animals? Can we foresee a robot creativity? After these sparse considerations, we shortly introduce the 16 papers16–31 collected in this Focus Issue. This Focus Issue presents interdisciplinary approaches to these problems, including modern methods from nonlinear dynamics, statistical physics, and mathematical statistics, as well as from cognitive and neuroscience. New kinds of experimental data, but also traditional ones, are combined and confronted with new modeling approaches and new data analysis techniques. The main intentions are to understand functionality of brain dynamics but also spatiotemporal dynamics of brain disorders and their identification and forecasts. Several model approaches are presented for understanding of brain functioning and of disorder. Rothkegel and Lehnertz23 studied structure formation in a small-world network composed of rather simple model neurons 共pulsecoupled nonleaky integrate-and-fire neurons兲 and find, in this nonregular network, multistable behavior, including local wave patterns, as well as global collective firing. This is an important first step to relate brain disorders 共e.g., epileptic seizures兲 to the topology of synaptic wiring. Traveling pulses are of fundamental importance in neuroscience because they transmit information, but they are also related to cell depolarizations in migraine or stroke. They are investigated in Ref. 24 by means of a hybrid model which combines volume and synaptic transmission. Methods for control of pulse propagation via moderate manipulations are developed. Zamora et al.31 presented a statistical analysis of corticocortical communication paths of a cat in order to understand how simultaneous segregation and integration of information is possible in the brain. It comes out that the modular struc-
Chaos 19, 015101 共2009兲
ture and the presence of highly connected hubs are crucial for the multisensory and complex information processing capabilities. Komarov et al.21 modeled the formation of slow brain rhythms with a minimal inhibitory circuit and investigate when it provides structurally stable solutions. Based on a subtle bifurcation analysis, it is shown that the condition for this is the existence of a stable heteroclinic channel. A neural network model for working memory is extended in Ref. 28 by a negative feedback. After decomposing the fast and slow dynamics and performing bifurcation analysis and simulations, they showed that this feedback is sufficient to explain the dynamics of reflex epilepsy. The influence of external stimuli on a network of chaotic oscillators was studied by Ciszak et al.17 It is shown that shortly below the onset of collective synchronization, a stimulus operating on only one oscillator is sufficient to generate a stable regime of synchronization. This provides an explanation of sudden transitions in the brain and the occurrence of conscious states. In Ref. 27, the typical brain dynamics is regarded as transitory. This leads to new ideas for typical scenarios of such a nonrandom and nonchaotic behavior. Then, nine hypotheses on the formation of dynamic memory and perception are presented. Next, various aspects of modeling cognitive and psychological phenomena are discussed. Each modeling is in the strong sense an inverse problem. This approach is consequently performed in Ref. 19 for cognitive modeling. The authors decomposed the model in three main steps and showed that the main problem—the determination of the synaptic weight matrices—is an ill-posed one. To overcome this serious problem, they designed an efficient regularization technique basing on Hebbian learning. Dynamic motor processes are a crucial element in various sensory systems to enhance perception; an outstanding case study for this is fixational eye movements.26 These eye movements represent self-generated noise. The authors studied the influence of external noise and show the constructive role of noise in visual perception. This perceptual performance, found experimentally, is described in a mathematical model. A challenging question is how can physiological processes give rise to psychological phenomena 共moods, cognitive modes, etc.兲. Allefeld et al.16 proposed a Markov coarse graining, which relates physiologically characterized microstates to psychologically characterized macrostates. Additionally, they developed an analysis technique to identify stable macrostates from EEG data and demonstrated its potential for epilepsy patients. To describe associated memory functioning and to represent conscious and unconscious mental processes, a complex network model is developed in Ref. 30. In this model, consciousness is related with symbolic and linguistic memory activity in the brain. As discussed in the previous contributions, brain activity is characterized by highly complex spatiotemporal dynamics. The measurements of this activity are usually noisy and nonstationary. These difficulties call for highly sophisticated techniques of data analysis. Vejmelka and Paluš29 proposed a special filtering technique to extract nonlinear oscillations from broadband signals. This is an important first step for further analysis, especially phase synchronization analysis. This way, EEG data during sleep are analyzed. Romano et al.22 derived a test statistics basing on twin surrogates to test
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Introduction to Focus Issue
for phase synchronization. Their new analytic expressions enable determination of the optimal parameters for the generation of twin surrogates, which is a highly relevant problem for analyzing experimental data. Applying this technique, it is shown that the left and right fixational eye movements are statistically significantly phase synchronized. The development of sleep is of fundamental importance in the maturation of brain functions. Based on in utero electrocorticogram data, different sleep states of fetal sheep are studied in Ref. 25 with a combination of a stability analysis and a bispectral analysis. This way, premature sleep states are clearly identified. Hamann et al.20 studied sleep by means of a synchronization analysis of the cardiovascular system. Based on this synchrogram technique, they can clearly distinguish different main sleep stages. This offers a new way for assessing sleep and sleep disorders by simply analyzing Holter recordings. The concept of synchronization is in combination with the unscented Kalman filter used for parameter estimation of nonlinear neuron models in Ref. 18. This method yields robust estimates even for strongly noisy data. F.T.A. acknowledges partial support from Ente CRF under the Project “Dinamiche cerebrali caotiche.” We are deeply grateful to Janis Bennett 共Assistant Editor, Chaos兲 for her invaluable help in preparing this Focus Issue. E. N. Lorenz, J. Atmos. Sci. 20, 130 共1963兲. E. Masi, M. Ciszak, G. Stefano, L. Renna, E. Azzarello, C. Pandolfi, S. Mugnai, F. Baluška, F. T. Arecchi, and S. Mancuso, Proc. Natl. Acad. Sci. U.S.A. 106, 4048 共2009兲. 3 M. I. Rabinovich, P. Varona, A. I. Selverston, and H. D. I. Abarbanel, Rev. Mod. Phys. 78, 1213 共2006兲. 4 M. I. Rabinovich, R. Huerta, P. Varona, and V. S. Afraimovich, PLoS Comput. Biol. 4, e1000072 共2008兲. 5 J. J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 79, 2554 共1982兲. 6 J. J. Hopfield, Scholarpedia J. 2, 1977 共2007兲. 7 E. Ott, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 64, 1196 共1990兲. 8 L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 共1990兲. 9 A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization—A Universal Concept in Nonlinear Sciences 共Cambridge University Press, Cambridge, UK, 2001兲. 10 A. Arenas, A. Diaz-Guilero, J. Kurths, Y. Moreno, and C. Zhou, Phys. Rep. 469, 93 共2008兲. 11 G. A. Carpenter and S. Grossberg, in The Handbook of Brain Theory and 1 2
Chaos 19, 015101 共2009兲 Neural Networks, 2nd ed., edited by M. A. Arbib 共MIT Press, Cambridge, MA, 2003兲, pp. 87–90. 12 W. Singer, Scholarpedia J. 2, 1657 共2007兲. 13 K. P. Kording and D. M. Wolpert, Trends Cogn. Sci. 10, 319 共2006兲; W. J. Ma, J. M. Beck, and A. Pouget, Curr. Opin. Neurobiol. 18, 1 共2008兲. 14 E. Rodriguez, N. George, J.-P. Lachaux, J. Martinerie, B. Renault, and F. J. Varela, Nature 共London兲 397, 430 共1999兲. 15 F. T. Arecchi, Eur. Phys. J. Spec. Top. 146, 205 共2007兲. 16 C. Allefeld, H. Atmanspacher, and J. Wackermann, “Mental states as macrostates emerging from EEG dynamics,” Chaos 19, 015102 共2009兲. 17 M. Ciszak, A. Montina, and F. T. Arecchi, “Control of transient synchronization with external stimuli,” Chaos 19, 015104 共2009兲. 18 B. Deng, J. Wang, and Y. Che, “A combined method to estimate parameters of neuron from a heavily noise-corrupted time series of active potential,” Chaos 19, 015105 共2009兲. 19 P. beim Graben and R. Potthast, “Inverse problems in dynamic cognitive modeling,” Chaos 19, 015103 共2009兲. 20 C. Hamann, R. P. Bartsch, A. Y. Schumann, T. Penzel, S. Havlin, and J. W. Kantelhardt, “Automated synchrogram analysis applied to heartbeat and reconstructed respiration,” Chaos 19, 015106 共2009兲. 21 M. A. Komarov, G. V. Osipov, J. A. K. Suykens, and M. I. Rabinovich, “Numerical studies of slow rhythms emergence in neural microcircuits: Bifurcations and stability,” Chaos 19, 015107 共2009兲. 22 M. Carmen Romano, M. Thiel, J. Kurths, K. Mergenthaler, and R. Engbert, “Hypothesis test for synchronization: Twin surrogates revisited,” Chaos 19, 015108 共2009兲. 23 A. Rothkegel and K. Lehnertz, “Multistability, local pattern formation, and global collective firing in a small-world network of nonleaky integrateand-fire neurons,” Chaos 19, 015109 共2009兲. 24 F. M. Schneider, E. Schöll, and M. A. Dahlem, “Controlling the onset of traveling pulses in excitable media by nonlocal spatial coupling and timedelayed feedback,” Chaos 19, 015110 共2009兲. 25 K. Schwab, T. Groh, M. Schwab, and H. Witte, “Nonlinear analysis and modeling of cortical activation and deactivation patterns in the immature fetal ECoG,” Chaos 19, 015111 共2009兲. 26 C. Starzynski and R. Engbert, “Noise-enhanced target discrimination under the influence of fixational eye movements and external noise,” Chaos 19, 015112 共2009兲. 27 I. Tsuda, “Hypotheses on the functional roles of chaotic transitory dynamics,” Chaos 19, 015113 共2009兲. 28 S. Verduzco-Flores, B. Ermentrout, and M. Bodner, “From working memory to epilepsy: Dynamics of facilitation and inhibition in a cortical network,” Chaos 19, 015115 共2009兲. 29 M. Vejmelka and M. Paluš, “Detecting nonlinear oscillations in broadband signals,” Chaos 19, 015114 共2009兲. 30 R. S. Wedemann, R. Donangelo, L. A. V. de Carvalho, “Generalized memory associativity in a network model for the neuroses,” Chaos 19, 015116 共2009兲. 31 G. Zamora-Lópes, C. Zhou, and J. Kurths, “Graph analysis of cortical networks reveals complex anatomical communication substrate,” Chaos 19, 015117 共2009兲.
CHAOS 19, 015102 共2009兲
Mental states as macrostates emerging from brain electrical dynamics Carsten Allefeld, Harald Atmanspacher, and Jiří Wackermann Department of Empirical and Analytical Psychophysics, Institute for Frontier Areas of Psychology and Mental Health, Freiburg, 79100 Germany
共Received 23 October 2008; accepted 29 December 2008; published online 31 March 2009兲 Psychophysiological correlations form the basis for different medical and scientific disciplines, but the nature of this relation has not yet been fully understood. One conceptual option is to understand the mental as “emerging” from neural processes in the specific sense that psychology and physiology provide two different descriptions of the same system. Stating these descriptions in terms of coarser- and finer-grained system states 共macro- and microstates兲, the two descriptions may be equally adequate if the coarse-graining preserves the possibility to obtain a dynamical rule for the system. To test the empirical viability of our approach, we describe an algorithm to obtain a specific form of such a coarse-graining from data, and illustrate its operation using a simulated dynamical system. We then apply the method to an electroencephalographic recording, where we are able to identify macrostates from the physiological data that correspond to mental states of the subject. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3072788兴 The question of how it is to be understood that physiological processes can give rise to psychological phenomena—moods, cognitive modes, sleep stages, being conscious/unconscious—has already been the subject of much philosophical and scientific debate, but has still not been satisfactorily settled. In this paper, we follow the line of thought that conceives of the domain of the mental as something that “emerges” from the physical, but argue that this only becomes a proper answer if the term emergence can be given a precise meaning. Following concepts that arose in theoretical physics, in particular statistical mechanics and nonlinear dynamics, we propose that the mental and the physical should be understood as different descriptions of the same system. A description is given in terms of the states the system can assume, and it becomes especially useful if it is possible to formulate a dynamical law, that is a rule that determines the change of the system state over time. In this case, the relation between the lower-level description in the form of (physiologically characterized) microstates and the higher-level description in the form of (psychologically characterized) macrostates is given by a so-called Markov coarsegraining. In order to give empirical support for the viability of these ideas, we turn to a special case in which the macrostates are metastable states, defined by areas in the state space within which the system stays for prolonged periods of time. We describe an algorithm to identify metastable states from empirical data, and we illustrate its operation using data from a simulated system. The method is then applied to a recording of brain electric (electroencephalographic, EEG) data from a patient suffering from a form of epilepsy characterized by frequent brief seizures during which the subject becomes mentally absent. We show that our algorithm is able to identify the segments of the recording belonging to normal and paroxysmal EEG, respectively. The method is therefore capable of identifying metastable macrostates from the physiological data which closely correspond to 1054-1500/2009/19共1兲/015102/12/$25.00
mental states of the subject, providing in this first test case support for the viability of our theoretical approach to the nature of the relation between physiology and psychology. I. INTRODUCTION
The existence of correlations between psychological and physiological phenomena, especially brain processes, is the basic empirical fact of psychophysiological research. Relations between mental processes, including modes of consciousness, and those occurring in its physical “substrate,” the central nervous system, are generally taken as a matter of course: They form the basis for the use of drugs in the treatment of mental disorders in psychiatry, they are applied as a research tool to shed light on the details of psychological mechanisms, and the explication of the neural structures underlying mental functioning forms the subject of cognitive neuroscience. Still, it remains unclear what the nature of the observed correlations is and what exactly is to be conceived as a neural correlate of a psychological phenomenon. One way to approach these issues is to interpret the mental as a domain emerging from an underlying physiological domain 共Broad, 1925; Beckermann et al., 1992兲. However, despite its long history reaching to recent scientific contributions 共e.g., Darley, 1994; Seth, 2008兲, the term emergence is not very well defined and it is used in a large number of different meanings 共cf. Stephan, 2002; O’Connor and Wong, 2006兲. In our understanding, emergence is a relation between different descriptions of the same system. In this view, the occurrence and correlation of psychological and physiological phenomena is due to the fact that the object of psychophysiological research 共the research subject兲 can be approached and examined in different ways. More specifically, emergence is to be conceived as a relation between different descriptions each of which is useful or adequate in its own
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manner. The question arises how there can be more than one adequate description for the same system, and what has to be the nature of their relation in order to permit this. In this paper, we present one possible answer to these questions, motivated by ideas on the emergence of mental states from neurodynamics introduced by Atmanspacher and beim Graben 共2007兲, where the two descriptions take on the form of a dynamical system. We introduce the further specification that the relation between the two associated state spaces is characterized by a Markov coarse-graining 共Sec. II兲, which leads us to consider metastable states as a particular form of emergent states. In order to demonstrate the practical viability of these ideas, we develop a method to identify metastable states from empirical data 共Sec. III兲, and illustrate the operation of the algorithm using data from a simulated system 共Sec. IV兲. In the application of the method to a recording of brain electrical activity, we are able to identify states closely corresponding to the mental states of a subject, based on the analysis of the EEG data alone 共Sec. V兲. II. EMERGENCE IN DYNAMICAL SYSTEMS
A descriptive approach that has proven very fruitful in physics and other fields of the natural sciences is utilizing the concept of a dynamical system 共Robinson, 1995; Chan and Tong, 2001兲. Such a description is formulated with respect to the states the system can assume, and a dynamical rule that defines the way the state of the system evolves over time. The possible system states form a state space, which in the most general case is just a set of identifiable and mutually distinguishable elements. 关This is in accordance with the concept of system states in cybernetics and related disciplines 共cf. Ashby, 1962兲, but is at variance with the use of the term in physics where a state space is generally taken to be spanned by a set of observables 共properties that can be precisely quantified兲. Such a less structured concept of state space is useful because it also covers cases where it is not obvious how to endow that space with a formal structure, for instance mental states. However, as Gaveau and Schulman 共2005兲 point out, introducing into a state space a dynamics in the form of transition probabilities 共see below兲 implicitly provides it with a metric structure.兴 For a well-defined relation between two such descriptions to hold, it is necessary that the two state spaces can be related to each other. 共This of course does not have to be the case; descriptions of the same system may also be incompatible with each other.兲 A simple possibility is that the system assumes a particular state in one description exactly if it is in any out of a certain set of states of the other description; that is to say, one state space is a coarse-graining or partition of the other state space. Because of this asymmetry between the two descriptions one may speak of a higher-level and a lower-level description, and refer correspondingly to macrostates and microstates of the system. The classic example in physics for this kind of interlevel relation is that between the phenomenological theory of thermodynamics, dealing with the macrostates of extended systems defined in terms of observables, such as temperature and pressure, and the theory of statistical mechanics, relating them to microstates defined in terms of the constituents of those systems. In this
context the terms macrostate and microstate derive from the circumstance that they refer to the properties of a “macroscopic” system versus those of its “microscopic” constituents. Though in many other cases the terms imply a difference in spatiotemporal scale, too, the important point is the difference in the amount of detail given by the descriptions. The description of a system is chosen by an observer, but it is also subject to objective constraints insofar as different descriptions may be differently adequate or useful. For a description as a dynamical system, the adequacy of a particular set of system states becomes apparent in the possibility to find a dynamical rule, ⌽⌬t, whereby the current state xt of the system determines its further evolution, xt+⌬t = ⌽⌬t共xt兲. Here t is a continuous or discrete time variable and ⌬t a time interval. A particular state space definition may therefore be called dynamically adequate if the specification of a state implies all the available information which is relevant for determining subsequent states, that is, if in this description the system possesses the Markov property 共cf. Shalizi and Moore, 2008兲. In this sense, the most general model of a dynamical system is the Markov process—a stochastic model which includes deterministic dynamics as a limiting case 共cf. Chan and Tong, 2001兲. It is important to note that this criterion for selecting a descriptive level implies a reference back to that same level; while employing a more fine-grained set of states may serve to improve the prediction of the future of a system in general, it will in most cases result in a loss of the Markov property with respect to these finer-grained states themselves. In other words, the Markov-property criterion distinguishes descriptive levels at which the system exhibits a selfcontained dynamics 共“eigendynamics”兲, independent of details present at other levels. This concept is akin to the idea of operational closure or autonomy in the theory of autopoietic systems 共Maturana and Varela, 1980; Varela, 1979兲, which alongside the separation from the environment also refers to the indifference of system operations towards the internal “microscopic” complexity of system elements 共cf. Luhmann, 1996兲. However, the topic of self-defined system boundaries is not addressed in this paper and accordingly, the term “system” is used in the unspecific sense of a section of reality which has been chosen for observation. This specification of the kind of descriptions sought for leads to a more specific concept of emergence as an interlevel relation. 关The stability conditions of Atmanspacher and beim Graben 共2007兲 are realized here by the Markov property, while contextual constraints 共Bishop and Atmanspacher, 2006兲 can be seen effective in the selection of a particular descriptive level out of those admissible.兴 Given a microscopic state description exhibiting the Markov property, an adequately higher-level description or coarse-graining should ideally preserve it. In the context of deterministic nonlinear systems, where the dynamics is defined by a map from a metric space onto itself, such a coarse-graining is called a Markov partition 共Adler, 1998; Bollt and Skufca, 2005兲; for the general case of stochastic dynamics we propose the term Markov coarse-graining 共cf. Gaveau and Schulman, 2005兲.
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Accordingly, states of a higher-level description may be called dynamically emergent states if they correspond to a Markov coarse-graining of a lower-level dynamics. Interpreting psychophysiological correlations as reflecting a relation of emergence between two levels of description as a dynamical system, the lower-level description is stated in terms of physiological, neural states, the higherlevel description in terms of mental states. At both levels a wide variety of descriptive approaches is possible, depending on the experimental methods used to assess the brain state on the one hand 共electrophysiology, imaging methods, brain chemistry, etc.兲 and the chosen set of psychological categories on the other hand 共conscious/unconscious, sleep stages, moods, cognitive modes, etc.兲. 关Since each mental state allows for multiple realizations at the neural level, mental states may be said to “supervene on” brain states 共cf. Kim, 1993兲—but this alone does not provide a sufficient characterization of their relation. Moreover, contrary to assumptions prevalent in the discussion 共cf. Chalmers, 2000兲 a neural correlate is not necessarily realized in a particular neural subsystem of the brain.兴 Applying the dynamical specification of emergence outlined above, emergent macrostates that are defined via a Markov coarse graining of the neural microstate dynamics are candidates for a further characterization as mental states. In order to empirically substantiate these ideas, macrostates obtained from the dynamics that has been observed in neurophysiological data are to be related to mental states of subjects that have been determined by other means, such as behavioral assessment or verbal reports. In the following we undertake first steps towards this program. Since a general algorithm for finding Markov coarse-grainings is not known, we focus on the special case of metastable states. Because a system stays in such a state for prolonged periods of time and only occasionally switches into another, antecedent states provide practically no information on the subsequent evolution beyond that implied in the current state, so that the macrostate dynamics is approximately Markovian. The following section describes an algorithm to obtain metastable states from the microstate dynamics observed in empirical data. Later, in Sec. V, we are going to apply our method to brain electrical data. With regard to this we should already mention here that the concept of metastable macrostates in the application to EEG is similar to the notion of “brain functional microstates” introduced by Lehmann and coworkers about two decades ago. Within their approach, “microstates” are defined as brief periods of time during which the spatial distribution of the brain’s electrical field remains relatively stable 共Lehmann, 1971; Lehmann et al., 1987兲. Transitions between such states are characterized by an abrupt change of the field topography, allowing the stream of EEG data to decompose into segments of the order of magnitude of 10– 100 ms duration, which can usually be grouped into a small number 共⬍10兲 of classes. Note, however, that the “microstate analysis” of Lehmann et al. results in a
coarse-grained description of the brain’s electrical activity, which in our nomenclature is a definition of macrostates. III. IDENTIFYING METASTABLE MACROSTATES FROM DATA
Metastable states correspond to the “almost invariant sets” of a dynamical system, i.e., subsets of the state space which are approximately invariant under the system’s dynamics. Since we are dealing with empirical data where there is generally no precise theoretical knowledge of the dynamics, it has to be determined from the data. Via a finite set of microstates resulting from a discretization of the state space 共Sec. III A兲, the time evolution operator ⌽⌬t is estimated in the form of a matrix of transition probabilities P 共Sec. III B兲. Metastable states are then determined using an algorithm to find the almost invariant sets of a Markov process 共Sec. III C兲. Additionally, an estimate of the optimal number of macrostates is obtained via an analysis of the characteristic time scales of the dynamics 共Sec. III D兲. Our algorithm builds on work by Deuflhard and Weber 共2005兲, Gaveau and Schulman 共2005兲, and Froyland 共2005兲, and reuses an idea of Allefeld and Bialonski 共2007兲. A. Discretization of the microstate space
In order to represent the observed microstate dynamics as a finite-state Markov process, the state space defined by K variables 共x1 , x2 , . . . , xK兲 = x has to be discretized, resulting in a set of compound microstates which forms the basis for further analysis. Since the data set may be high-dimensional and of varying density in different areas of the state space, we need a flexible algorithm which adapts the size and shape of microstate cells to local properties of the distribution of data points. This procedure has to meet two competing goals: It should capture as much detail as possible in order to faithfully represent the underlying continuous dynamics within its discretized version; but since transition probabilities between cells are to be estimated, the number of data points per cell should not fall below a certain minimum. Moreover, the extensions of the cells in the directions of the different variables should be of roughly the same size—assuming that the variables spanning the state space permit a comparison of distances along different directions; where this is not the case, it is advisable to map all variables onto the same range of values before performing the discretization. To achieve this, we use a recursive bipartitioning approach 共Fig. 1兲: For a given set of n data points S = 兵xm其, m = 1 . . . n, the direction of maximal variance is determined, i.e., a unit vector e, 兩e兩 = 1, such that varm 共xm · e兲 obtains its maximum value. Using the median M of the data points’ positions along this direction as a threshold value, the set is divided into two subsets, S1 = 兵xm兩xm · e 艋 M其, S2 = 兵xm兩xm · e ⬎ M其. The procedure is repeated for each of the resulting subsets, up to a recursion depth of b steps. This algorithm leads to a
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3 2 1 0 −1 −2 −3 −3
−2
−1
0
1
2
3
FIG. 1. 共Color online兲 Discretization by recursive bipartitioning, illustrated with a set of data points drawn from a two-dimensional normal distribution stretched out along the main diagonal. Cuts occurring earlier in the procedure are indicated by thicker lines.
practically identical number of data points per cell 共either n / 2b or n / 2b, where the brackets denote the floor and ceiling functions, respectively兲 which can be adjusted via the parameter b. It provides a high level of detail in those areas of the state space where the system spends most of the time, and it avoids too elongated cells by applying cuts perpendicular to the current main extension.
invariant sets, each forming an irreducible process of its own which itself may be subjected to a search for almost invariant subsets. 共Another possible problem is that there may be states a transition into or out of which is never observed, because they only occur at the beginning or end of the given data segments. Along with the general possibility of transient states, this issue is resolved in a natural way by the reversibilization step described below.兲 We assume moreover that the process is aperiodic, which is already the case if only one diagonal element Pii is different from zero. For a finite-state Markov process these two properties amount to ergodicity, which implies that there exists a unique invariant probability distribution over microstates, with P = , which is also the limit distribution approached from every initial condition. The analysis of the dynamical properties of a Markov process leading to the identification of its metastable states is strongly facilitated if it is reversible, i.e., if the dynamics is invariant under time reversal: Pij j = P ji i for all i , j. This property cannot usually be assumed for an arbitrary empirically observed process. But since the property of metastability, the tendency of the system to stay within certain regions of the state space for prolonged periods of time, is itself indifferent with respect to the direction of time 共Froyland, 2005兲, we can base the search for the corresponding almost invariant sets on the transition matrix for the reversibilized process R, Rij =
Pij = Pr共m+1 = i兩m = j兲, which may be estimated according to Pˆij =
cij , 兺 i⬘c i⬘ j
where cij = #共m+1 = i ∧ m = j兲 is the number of observed transitions from state j to state i. We assume that the Markov process 共cf. Feller, 1968, Chap. XV兲 described by P is irreducible, i.e., that it is possible to reach any state from any other state. If this is not the case, the system has not only almost invariant but proper
冊
instead. This operator can be directly estimated according to Rˆij =
B. Microstate dynamics
Via the bipartitioning procedure, each data point xm 共m = 1 . . . n兲 is assigned to one out of a finite set of microstates, identified by an index 苸 兵1 . . . N其 共N = 2b兲. The observed sequence of data points 共where the index m enumerates samples taken at consecutive time points兲 is thereby transformed into a sequence of microstate indices m. Considering this sequence of compound microstates as a realization of a finite-state Markov process, the underlying dynamics is described by a discrete transfer operator P, an N ⫻ N-matrix of transition probabilities between states,
冉
1 P jii Pij + , 2 j
cij + c ji , 兺i⬘共ci⬘ j + c ji⬘兲
i.e., by counting transitions forwards and backwards in time, and the corresponding invariant probability distribution determined as
ˆ i =
兺 j共cij + c ji兲 . 兺i⬘兺 j共ci⬘ j + c ji⬘兲
In the following we will use the symbols R and to denote these estimated quantities. C. Almost invariant sets
To identify almost invariant sets we employ the PCCA+ algorithm which was developed by Deuflhard and Weber 共2005兲 to find metastable states in the conformation dynamics of molecules. In this section we outline the main ideas of that approach which are necessary to understand the operation of the method, while for further details on the implementation and mathematical background the reader is referred to their paper. Our starting point is the ergodic and reversible Markov process characterized by the N ⫻ N-transition matrix R along with its invariant probability distribution . Due to the reversibility of the process 共symmetry of the stationary flow Rij j兲, the left and right eigenvectors Ak, pk and eigenvalues k of R,
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A kR = kA k,
Rpk = kpk,
k = 1 . . . N,
are real-valued. Resolving the scaling ambiguity left by the orthonormality relation Ak pl = ␦kl by the choice pik = i Aki leads to the normalization equations
兺i
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p2ik =1 i
and
兺i iA2ki = 1,
and the transition matrix can be given the spectral representation R = 兺 kp kA k . k
We assume that the eigenvalues are sorted in descending order, 1 艌 2 艌 ¯ 艌 N. The unique largest eigenvalue 1 = 1 belongs to the invariant probability distribution, p1 = , while the corresponding left eigenvector has constant coefficients, A1i = 1. If the process R possesses q almost invariant sets, it can ¯ that be seen as the result of a perturbation of a process R possesses q perfectly invariant sets. Any 共normalized兲 ele¯ belonging to ment of the right eigenvector subspace of R eigenvalues ¯1 = ¯ = ¯q = 1 gives an invariant probability distribution of the process. Moreover, if the invariant sets are described by characteristic functions ¯l 共l = 1 . . . q兲, such that ¯l共i兲 = 1 if state i belongs to invariant subset l, 0 otherwise, then any linear combination of them is a left eigenvector of ¯ for eigenvalue 1. Conversely, from any linearly indepenR dent set of left eigenvectors for eigenvalue 1, ¯ ,A ¯ , ... ,A ¯ 其, the characteristic functions of the invariant 兵A 1 2 q sets can be recovered via suitable linear combinations. Through the perturbation, the multiple eigenvalue 1 becomes a cluster of large eigenvalues 1 , 2 , . . . , q close to 1 = 1, and the invariant sets become almost invariant sets. They are described by almost characteristic functions l共i兲, attaining values in the range 关0, 1兴 which may be interpreted as quantifying the degree to which state i belongs to almost invariant set l. In analogy to the unperturbed case, these functions are constructed as linear combinations of the left eigenvectors belonging to the q large eigenvalues, q
l共i兲 = 兺 ␣klAki,
l = 1 . . . q,
k=1
defined by coefficients ␣ = 共␣kl兲. Admissible are those regular transforms that conform to the constraints • partition of unity: 兺ll共i兲 = 1 for all i, and • non-negativity: l共i兲 艌 0 for all i , l. The PCCA+ algorithm optimizes the transform ␣ with respect to an objective function to be maximized; we choose here the maximum scaling function I共␣兲 = 兺 max l共i兲, l
i
which favors attributions of states i to almost invariant sets l that are as clearcut as possible.
The input data for the optimization are the dominant left eigenvectors Ak, k = 1 . . . q. The first eigenvector is trivially A1 = 共1 , . . . , 1兲, but the remaining eigenvector coefficients can be geometrically interpreted as attributing to each microstate i a position in a 共q − 1兲-dimensional left eigenvector space with position vectors o共i兲 = 共Aki兲,
k = 2 . . . q.
Within this space, the optimization procedure appears as fitting a q-simplex as closely as possible around the microstate points. In a system with pronounced metastable macrostates each of them appears as a cluster of microstates located at the boundary of the point cloud, and the optimization procedure matches these q clusters to one of the vertices of the q-simplex. The values of the almost characteristic functions l共i兲 then attain the geometric meaning of barycentric coordinates of the data points with respect to the locations of the simplex vertices vl: q
o共i兲 = 兺 l共i兲vl . l=1
Finally, metastable macrostates corresponding to almost invariant sets of microstates are identified by attributing each microstate i to that macrostate l 苸 兵1 . . . q其 for which the almost characteristic function l共i兲 attains the highest value 共or, to whose defining vertex it is closest in terms of barycentric coordinates兲. D. Macrostates and time scales
If no prior information on the number of metastable states to be identified is available, it is desirable to obtain an estimate from the data set itself. Since the existence of q almost invariant sets leads to q large eigenvalues, a criterion based on gaps in the eigenvalue spectrum is the natural choice. However, the concrete values in the spectrum of R depend on the step size of the underlying discrete time, which is implicitly given with the input data. Changing the time scale from 1 to steps, the process has to be described by the transition matrix R, whose spectral representation R = 兺 kpkAk k
is essentially the same as that of R, but with eigenvalues raised to the power . The question which eigenvalues or which gap in the eigenvalue spectrum is to be considered “large” therefore depends on the chosen time scale. As Gaveau and Schulman 共2005兲 note, a coarse-graining of the state space always implies a corresponding “coarse-graining” or rather change of scale with respect to the time axis. We propose 共cf. Allefeld and Bialonski, 2007兲 a measure of the size of spectral gaps that is invariant under a rescaling of the time axis. This is achieved by transforming eigenvalues into associated characteristic time scales, T共k兲 = −
1 , log兩k兩
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1.5
and introducing the time-scale separation factor as the ratio of subsequent time scales,
1
log兩k+1兩 T共k兲 = . T共k + 1兲 log兩k兩
Substituting k for k in this equation, the resulting factors cancel out, so that F共k兲 provides a measure of the spectral gap between eigenvalues k and k+1 that is independent of the time scale. Using this measure, the number of macrostates q is estimated as the value of k for which F共k兲 becomes maximal. The choice q = 1 leading to a single macrostate comprising all microstates has thereby to be excluded, because it is always associated with the largest time-scale separation factor, F共1兲 → ⬁. If several larger gaps exist, a ranking list of possible q-values may be compiled, where each value leads to a different possible coarse-graining of the system into macrostates. This way different layers of the system’s dynamical structure are recovered, which 共extending Deuflhard and Weber’s approach兲 may be considered as the result of multiple superimposed perturbations. An example of this is given in the following section, where the method is illustrated using data from a simulated system. IV. Example: A system with four metastable macrostates
To illustrate the operation of the algorithm we apply it to data from a simulated system, where we can interpret the analysis results with respect to our precise knowledge of the underlying dynamics. We use a discrete-time stochastic system in two dimensions, 共x1 , x2兲, where the change over each time step is given by
0.5 0
x2
F共k兲 =
−0.5 −1 −1.5 −1.5
−0.5
0 x1
0.5
1
1.5
FIG. 2. 共Color online兲 Data points from a simulation run of a system with four metastable macrostates over 106 time steps, and a part of the connecting trajectory. Straight lines indicate the cell borders of the partition into 4096 microstates obtained via the bipartition algorithm.
The largest eigenvalues of R are plotted in Fig. 3共a兲, revealing a group of four large eigenvalues 共⬎0.995兲, which itself is subdivided into two groups of two eigenvalues each. This picture becomes clearer after the transformation into time scales T共k兲. In Fig. 3共b兲 they are displayed on a logarithmic scale, such that the magnitude of time-scale separation factors F共k兲 becomes directly visible in the vertical distances between subsequent data points. The largest a)
⌬xi = a共xi − 2x3i 兲 + bii ,
1
|λk|
0.99 0.98 0.97 0.96 b) 1000 500 T(k)
with a = 0.01, 共1 , 2兲 standard normal two-dimensional white noise, b1 = 0.03, and b2 = 0.05. The first term of the right-hand side of this equation describes an overdamped movement within a double-well potential along each dimension, leading to four attracting fixed points at 共x1 , x2兲 = 共⫾1 / 冑2 , ⫾ 1 / 冑2兲. Without the stochastic second term, the system would be decomposable into four invariant sets, separated by the two coordinate axes. But due to the noise the system performs a random walk, staying for prolonged periods of time in the vicinity of one of the attracting points, but occasionally wandering into another point’s basin of attraction. These switches occur more frequently along x2 because the noise amplitude is larger in that direction, b2 ⬎ b1. This system can be seen as an example of multistability, a type of nonlinear dynamics that has attracted much attention recently 关for an overview, see Feudel 共2008兲兴. Data resulting from a simulation run of this system are shown in Fig. 2. A section of the connecting trajectory illustrates how the system state moves through the state space, entering and leaving the cells of the microstate partition. Counting these transitions between cells leads to an estimate of the reversibilized transition matrix R.
−1
100 50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 k ← F(2) = 4.55 ← F(4) = 5.02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 k FIG. 3. 共Color online兲 Eigenvalue spectrum of the transition matrix R of the system with four metastable macrostates. 共a兲 The eigenvalues of largest magnitude. 共b兲 Logarithmic time scales; locations and values of the two largest time-scale separation factors are indicated.
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1.5 1
1
o3
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x2
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0 −0.5
1
0
o
2
−1
−2
−1
o
0
1
2
−1
1
FIG. 4. 共Color online兲 Eigenvector space 共o1 , o2 , o3兲 of the system with four metastable macrostates for q = 4. Each dot representing a microstate is colored according to which vertex of the enclosing tetrahedron is closest, defining the four metastable macrostates.
separation factor is F共4兲 = 5.02, closely followed by F共2兲 = 4.55, indicating that a partitioning of the state space into q = 4 macrostates is optimal, while searching for two different macrostates may also yield a meaningful result. The identification of almost invariant sets of microstates defining the metastable macrostates is performed within the three-dimensional eigenvector space 共o1 , o2 , o3兲. Figure 4 reveals that the points representing microstates are located on a saddle-shaped surface stretched out within a 4-simplex or tetrahedron. The algorithm identifies the vertices of the tetrahedron and attributes each microstate to that macrostate whose defining vertex is closest, resulting in the depicted separation into four sets. In Fig. 5 this result is retranslated into the original state space of Fig. 2, by coloring the data points of each microstate according to the macrostate it is assigned to. The identified metastable states coincide roughly with the basins of attraction of the four attracting points, i.e., the almost invariant sets of the system’s dynamics. From Fig. 4 we can also assess which macrostate definitions would be obtained by choosing q = 2, the next-best choice for the number of metastable states according to the time-scale separation factor criterion. In this case the eigenvector space is spanned by the single dimension o1, along which the two vertices of the tetrahedron on the left and right side, respectively, coincide. This means that the two resulting macrostates each consist of the union of two of the macrostates obtained for q = 4. With respect to the state space, these two macrostates correspond approximately to the areas x1 ⬎ 0 and x1 ⬍ 0. This result can be understood from the system’s dynamics, since because of the smaller probability of transitions along x1 these two areas of the state space form almost invariant sets, too. As can be seen from this example, the possibility to select different q-values of comparably good rating may allow us to recover different dynamical levels of a system, giving rise to a hierarchical structure of potential macrostate definitions.
−1.5 −1.5
−1
−0.5
0 x1
0.5
1
1.5
FIG. 5. 共Color online兲 Micro- and macrostates of the system with four metastable macrostates. Lines indicate the cell borders of the partition of the state space into microstates, while the coloring of data points shows the attribution of microstates to one of the four metastable macrostates.
V. APPLICATION TO EEG DATA
For the purpose of a first application of the algorithm to neurophysiological data, we chose an electroencephalographic 共EEG兲 recording from a patient suffering from petitmal epilepsy, a condition characterized by the occurrence of frequent short 共several seconds兲 epileptic episodes, during which the patient becomes irresponsive 共cf. Niedermeyer, 1993兲. This kind of data is favorable for our methodological approach because we can expect two clearly distinct states to be present—“normal” EEG/mentally present and paroxysmal episodes/mentally absent—and because it is possible to observe many transitions between these states in a recording of moderate size. The data set consists of a section of 89 min length from the patient’s monitoring EEG. It was recorded from the 19 electrode positions of the international 10-20 system 共American Electroencephalographic Society, 1991兲 at a sampling rate of 250 Hz, digitally bandpass-filtered 共2 – 15 Hz兲, and transformed to the average reference. Due to artifact removal by visual inspection the amount of data available for analysis was reduced to 71 min total length 共1 064 435 data points兲. In the preceding simulation example we know by definition that the given values of the system variables immediately specify its dynamical state, and therefore can be directly processed by the algorithm for the identification of metastable states. With measurement data like EEG the situation is not so clear. The data “as is” may be accepted as a specification of the system state, but any further processed version of them fulfills this function as well and may for some reason be even more suitable. This means that empirical data pose the problem of how to define the input state space for the analysis. For low-dimensional nonlinear deterministic dynamical systems, techniques have been developed to reconstruct the
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a)
300 200
← F(3) = 2.15
100 50 1
3
5
7
9 11 13 15 17 k
20
b)
15 10 o
2
state space of the system, or a higher-dimensional space comprising it, from scalar time series via the method of timedelay embedding 共Takens, 1981; Kantz and Schreiber, 1997兲. However, these techniques are not appropriate for our purposes. First, the dataset is already multidimensional and using the embedding approach we would have to either blow up the dimensionality even more, thereby introducing a high amount of redundancy, or discard many of the input data channels, possibly losing crucial information. And secondly, previous attempts to demonstrate low-dimensional nonlinear structure in EEG data had only limited success 共cf. Theiler and Rapp, 1996; Paluš, 1996兲. Instead, we pursue the following strategy: In the first step, we use the original 19-dimensional data space as the input state space. Guided by the results obtained in this way as well as by independent observations on the behavior of multichannel EEG, in a second step we develop a preprocessing procedure defining a more abstract input state space.
T(k) / ms
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A. Original data state space
Using the recursive bipartitioning algorithm, the data points were assigned to 32 768 different compound microstates 共32 or 33 points in each cell兲. The resulting timescale spectrum 关Fig. 6共a兲兴 exhibiting a large separation factor F共3兲 = 2.15 suggests a search for three metastable macrostates in a two-dimensional eigenvector space 关Fig. 6共b兲兴. This is supported by the 3-simplex shape of the distribution of microstate positions within this space. The arrangement of the areas belonging to the identified macrostates in the input data space is shown in Fig. 7, where the 19-dimensional space is represented using the first three PCA components of the data. The most prevalent state accounting for about 99% of the data points appears here as a centrally located spherical area, with the two other states forming handlelike appendices at opposite sides. The role of these three macrostates becomes clearer considering the transitions between them over time, in comparison with the underlying EEG time series 关Fig. 8共a兲兴. Within periods of normal electroencephalographic activity the system stays within the “main” macrostate, while during seizures switches between all three states occur regularly, corresponding to an oscillation along the PC1 axis of Fig. 7. This macrostate dynamics reflects the spike-wave oscillatory activity visible in the EEG channels shown in the lower panel of Fig. 8, which are characteristic for paroxysmal episodes. With this first result, the attempt at identifying emergent macrostates using the EEG data space is only partially successful: The occurrence of states correlates strongly with those features of the underlying process which are psychophysiologically most important, and also most prominent in visual inspection of the data. However, the two states expected are not directly recovered by the EEG analysis. Instead of one persistent state during paroxysmal episodes, we find rapid oscillatory changes between states including the one associated with normal EEG. This indicates that the input state space is not yet optimally defined.
5 0 −5 −5
0
5
o1
10
15
20
FIG. 6. 共Color online兲 Analysis results for the original EEG data state space. 共a兲 Time-scale spectrum; a large separation factor indicates three metastable states. 共b兲 Microstate positions in two-dimensional eigenvector space forming a triangular structure, and the resulting three metastable macrostates.
B. Amplitude vector state space
This finding can be understood from the fact that electroencephalographic activity in general is so strongly shaped by a predominant oscillatory layer of the dynamics—not only during epileptic episodes but also in normal EEG, particularly in the form of the alpha rhythm—that it is hard to discern more subtle dynamical features. To recover those fea-
FIG. 7. 共Color online兲 Analysis results for the original EEG data state space: Location of data points belonging to the three identified metastable states. The 19-dimensional state space is represented using the first three PCA components of the data.
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a) b) c) F3 F4 P3 P4 2050
2055
2060
2065
2070 t/s
2075
2080
2085
2090
FIG. 8. 共Color online兲 Analysis results illustrated using a segment of 40 s length. Upper panels: Macrostate dynamics over time, resulting from different input state space definitions. 共a兲 Original EEG data state space. 共b兲 Amplitude vector state space. 共c兲 As in 共b兲, but using normalized amplitude vectors. Lower panel: EEG time series at four selected recording sites. Paroxysmal episodes are characterized by short bursts of spike-wave activity.
tures, we need a preprocessing step that eliminates the oscillatory character of the data but retains the more slowly changing parameters of the oscillation. As observed by Wackermann 共1994兲, the trajectory formed by multichannel EEG within the data space can be approximated by a movement along an elliptical orbit with slowly changing orientation and shape 共Fig. 9兲. By locally matching ellipses to the data, a global instantaneous phase and amplitude can be defined, where the amplitude corresponds to the two main semiaxis vectors of the ellipse. 共For a full account of the calculation see the Appendix.兲 For simplicity we only use the major semiaxis vector, which specifies the direction and strength of the momentarily dominant oscillatory component, to define an amplitude vector state space as the input state space for the algorithm. 关This approach is similar to one of the strategies employed in the “spatial analysis” of EEG 共Lehmann, 1987兲, to select only those EEG potential maps 共data vectors兲 which occur at local maxima of the “global field strength” 共the norm of the data vectors兲.兴
With the specification of the system state via the major amplitude vector an ambiguity arises, because vectors of opposite orientation are equivalent. This is resolved by enforcing positive sign for the first vector component during the assignment of data points to microstates. For visualization 共Fig. 11兲 the axis vectors are used as they come out of the calculation described in the Appendix, that is with basically random orientation. The resulting time-scale spectrum is shown in Fig. 10共a兲. The largest separation factor of F共2兲 = 4.23 now gives a more definite indication of the number of macrostates than for the original data state space. In the corresponding onedimensional eigenvector space 关Fig. 10共b兲兴 the two macrostates are trivially defined by a cut at the center of the range of values. In Fig. 11, the location of data points belonging to the two macrostates is shown using the first three PCA compo-
a) T(k) / ms
1000
← F(2) = 4.23
500 200 100
0
1
3
5
b)
0
7
9 11 13 15 17 k
0
FIG. 9. 共Color online兲 Amplitude vector state space. The trajectory corresponding to a multivariate oscillatory signal like EEG takes on the form of an elliptical orbit with slowly varying parameters. Locally matching ellipses to the trajectory, the instantaneously dominant oscillatory component can be characterized by the major semiaxis vectors 共straight radial lines兲, resulting in a description of the system’s oscillatory state which itself evolves in a nonoscillatory way 共black curve兲.
−8
−6
−4 o
−2
0
1
FIG. 10. 共Color online兲 Analysis results for the amplitude vector state space. 共a兲 Time-scale spectrum indicating the presence of two metastable states. 共b兲 Microstate positions in one-dimensional eigenvector space and the resulting two metastable macrostates.
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FIG. 11. 共Color online兲 Analysis results for the amplitude vector state space: Location of data points belonging to the two identified metastable states in a representation of the state space using the first three PCA components.
nents of the data points in the amplitude vector state space. Again, a prevalent macrostate 共accounting for 98% of the data points兲 fills a spherically shaped central area, while two appendices protruding on opposite sides together constitute the second macrostate. Despite the fact that the overall shape of the data cloud is similar to that shown in Fig. 7, the reader should keep in mind that the two diagrams depict differently defined state spaces represented with respect to a different set of dimensions. Figure 8共b兲 demonstrates that the revision of the input state definition successfully eliminates the oscillatory switching between states during paroxysmal episodes. Starting from the amplitude vector input state space, the algorithm for the identification of metastable states is able to consistently associate normal EEG with one macrostate, and—except for short relapses—epileptic EEG with another macrostate. The macrostate structure of the amplitude vector state space shown in Fig. 11 suggests that the distinction of the two macrostates relies only on the length of the amplitude vector. To check this, we tested the performance of the algorithm when normalized amplitude vectors are used. The time-scale spectrum 共Fig. 12兲, with a maximal separation factor F共6兲 = 1.21 not substantially larger than the rest, indicates that the identification of macrostates is severely impaired under these circumstances. Even so, an examination of the state dynamics over time 关Fig. 8共c兲兴 reveals that there are still two states that are mainly attained during epileptic episodes.
T(k) / ms
150 100
← F(6) = 1.21
75 50
Relations between mental 共psychological兲 and neural 共physiological兲 phenomena form the generally accepted basis for work in various disciplines, such as psychiatry, psychophysiology, and cognitive neuroscience. While a large body of knowledge has been gathered in these fields, the conceptual question of how mind and brain are related in precise terms is still largely unresolved. Starting from the notion of the mental as “emerging” from neural processes, we argue that this relation of emergence should be understood as one between different descriptions of the same system. Utilizing concepts from the theory of dynamical systems for the formulation of descriptions, we propose that the relation between descriptive levels should take on the form of a partition or coarse-graining of the state space that is characterized by a preservation of the Markov property. To empirically test the validity of our approach, we turn to a form of such a Markov coarse-graining which can be algorithmically obtained; that of metastable states. We describe how metastable macrostates of a dynamics observed in empirical data can be identified based on the spectral analysis of the transition matrix governing the microstate dynamics, and illustrate its operation with simulation data. We apply the method to a recording of electroencephalographic 共EEG兲 data from a human subject suffering from petit-mal epilepsy. Combined with a suitable preprocessing procedure, the algorithm is able to automatically identify metastable states from the data which closely correspond to the mental states of the subject 共mentally present/absent兲. This first application substantiates the practical viability of our approach and appears promising for the future application of the method to more challenging forms of data. ACKNOWLEDGMENTS
The authors would like to thank P. beim Graben for discussions, G. Froyland for helpful hints, and V. Krajča and S. E. Petránek 共University Hospital Na Bulovce, Prague, Czech Republic兲 for providing us with the EEG recording. APPENDIX: INSTANTANEOUS AMPLITUDE AND PHASE FOR MULTIVARIATE TIME SERIES
The local oscillatory behavior of a real-valued univariate signal x共t兲 is commonly characterized using the corresponding complex-valued analytic signal z共t兲 共Gabor, 1946兲. It is obtained by combining x共t兲 with an imaginary part, z共t兲 = x共t兲 + iy共t兲, which is defined as the Hilbert transform of x, y共t兲 = Hx共t兲 =
1
3
5
7
9 11 13 15 17 k
FIG. 12. 共Color online兲 Time-scale spectrum obtained using normalized axis vectors. The largest separation factor 共for six macrostates兲 is only marginally larger than the other occurring values, indicating that no adequate definition of metastable states is possible.
1 P.V.
冕
⬁
−⬁
x共t⬘兲 dt⬘ , t − t⬘
where P.V. denotes the Cauchy principal value of the integral. Under the condition that x共t兲 is dominated by a single frequency component, its instantaneous amplitude A共t兲 and phase 共t兲 can be determined via the analytic signal according to
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independent兲 instantaneous phase 共t兲, such that for 共t兲 苸 兵0 , 21 , , 23 其 or equivalents, x共t兲 coincides with one of the main semiaxis vectors or its negative. This is achieved by choosing
φ b
x
共t兲 =
a
y
FIG. 13. 共Color online兲 Determination of local ellipse axes. The trajectory formed by the multivariate signal is locally matched to an elliptical orbit, which is defined by the data vector x at a given instant and the corresponding vector y from the signal’s channel-wise Hilbert transform as conjugate semiaxis vectors. Main semiaxis vectors of the ellipse, a and b, are obtained using the associated multivariate instantaneous phase .
A共t兲 = 兩z共t兲兩,
共t兲 = arg z共t兲,
so that x共t兲 = A共t兲cos 共t兲 or z共t兲 = A共t兲exp共i共t兲兲. The terms amplitude A and phase as they are used here can be interpreted such that they specify the parameters of a strictly periodic sinusoidal oscillation which locally matches the behavior of the observed signal x共t兲 at a given instant t. In particular, 共t兲 attains the value 0 共or equivalently, an integer multiple of 2兲 whenever the actual value of x共t兲 coincides with the associated instantaneous amplitude A共t兲. These properties of the analytic signal can also be utilized to determine the parameters of the locally matching oscillation for a multivariate signal x共t兲 = 共xi共t兲兲 共i = 1 . . . K兲. We assume that each component signal xi共t兲 is dominated by a single frequency and that the frequencies of different signals are similar. Using y共t兲 to denote the channel-wise Hilbert transform of x共t兲 and z共t兲 for its channel-wise completion to the analytic signal, the local extension of the signal’s oscillatory behavior for instant t is obtained with zt共兲 = z共t兲exp共i兲, parametrized by 苸 关0 , 2兴. Its real part xt共兲 = x共t兲cos − y共t兲sin gives the multivariate oscillation that locally matches the behavior of the signal at instant t; its trajectory is an elliptical orbit with conjugate axes specified by the vectors x共t兲 and y共t兲. From these conjugate axes, the main semiaxis vectors a共t兲 and b共t兲 of the local ellipse can be calculated 共Fig. 13兲. It proves useful to do so via introducing a global 共channel-
2x共t兲 · y共t兲 1 arctan . 2 兩x共t兲兩2 − 兩y共t兲兩2
Since the resulting values in the range 关− 4 , 4 兴 cover only one quarter of a cycle, the outcome may be transformed into an equivalent but more useful representation via a standard “unwrapping” procedure 共adding or subtracting 2 at discontinuity points兲 to enforce a smooth evolution of 共t兲. Using this result, main semiaxis vectors of the locally matching ellipse at instant t are obtained by going backwards along xt共兲 by an amount of 共t兲 or forwards by 2 − 共t兲, i.e., a共t兲 = xt共− 共t兲兲 and b共t兲 = xt
冉
冊
− 共t兲 . 2
If 共t兲 has been adjusted for a smooth evolution over time, the same can be expected from the resulting a共t兲 and b共t兲. It is, however, not clear from this definition which one of these vectors specifies the major and minor axis of the ellipse, respectively, and it is possible that over the course of time the two vectors change roles. For a specific application of this result, further processing may therefore be necessary. Complementary to the generalization of the instantaneous phase concept, a multivariate instantaneous amplitude A共t兲 can be defined such that z共t兲 = A共t兲exp共i共t兲兲, which is given by A共t兲 = a共t兲 − ib共t兲. The channel-wise modulus of this quantity corresponds to the instantaneous amplitudes Ai共t兲 of the component signals, while the argument comprises the phase differences between the global and the component signal instantaneous phases, i共t兲 − 共t兲. Adler, R. L., “Symbolic dynamics and Markov partitions,” Bull., New Ser., Am. Math. Soc. 35, 1 共1998兲. Allefeld, C. and Bialonski, S., “Detecting synchronization clusters in multivariate time series via coarse-graining of Markov chains,” Phys. Rev. E 76, 066207 共2007兲. The American Electroencephalographic Society, “Guidelines for standard electrode position nomenclature,” J. Clin. Neurophysiol. 8, 200 共1991兲. Ashby, W. R., “Principles of the self-organizing system,” in Principles of Self-Organization: Transactions of the University of Illinois Symposium, edited by H. von Foerster and G. W. Zopf, Jr. 共Pergamon, Oxford, 1962兲, pp. 255–278. Atmanspacher, H. and beim Graben, P., “Contextual emergence of mental states from neurodynamics,” Chaos Complexity Lett. 2, 151 共2007兲. Beckermann, A., Flohr, H., and Kim, J., Emergence or Reduction? 共de Gruyter, New York, 1992兲. Bishop, R. C. and Atmanspacher, H., “Contextual emergence in the description of properties,” Found. Phys. 36, 1753, 2006. Bollt, E. M. and Skufca, J. D. , “Markov partitions,” in Encyclopedia of Nonlinear Science, edited by A. Scott 共Routledge, New York, 2005兲. Broad, C. D., The Mind and Its Place in Nature 共Kegan Paul, London, 1925兲.
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Allefeld, Atmanspacher, and Wackermann
Chalmers, D. J., “What is a neural correlate of consciousness?,” in Neural Correlates of Consciousness: Empirical and Conceptual Questions, edited by T. Metzinger 共MIT, Cambridge, 2000兲. Chan, K.-S. and Tong, H., Chaos: A Statistical Perspective 共Springer, Berlin, 2001兲. Darley, V., “Emergent phenomena and complexity,” in Artificial Life IV, edited by R. A. Brooks and P. Maes 共MIT Press, Cambridge, 1994兲, pp. 411–416. Deuflhard, P. and Weber, M., “Robust Perron cluster analysis in conformation dynamics,” Linear Algebr. Appl. 398, 161 共2005兲. Feller, W., An Introduction to Probability Theory and Its Applications, 3rd ed. 共Wiley, New York, 1968兲, Vol. I. Froyland, G., “Statistically optimal almost-invariant sets,” Physica D 200, 205 共2005兲. Feudel, U., “Complex dynamics in multistable systems,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 18, 1607 共2008兲. Gabor, D., “Theory of communication,” J. Inst. Electr. Eng., Part 3 93, 429 共1946兲. Gaveau, B. and Schulman, L. S., “Dynamical distance: Coarse grains, pattern recognition, and network analysis,” Bull. Sci. Math. 129, 631 共2005兲. Kantz, H. and Schreiber, T., Nonlinear Time Series Analysis 共Cambridge University Press, Cambridge, 1997兲. Kim, J., Supervenience and Mind 共Cambridge University Press, Cambridge, 1993兲. Lehmann, D., “Multichannel topography of human alpha EEG fields,” Electroencephalogr. Clin. Neurophysiol. 31, 439 共1971兲. Lehmann, D., “Principles of spatial analysis,” in Methods of Analysis of Brain Electrical and Magnetic Signals, edited by A. S. Gevins and A. Rémond 共Elsevier, New York, 1987兲, pp. 309–354. Lehmann D., Ozaki, H., and Pal, I., “EEG alpha map series: Brain microstates by space-oriented adaptive segmentation,” Electroencephalogr. Clin. Neurophysiol. 67, 271 共1987兲.
Chaos 19, 015102 共2009兲 Luhmann N., Social Systems 共Stanford University Press, Stanford, 1996兲. Maturana, H. R. and Varela, F. J., Autopoiesis and Cognition: The Realization of the Living 共Reidel, Dordrecht, 1980兲. Niedermeyer, E., “Abnormal EEG patterns: Epileptic and paroxysmal,” in Electroencephalography: Basic Principles, Clinical Applications and Related Fields, 3rd ed. 共Williams & Wilkins, New York, 1993兲, pp. 217–240. O’Connor, T. and Wong, H. Y., “Emergent properties,” in The Stanford Encyclopedia of Philosophy, edited by E. N. Zalta 共Winter 2006兲, http:// plato.stanford.edu/archives/win2006/entries/properties-emergent/. Paluš, M., “Nonlinearity in normal human EEG: Cycles, temporal asymmetry, nonstationarity and randomness, not chaos,” Biol. Cybern. 75, 389 共1996兲. Robinson, C., Dynamical Systems. Stability, Symbolic Dynamics, and Chaos 共CRC, Boca Raton, 1995兲. Seth, A. K., “Measuring emergence via nonlinear Granger causality,” in Artificial Life XI: Proceedings of the Eleventh International Conference on the Simulation and Synthesis of Living Systems, edited by S. Bullock et al. 共MIT, Cambridge, 2008兲, pp. 545–553. Shalizi, C. R. and Moore, C., “What is a macrostate? Subjective observations and objective dynamics,” preprint 2008, arXiv:cond-mat/0303625. Stephan, A., “Emergentism, irreducibility, and downward causation,” Grazer Philosophische Studien 65, 77 共2002兲. Takens, F., “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, edited by D. A. Rand and L. S. Young 共Springer, Berlin, 1981兲, pp. 366–381. Theiler, J. and Rapp, E., “Re-examination of the evidence for lowdimensional, nonlinear structure in the human electroencephalogram,” Electroencephalogr. Clin. Neurophysiol. 98, 213 共1996兲. Varela, F. J., Principles of Biological Autonomy 共North-Holland, Amsterdam, 1979兲. Wackermann, J., “Segmentation of EEG map series in n-dimensional state space,” Brain Topogr. 6, 246 共1994兲.
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Inverse problems in dynamic cognitive modeling Peter beim Graben1,a兲 and Roland Potthast2 1
School of Psychology and Clinical Language Sciences, University of Reading, Reading, Berkshire RG6 6AH, United Kingdom 2 Department of Mathematics, University of Reading, Reading, Berkshire RG6 6AH, United Kingdom
共Received 28 November 2008; accepted 11 February 2009; published online 31 March 2009兲 Inverse problems for dynamical system models of cognitive processes comprise the determination of synaptic weight matrices or kernel functions for neural networks or neural/dynamic field models, respectively. We introduce dynamic cognitive modeling as a three tier top-down approach where cognitive processes are first described as algorithms that operate on complex symbolic data structures. Second, symbolic expressions and operations are represented by states and transformations in abstract vector spaces. Third, prescribed trajectories through representation space are implemented in neurodynamical systems. We discuss the Amari equation for a neural/dynamic field theory as a special case and show that the kernel construction problem is particularly ill-posed. We suggest a Tikhonov–Hebbian learning method as regularization technique and demonstrate its validity and robustness for basic examples of cognitive computations. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3097067兴 Inverse problems, the determination of system parameters from observable or theoretically prescribed dynamics, are prevalent in the cognitive neurosciences. In particular, the dynamical system approach to cognition involves learning procedures for neural networks or neural/dynamic fields. We present dynamic cognitive modeling as a three tier top-down approach comprising the levels of (1) cognitive processes, (2) their state space representation, and (3) dynamical system implementations that are guided by neuroscientific principles. These levels are passed through in a top-down fashion: (1) cognitive processes are described as algorithms sequentially operating on complex symbolic data structures that are decomposed using so-called filler/role bindings; (2) data structures are mapped onto points in abstract vector spaces using tensor product representations; and (3) cognitive operations are implemented as dynamics of neural networks or neural/dynamic fields. The last step involves the solution of inverse problems, namely, training the system’s parameters to reproducing prescribed trajectories of cognitive operations in representation space. We show that learning tasks for neural/dynamic field models are particularly ill-posed and propose a regularization technique for the common Hebb rule, resulting into modified Tikhonov–Hebbian learning. The methods are illustrated by means of three instructive examples for basic cognitive processing, where we show that Tikhonov–Hebbian learning is a quick and simple training algorithm, not requiring orthogonality or even linear independence of training patterns. In fact, the regularization is robust against linearly dependent patterns as they could result from oversampling.
a兲
Electronic mail: p [email protected].
1054-1500/2009/19共1兲/015103/21/$25.00
I. INTRODUCTION
Investigating nonlinear dynamical systems is an important task in the sciences. In the ideal case, one has a theoretical model in form of differential 共or integrodifferential兲 equations and computes their analytical solutions. However, this approach is often not tractable for complex nonlinear systems. Here, analytical techniques, such as stability, bifurcation, or synchronization analysis, provide insights into the structural properties of the flow in phase space. If these methods are applicable only up to some extent, numerical solution of the model equations, i.e., determining the system’s trajectories through phase space for given initial and boundary conditions, is of great importance. This forward problem for nonlinear dynamical systems is nowadays well understood and appreciated in science. By contrast, the peculiarities and possible pitfalls of the inverse problem of determining the system’s equations 共or the system’s parameters for a given class of equations兲 from observed or prescribed trajectories are less acknowledged in nonlinear dynamical system research today. Inverse problems are typically ill-posed. This notion does not only refer to the fact that there is, in general, no unique parametrization for a prescribed solution, but moreover, that such parametrizations are highly unstable and extremely sensitive to training data. Related to dynamical system models of cognitive processes,1–4 inverse problems are prevalent in several training algorithms for artificial neural networks 共ANNs兲.5,6 It is the aim of the present study to investigate these problems, in principle, by means of continuum approximations for neural networks, which are known as neural or dynamic field models. We show that learning tasks for such fields are particularly ill-posed and suggest a regularization technique for the common Hebb rule, resulting into modified Tikhonov– Hebbian learning.
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(a)
((a, b), (a, b))
(b)
(c) a
S
(d)
(e)
S a
b
a
b
(a, (b, (a, b)), b)
S b
a
b
FIG. 2. Complex symbolic data structures for mental representations. 共a兲 simple list, 共b兲 list of simple lists, 共c兲 the corresponding tree for 共b兲, 共d兲 the same tree with node labels S, 共e兲 an even more complex frame of nested lists.
FIG. 1. Three tier top-down approach to inverse dynamic cognitive modeling.
This paper is structured as follows. Section II reviews cognitive modeling and inverse problems related to this field. Inspired by seminal work of Marr and Poggio,7 we introduce dynamic cognitive modeling as a three tier top-down approach, as illustrated in Fig. 1. 共1兲 At the highest level of mental states and processes, the relevant data structures and algorithms from the prospect of the computer metaphor of the mind8 are determined in order to obtain symbolic patterns and transformation rules that are described by the so-called filler/role bindings.9–14 共2兲 These symbolic structures and processes are mapped onto points and continuous trajectories15 in abstract vector spaces, respectively, by tensor product representations.9–14 Interestingly, the word “representation” does not only refer to models of mental representations here. It assumes a precise meaning in terms of mathematical representation theory:16,17 cognitive operations are represented as operators in neural representation spaces. 共3兲 Cognitive representations are implemented by nonlinear dynamical systems obeying guiding principles from the neurosciences. Here, we discuss the inverse problem for a large class of continuous neural networks 共so-called neural or dynamic field models兲18–34 described by integrodifferential equations. In Secs. III and IV we demonstrate how inverse problems for dynamical system models of cognition can be regularized by a Tikhonov–Hebbian learning rule for neural field models. Three examples are constructed in Sec. III and presented in Sec. IV. Section V gives a concluding discussion. II. DYNAMIC COGNITIVE MODELING
This section introduces dynamic cognitive modeling as a three tier hierarchy, comprising 共1兲 cognitive processes and symbolic structures, 共2兲 their state space representations, and 共3兲 their implementation by neurodynamics,7 which is pursued from the top level 共1兲 down to the bottom level 共3兲. A. Cognitive processes
According to the computer metaphor of the mind,8 cognition is essentially symbol manipulation obeying combinatorial rules.35,36 A paradigmatic concept for classical cognitive science is the Turing machine as a formal computer.37
However, the Turing machine is of only marginal interest in cognitive psychology38 and psycholinguistics39 as it has unlimited computational resources in the form of a randomly accessible bi-infinite memory tape. Therefore, less powerful devices such as pushdown automata or nested stack automata37,40 are of greater importance for computational models of cognitive processes. A pushdown automaton has a one-sided infinite memory tape that is accessible in “last in/ first out” manner, i.e., the device has only access to the topmost symbol stored at the stack. Turing machines, pushdown automata, or nested stack automata define particular formal languages, namely, the recursively enumerable languages, context-free languages, and indexed languages, respectively.37 Especially the latter both are relevant in the field of computational psycholinguistics because sentences can be described by phrase structure trees and some restricted operations acting on them.40–44 1. Data structures and algorithms
The first step in devising a computational dynamical system model is specification of the relevant data structures and algorithms where the former instantiates a model for mental representations, while the latter defines the mental or cognitive computations. In computer science, complex symbolic data structures are e.g., lists, trees, nested lists, often called “frames,” or even lists of trees, etc. Figure 2 depicts some examples for data structures. The first example in Fig. 2共a兲 is a list of three items, the symbols a and b 共note that symbolic expressions are printed in Roman font subsequently兲, where a takes the first and third position, while b occupies the second position of the list. Assuming a pushdown automaton that has only access to the first symbol a in the list, we have a simple description of a stack tape. Such an automaton can achieve symbolic operations, e.g., “push,” where ␣⬘ = push共a , ␣兲 places a new symbol a at the top of the stack, denoted ␣ = 共a1 , a2 , . . . , an兲, such that ␣⬘ = 共a , a1 , a2 , . . . , an兲. Starting with an empty list, we could, e.g., apply push three times resulting into the state transitions
␣0 = 共 兲,
␣1 = 共a兲 = push共a, ␣0兲,
␣2 = 共a,a兲 = push共a, ␣1兲,
␣3 = 共a,a,a兲 = push共a, ␣2兲.
共1兲
Another important interpretation of lists such as in Fig. 2共a兲 is related to logical inferences. Replacing, e.g., the symbol a by 0 and b by 1, yields the list 共0,1,0兲 that can be regarded as one row of a logical truth table, where the first two items are
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TABLE I. Truth table of the logical equivalence relation A equiv B = C. A
B
C
0 0 1 1
0 1 0 1
1 0 0 1
the inputs and the third is the output of a logical function. In our case, the list 共0, 1, 0兲 is the second row of the truth table of the logical equivalence relation equiv, defined in Table I. Figure 2共b兲 shows a complex lists containing two simple lists. This structure can be regarded as a tree shown in Fig. 2共c兲. Figure 2共d兲 introduces the concept of a labeled tree where to each node a symbolic label is assigned. From a labeled tree one can derive a context-free grammar 共CFG兲 of production rules by interpreting each tree branching as a rule of the form X → YZ, where X denotes the mother node, and Y, and Z its immediate daughters. Therefore, example in Fig. 2共d兲 gives rise to the CFG T = 兵a,b其,
N = 兵S其,
P=
再
共1兲 S → S S 共2兲 S → a b
冎
,
共2兲
where T = 兵a , b其 is called the set of terminal symbols, N = 兵S其 that of nonterminal symbols, with the distinguished start symbol S, and the production rules expand one nonterminal at the left-hand side into a string of nonterminals or terminals at the right-hand side. Applying the rules from a CFG recursively describes a tree generation dynamics as, e.g., depicted in Fig. 10 in Sec. III D. Finally, Fig. 2共e兲 presents an even more complex expression of nested lists that might be regarded as a model for cognitive frames. Coming back to the general paradigm of a Turing machine, such an automaton is formally defined as a 7-tuple M TM = 共Q,N,T, ␦,q0,b,F兲,
共3兲
where Q is a finite set of machine control states, N is another finite set of tape symbols, containing a distinguished “blank” symbol b, T 傺 N \ 兵b其 is the set of admitted input symbols,
␦:Q ⫻ N → Q ⫻ N ⫻ 兵L,R其
共4兲
is a partial state transition function 共the “machine table”兲 determining the action of the machine when q 苸 Q is the current state at time t and a 苸 N is the current symbol being read from the memory tape. The machine moves then into another state q⬘ 苸 Q at time t + 1 replacing the symbol a by another symbol a⬘ 苸 N and shifting the tape either one place to the left 共L兲 or to the right 共R兲. Figure 3 illustrates such a state transition. Finally, q0 苸 Q is a distinguished initial state and F 傺 Q is a set of “halting states” assumed by the machine when a computation terminates.37 In order to describe the machine’s behavior as deterministic dynamics in symbolic “phase space,” one introduces the notion of a state description, which is a triple s = 共␣,q, 兲
共5兲
with ␣ ,  苸 Nⴱ 共i.e., they are lists of tape symbols from N of arbitrary, yet finite, length, delimited by blank symbols b兲.
FIG. 3. Example state transition from 共a兲 to 共b兲 of a Turing machine with ␦共1 , a兲 = 共2 , b , L兲.
Then, the transition function can be extended to state descriptions by
␦ⴱ:S → S, ⴱ
s⬘ = ␦ⴱ共s兲,
共6兲
ⴱ
where S = N ⫻ Q ⫻ N now plays the role of a phase space of a discrete dynamical system. We discuss consequences of this view in the next section. 2. Filler/role bindings
However, first we introduce a general framework for formalizing arbitrary data structures and symbolic operations suggested by Smolensky and co-workers9–12 and recently deployed by beim Graben et al.13,14 This filler/role binding decomposition identifies the particular symbols occurring in complex expressions with the so-called fillers f 苸 F, where F is some finite set of cardinality NF. Applied to the Turing machine, we can therefore choose F = N, the set of tape symbols, including blank and input symbols. Fillers are bound to symbolic roles r 苸 R, where R is another finite 共or countable兲 set of possible roles. Examples for such roles are 共1兲 slots for list positions R = 兵ri 兩 1 ⱕ i ⱕ n其 indicating the ith position in a list of length n; 共2兲 slots for tree positions R = 兵r1 , r2 , r3其, where r1 = mother,
r2 = left daughter,
r3 = right daughter, 共7兲
as indicated in Fig. 4; and 共3兲 slots in arbitrary cognitive frames as in Fig. 2共e兲. Considering the example from Fig. 2共a兲 yields fillers F = 兵a , b其 and roles R = 兵r1 , r2 , r3其 for the three list positions. A filler/role binding is now a set of pairs 共f i , r j兲 when filler f i occurs at 共is “bound” to兲 role r j. Thus, the filler/role decomposition for example Fig. 2共a兲 is given as
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r1
r2
However, another decomposition is more obvious. Here, the state description is regarded as a concatenation product
␥ = ␣⬘ · q · 
共14兲
of the strings ␣,  with the state q in the given order, where ␣⬘ is the string ␣ in reverted order, ␣⬘ = anan−1 ¯ a1.13 Introducing the notion of “dotted sequences,”45 yields a two-sided list of tape and control state symbols from F = N 艛 Q, where the dot “.” indicates the position of the control state q 苸 Q at the tape,
r3
␥ = anan−1 ¯ a1q . b1b2 ¯ bm .
FIG. 4. Elementary role positions of a labeled binary tree.
共15兲
46,47
f ␣ = 兵共a,r1兲,共b,r2兲,共a,r3兲其.
共8兲
Such a relation can be regarded as a complex filler that could be recursively bound to roles again. Therefore, we obtain the filler/role decomposition for example Fig. 2共b兲 as a set of pairs of sets of pairs f T = 兵共兵共a,r1兲,共b,r2兲其,r1兲,共兵共a,r1兲,共b,r2兲其,r2兲其,
共9兲
where the complex filler, namely, the list 兵共a , r1兲 , 共b , r2兲其 is bound to both roles r1 and r2 recursively. This is also the correct decomposition of the tree from Fig. 2共c兲. For describing the tree from Fig. 2共d兲, the node labels have to be taken into account. As fillers we choose the labels F = 兵S , a , b其, whereas the roles R = 兵r1 , r2 , r3其 are given through Eq. 共7兲. In a bottom-up manner, we first decompose the leftmost subtree L by assigning filler S to the root node r1, filler a to the left daughter node r2 and filler b to the right daughter node r3, obtaining the complex filler f L = 兵共S,r1兲,共a,r2兲,共b,r3兲其.
共10兲
This is also the correct decomposition of the right subtree f R, such that f R = f L. Then, f L is bound to the left daughter r2 of the next tree level, f R is bound to its right daughter r3 and its root node r1 is occupied by the simple filler S again, such that f T = 兵共S,r1兲,共f L,r2兲,共f R,r3兲其.
共11兲
The complete filler/role binding of the tree Fig. 2共d兲 is hence f T = 兵共S,r1兲,共兵共S,r1兲,共a,r2兲,共b,r3兲其,r2兲, 共兵共S,r1兲,共a,r2兲,共b,r3兲其,r3兲其.
共12兲
Application of filler/role binding to state descriptions of Turing machines is possible in several ways. Most straightforwardly, one assigns three role positions RS = 兵r1 , r2 , r3其 to the three slots in the state description 关Eq. 共5兲兴. Then, the strings ␣ ,  苸 Nⴱ are regarded as complex fillers, namely, lists of tape symbols F = N with a countable number of slots RL = 兵si 兩 i 苸 N其. Additionally, the machine states are represented by another set of fillers FQ = Q that only bind to role r2. Thereby, one possible Turing machine filler/role decomposition is f s = 兵共兵共a1,s1兲,共a2,s2兲, . . . ,共an,sn兲其,r1兲,共q,r2兲, 共兵共b1,s1兲,共b2,s2兲, . . . ,共bm,sm兲其,r3兲其, with ai , b j 苸 F and n , m 苸 N.
共13兲
Moore proved that this description of a Turing machine leads to generalized shifts investigated in symbolic dynamics.45–48 Then, the filler/role decomposition of a Turing machine is that of a simple list f ␥ = 兵共an,sn−1兲,共an−1,sn−2兲, . . . , 共a1,s−1兲,共q,s0兲,共b1,s1兲,共b2,s2兲, . . . ,共bm,sm兲其,
共16兲
where we have introduced integer list positions as roles R = 兵si 兩 i 苸 Z其. A more axiomatic framework for the filler/role binding was presented by beim Graben et al.14 B. State space representations
The filler/role decomposition of complex symbolic data structures is a first step toward their state space representation. This is achieved by the tensor product representation independently invented by Smolensky and co-workers9–12 and Mizraji.49,50 The tensor product calculus is a universal framework to describe different state space representations for dynamic cognitive modeling. We first review its general algebraic framework and discuss particular representations in the subsequent subsections. In order to employ the tensor product representation, the respective fillers and roles, f i 苸 F and r j 苸 R, are mapped onto vectors from two vector spaces VF and VR by a function
:F⬁ → VF 艛 VR,
共f ⴱ兲 = fⴱ ,
共17兲
where F⬁ contains the roles and the simple fillers 共F 艛 R 傺 F⬁兲 but also all complex fillers f ⴱ 共for the exact definition of F⬁, see Ref. 14兲 such that fi = 共f i兲 苸 VF represents a filler and r j = 共r j兲 苸 VR represents a role. A filler/role binding is then represented by the direct sum of tensor products
共共f i,r j兲兲 = 共f i兲 丢 共r j兲,
共18兲
共兵共f i1,r j1兲,共f i2,r j2兲其兲 = 共f i1兲 丢 共r j1兲 丣 共f i2兲 丢 共r j2兲.
共19兲
As a consequence, the image F = 共F⬁兲 is isomorphic to the Fock space ⬁
n
n=1
k=1
F = 丣 VF 丢 丢 VR of many particle quantum systems.11,17 Applying this mapping to the examples from Fig. 2 yields the following representations. The simple list 关Eq. 共8兲兴 from Fig. 2共a兲 is represented by a vector
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共␣兲 = a 丢 r1 丣 b 丢 r2 丣 a 丢 r3
共20兲
with filler vectors a = 共a兲 and b = 共b兲. 关For the sake of clarity, we write 共␣兲 instead of 共f ␣兲 in the sequel which would be the precise notation as is applied to the filler/role binding f ␣ and not to the symbolic expression ␣ itself.兴 Correspondingly, the nested list from Fig. 2共b兲 and the tree from Fig. 2共c兲 关Eq. 共9兲兴 are represented by tensor products of higher rank
共T兲 = 共a 丢 r1 丣 b 丢 r2兲 丢 r1 丣 共a 丢 r1 丣 b 丢 r2兲 丢 r2 = a 丢 r1 丢 r1 丣 b 丢 r2 丢 r1 丣 a 丢 r1 丢 r2 丣 b 丢 r2
In order to avoid sparse high-dimensional state space representations, combinations of vectorial with purely numerical encodings, the so-called fractal encodings were used by Siegelmann and Sontag51 and Tabor.52,53 Consider again the list example 关Eq. 共20兲兴 from Fig. 2共a兲 with two fillers F = 兵a , b其 and a countable number of list position roles R = 兵r j 兩 j 苸 N其. Then the assignments 共a兲 = 0, 共b兲 = 2, and 共r j兲 = 3−j yield a numerical representation of any list ␣ of n symbols from F by triadic numbers n
共␣兲 = 兺 a j3−j
In the same way, we obtain the tensor product representation of the tree from Fig. 2共d兲, derived in Eq. 共12兲 as
共T兲 = S 丢 r1 丣 共S 丢 r1 丣 a 丢 r2 丣 b 丢 r3兲 丢 r2 共S 丢 r1 丣 a 丢 r2 丣 b 丢 r3兲 丢 r3 .
n
x = 共␣兲 = 兺 a jg−j, j=1
Dolan and Smolensky,9 Smolensky,10 Smolensky and Legendre,11 Smolensky,12 Mizraji,49,50 and more recently, beim Graben et al.13 used finite-dimensional arithmetic vector spaces VF = R p, VR = Rq 共p , q 苸 N兲, and their Kronecker tensor products to obtain arithmetic vector space representations that can be regarded as activation states of neural networks or connectionist architectures. Setting, e.g.,
冉冊 0
,
b=
冉冊 0 1
冢冣 冢冣 冢冣 1
,
0
r1 = 0 , 0
0
r2 = 1 , 0
r3 = 0 1
yields the tensor product representation of the simple list 关Eq. 共20兲兴 from example in Fig. 2共a兲
共␣兲 =
=
冉冊 1 0
冢冣 1
丢
0
丣
0
冉冊 0 1
冢冣 0
丢
1 0
丣
冉冊 1 0
冢冣 冢冣 冢冣 冢冣 1
0
1
2
0
0
0
0
0 0
丣
0 0
丣
0 0
=
0 0
0
1
0
1
0
0
0
0
冢冣
r1 =
0 1
共23兲
j=1
冉冊 1 0
,
r2 = 1,
r3 =
冉冊 0 1
,
where the role of r2 for the control states q 苸 Q was simply taken as the scalar constant one. Moreover, the p control states were represented in a local way by p canonical basis vectors 共q兲 = ek of R p. The complete tensor product representation of a Turing machine state s is thus s = 共s兲 = 共x,0, . . . ,1, . . . ,0,y兲T
共24兲
with 1 in the k + 1th position encoding the kth control state q. Other higher-dimensional fractal representations are obtained for arbitrary filler symbols. Assigning to each filler f i 苸 F a different vector 共f i兲 = fi 苸 R p and using the onedimensional role representation 共r j兲 = 2−j entails a p-dimensional Sierpinski sponge
0
丢
m
y = 共兲 = 兺 b jg−j
with an appropriate base number g 苸 N. As role vectors for the state description, they chose
1. Arithmetic representations
a=
共22兲
which constitutes exactly the Cantor set as representation space. Siegelmann and Sontag51 used such fractal encoding for the Turing machine tape sequences ␣,  in the state description Eq. 共5兲, yielding
共21兲
Also the tensor product representations for the different filler/role decompositions of Turing machine state descriptions are constructed similarly.
1
with a j 苸 兵0,2其,
j=1
丢 r2 .
丣
2. Fractal representations
S=
再
n
x 苸 关− 0.5,0.5兴 p兩x = 兺 2−jf j, n 苸 N j=1
冎
共25兲
as representation space.52,53 Tabor,52–54 e.g., represented three symbols 兵a,b,c其 by planar vectors .
Obviously, the dimensionality of this representation is exponentially increasing with increasing recursion, thus leading to an almost meaningless vacuum, where symbolic states are sparsely scattered around 共see, however, Refs. 11 and 12 for possible solutions兲.
a = 2−2
冉 冊 1
−1
,
b = 2−2
冉 冊 −1 −1
,
c = 2−2
冉 冊 −1 1
.
共26兲
The state space resulting from this assignment in combination with Eq. 共25兲 is the Sierpinski gasket shown in Fig. 5. Since the Sierpinski gasket can be generated by an iterated function system, Tabor52–54 proposed a general class of computational dynamical systems, called dynamical automata. Formally, a dynamical automaton is defined as an 8-tuple
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TABLE II. Admissible input mappings 共共Ci , a j , f k兲 = 1兲 for PDDA defined in Eq. 共30兲 and displayed in Fig. 5. Compartment共s兲 C1 C3 Any
x0 =
FIG. 5. 共Color online兲 PDDA with stack tape represented by the Sierpinski gasket in the fractal encoding Eq. 共25兲. The vectors a, b, and c are defined in Eq. 共26兲. The blue lines demarcate the partition compartments in Eq. 共30兲.
M DA = 共X,F,P,T, ,x0,A兲,
共27兲
where X is a metric space 共the automaton’s phase space兲, F is a finite set of functions, f k : X → X, 1 ⱕ k ⱕ K, P is a partition of X into M pairwise disjoint compartments, Ci 苸 P, that cover the whole phase space X, T is a finite input alphabet, x0 苸 X is the initial state, and
:P ⫻ T ⫻ F → 兵0,1其
共28兲
is the input mapping specifying for each compartment Ci 苸 P and each input symbol a j 苸 T whether a function f k 苸 F is applicable 共共Ci , a j , f k兲 = 1兲 or not 共共Ci , a j , f k兲 = 0兲 when the current state x 苸 Ci. Finally, A 傺 X is a region of accepting states of the dynamical automaton. 共We refer to the more general definition in Refs. 52 and 54 here.兲 Using this description, Fig. 5 illustrates a particular subclass, a pushdown dynamical automaton 共PDDA兲 recognizing the context-free grammar T = 兵a,b,c,d其,
N = 兵S,A,B,C,D其,
共29兲 P = 兵S → ABCD,S → ⑀,A → aA,A → a,B → bB,B → b, C → cC,C → c,C → aS,C → a,D → dS,D → d其, by setting X = 关− 0.5,0.5兴2 \ 关0,0.5兴2 ,
再
F = f 1共x兲 = x +
冉冊 冉 冊冎 0 2
1 1 f 3共x兲 = x − 1 2
, f 2共x兲 = 2x +
冉 冊 2
−2
,
,
共30兲 P = 兵C1 = 关− 0.5,0兴 ⫻ 关− 0.5,0兴,C2 = 关0,0.5兴 ⫻ 关− 0.5,0兴, C3 = 关− 0.5,0兴 ⫻ 关0,0.5兴其, T = 兵a,b,c其 for see Table II,
冉冊 0 0
,
Input
State transition
b c a
f1 f2 f3
A = 兵x0其.
Table II presents the admitted input mappings where 共Ci , a j , f k兲 = 1. Dynamical automata cover a broad range of computational dynamical systems. If, e.g., the number of functions K in F equals the number of input symbols N in T and 共Ci , a j , f j兲 = 1 for all compartments Ci 苸 P, function f j is uniquely associated with symbol a j. In this case, F is an iterated function system and the dynamical automaton is called dynamical recognizer.55–57 If, on the other hand, the phase space is the unit square X = 关0 , 1兴2, the partition P is rectangularly generated by Cartesian products of intervals of the x- and y-axes containing the same number of cells as the function set F, and if, furthermore, these functions f k are piecewise affine linear and if eventually 共Ci , a j , f i兲 = 1 for all input symbols a j 苸 T, then the functions f k are piecewise affine linear branches of one unique nonlinear map f : X → X and the symbolic dynamics of f is a generalized shift, as discussed in Sec. II A 1.45–47 The resulting dynamical automaton does not longer process input symbols directly but rather according to an autonomous nonlinear dynamics given by f. These systems have been called nonlinear dynamical automata.4,13,58 Finally, if the number of functions K = 2M, where M is the number of compartments of the partition, and if the criteria for dynamical recognizers and for nonlinear dynamical automata are both satisfied, one can chose the f k in such a way to obtain an interacting nonlinear dynamical automaton.13,59 3. Gödel representations
Nonlinear dynamical automata as a subclass of dynamical automata are explicitly constructed by two-dimensional tensor product representations. These Gödel encodings are obtained from scalar representations of fillers as integer numbers and fractal roles, respectively. Let us again consider the list example from Fig. 2共a兲 with two fillers F = 兵a , b其 and a countable number of list position roles R = 兵r j 兩 j 苸 N其. Setting 共a兲 = 0, 共b兲 = 1 and 共r j兲 = 2−j entails then a representation of a list ␣ with symbols from F as binary numbers n
共␣兲 = 兺 a j2−j
with a j 苸 兵0,1其.
共31兲
j=1
More generally, a set of N fillers is mapped by the Gödel encoding onto N − 1 positive integers. A string or list ␣ of length n of these symbols is then represented by an N-adic rational number
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共␣兲 = 兺 a jN−j .
6
共32兲
j=1
? 6
The main advantage of Gödel codes is that they could be naturally extended to lists of infinite length, such that 共f ␣兲 = 兺⬁j=1a jN−j is then a real number in the unit interval 关0, 1兴. Moore46,47 used a Gödel encoding for the state description 关Eq. 共15兲兴 of a Turing machine by a generalized shift. Decomposing a dotted bi-infinite sequence
␥ R = b 1b 2b 3 ¯
FIG. 6. Tree roles in a spin-one term schema.
共33兲
x = 共␥L兲 = 共q兲N−1 + 兺 共a j兲N−j−1 , j=1
共34兲
y = 共␥R兲 = 兺 共b j兲N−j j=1
yields the so-called symbologram representation of the symbol sequence ␥ and thus of the state description of a Turing machine in the unit square.60,61 The generalized shift is thereby represented by a piecewise affine linear map whose branches are defined at the domains of dependence of the shift, thus entailing a nonlinear dynamical automaton. Gödel representations were used by beim Graben and co-workers4,13,58,59 in the field of computational psycholinguistics. 4. Functional representations
The high dimensionality and sparsity of arithmetic tensor product representations should be avoided in dynamic cognitive modeling when recursion or parallelism are involved.13,14 In such cases, a further generalization of dynamical automata, where the metric space X is an infinitedimensional Banach or Hilbert space, appears to be appropriate. Such functional representations were suggested for quantum automata by Moore and Crutchfield.57 Related approaches represent compositional semantics through Hilbert space oscillations,62–65 or linguistic phrase structure trees through spherical harmonics.14 In order to construct a functional representation for our simple example from Fig. 2共a兲, we assign particular basis functions
共b兲 = f b共x兲 = x
共35兲
and
共r1兲 = g1共y兲 = sin y,
共36兲
共r3兲 = g3共y兲 = sin 3y
共39兲
= sin y + x sin 2y + sin 3y.
n
n
共38兲
共␣兲 = f a共x兲g1共y兲 + f b共x兲g2共y兲 + f a共x兲g3共y兲
and computing their Gödel numbers
共r2兲 = g2共y兲 = sin 2y,
ρ(r2) = |1, −1
?
Accordingly, the functional tensor product representation of the list ␣ = 共a , b , a兲 turns out to be
into two one-sided infinite sequences
共a兲 = f a共x兲 = 1,
ρ(r1) = |1, 0
共b兲 丢 共r2兲 = 共f b 丢 g2兲共x,y兲 = f b共x兲g2共y兲 = x sin 2y.
␥ = ¯ a 2a 1q . b 1b 2 ¯ ␥L = qa1a2 ¯ ,
ρ(r3) = |1, 1
共37兲
to fillers and roles, respectively. Then, the tensor product in function space leads to functions of several variables, e.g., given as
As another example, we give a functional representation of the logical equivalence relation discussed in Sec. II A 1. Here, we encode the inputs A and B to the truth Table I as fillers A , B 苸 兵0 , 1其. The first two input positions are represented by one-dimensional Gauss functions centered around sites y 1 , y 2 苸 R. In addition, we introduce a third input G = 1, acting as a gating variable, bound to another Gaussian centered around a third site y 3 苸 R. All three inputs are linearly superimposed in the one-dimensional tensor product representation as 2
2
2
共␣兲 = Ae−R兩y − y1兩 + Be−R兩y − y2兩 + Ge−R兩y − y3兩 ,
共40兲
where R is a characteristic spatial scale. In Sec. III C we use a second spatial dimension x 苸 R to implement logical inference through traveling pulses with lateral inhibition in a neural field. Choosing basis functions for functional tensor product representations naively could lead to an explosion of the number of independent variables. This can be avoided by selecting suitable basis functions with particular recursion properties. One of these systems is spherical harmonics used in the angular momentum algebra of quantum systems. Here, the Clebsch–Gordan coefficients allow an embedding of tensor products for coupled spins into the original single particle space.66 We demonstrate this construction for our example tree T from Fig. 2共d兲 关Eq. 共21兲兴. In a first step, we identify the three fillers F = 兵S , a , b其 with three roots of unity
k =
2k 3
共41兲
leading to the harmonic oscillations in the variable x, f k共x兲 = eikx .
共42兲
Then, we regard the tree in Fig. 4 as a “deformed” term schema for a spin-one triplet as in Fig. 6. Figure 6 indicates that the three role positions r1, r2, r3 关Eq. 共7兲兴 in a labeled binary tree are represented by three z-projections of a spin-one particle,
共r2兲 = 兩1,− 1典,
共r1兲 = 兩1,0典,
共r3兲 = 兩1,1典,
共43兲
which have an L2共S兲 representation by spherical harmonics
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兩j,m典 = Y jm共y兲
with y = 共 , 兲 and 苸 关0 , 兴, 苸 关0 , 2关. For treating complex phrase structure trees, we compute the tensor products of role vectors 共ri兲 丢 共r j兲. Inserting the spin eigenvectors from Eq. 共43兲 yields expressions such as 兩j1,m1典 丢 兩j2,m2典 = 兩j1,m1, j2,m2典,
共45兲
well known from the angular momentum coupling in quantum mechanics.66 In quantum mechanics, the product states 关Eq. 共45兲兴 generally belong to different multiplets, which are given by the irreducible representations of the spin algebra sl共2兲. These are obtained by the Clebsch–Gordan coefficients in the expansions 兩j,m, j1, j2典 =
+ 具1,− 1,1,1兩1,0,1,− 1典兩1,− 1,1,1典
共44兲
兺 m ,m =m−m 1
2
+ 具2,− 1,1,1兩1,0,1,− 1典兩2,− 1,1,1典. The first Clebsch–Gordan coefficient 具0 , −1 , 1 , 1 兩 1,0,1,−1典 = 0 as a spin j = 0 particle does not permit an m = −1 projection. The two other Clebsch–Gordan coefficients are 具1 , −1 , 1 , 1 兩 1 , 0 , 1 , −1典 = 具2 , −1 , 1 , 1 兩 1 , 0 , 1 , −1典 = 1 / 冑2. Correspondingly, we obtain for 兩1,− 1典兩1,− 1典 = 兩1,− 1,1,− 1典 2
= 兺 具j,− 2,1,1兩1,− 1,1,− 1典兩j,− 2,1,1典 j=0
= 具0,− 2,1,1兩1,− 1,1,− 1典兩0,− 2,1,1典 + 具1,− 2,1,1兩1,− 1,1,− 1典兩1,− 2,1,1典
具j1,m1, j2,m2兩j,m, j1, j2典
+ 具2,− 2,1,1兩1,− 1,1,− 1典兩2,− 2,1,1典.
1
⫻兩j1,m1, j2,m2典,
共46兲
where the total angular momentum j obeys the triangle relation 兩j 1 − j 2兩 ⱕ j ⱕ j 1 + j 2 .
共47兲
In order to describe recursive tree generation by wave functions of only one 共spherical兲 variable y = 共 , 兲, we invert Eq. 共46兲, leading to
Here, the first two Clebsch–Gordan coefficients vanish because spins j = 0 and j = 1 forbid m = −2. Therefore, only 具2 , −2 , 1 , 1 兩 1 , −1 , 1 , −1典 = 1 accounts for this state. Finally, we consider 兩1,1典兩1,− 1典 = 兩1,1,1,− 1典 2
= 兺 具j,0,1,1兩1,1,1,− 1典兩j,0,1,1典 j=0
j1+j2
兩j1,m1, j2,m2典 =
兺
j=兩j1−j2兩
= 具0,0,1,1兩1,1,1,− 1典兩0,0,1,1典
具j,m, j1, j2兩j1,m1, j2,m2典兩j,m, j1, j2典
+ 具1,0,1,1兩1,1,1,− 1典兩1,0,1,1典 共48兲
with the constraint m = m1 + m2. Equation 共48兲 has to be applied recursively for obtaining the role positions of more and more complex phrase structure trees. Finally, a single tree is represented by its filler/role bindings in the basis of spherical harmonics after contraction over 兩j1 , j2典,
共T兲 = 兺 a jkm f k共x兲Y jm共y兲,
共49兲
jkm
where the coefficients a jkm = 0 if filler k is not bound to pattern Y jm. Otherwise, the a jkm encode the Clebsch–Gordan coefficients in Eq. 共48兲. Now we are able to construct the functional tensor product representation for the tree in Fig. 2共d兲. Its representation in algebraic form was obtained in Eq. 共21兲. Inserting the spin representation for the roles yields
共S兲兩1,0典 + 共S兲兩1,0,1,− 1典 + 共a兲兩1,− 1,1,− 1典 + 共b兲兩1,1,1,− 1典 + 共S兲兩1,0,1,1典 + 共a兲兩1,− 1,1,1典 + 共b兲兩1,1,1,1典. Expressing the tensor products by Eq. 共48兲 yields first 兩1,0典兩1,− 1典 = 兩1,0,1,− 1典 2
= 兺 具j,− 1,1,1兩1,0,1,− 1典兩j,− 1,1,1典 j=0
= 具0,− 1,1,1兩1,0,1,− 1典兩0,− 1,1,1典
共50兲
+ 具2,0,1,1兩1,1,1,− 1典兩2,0,1,1典. Here, m = 0 is consistent with j = 0 , 1 , 2 such that all three terms have to be taken into account through 具0 , 0 , 1 , 1 兩 1 , 1 , 1 , −1典 = 1 / 冑3, 具1 , 0 , 1 , 1 兩 1 , 1 , 1 , −1典 = 1 / 冑2, and 具2 , 0 , 1 , 1 兩 1 , 1 , 1 , −1典 = 1 / 冑6. Thus, we have constructed the functional representation of the left subtree of Fig. 2共d兲. The corresponding expressions for the right subtree were derived in Ref. 14. The complete spherical wave representation of the tree in Fig. 2共d兲 is then
共T兲 = f S共x兲Y 1,0共y兲 +
f S共 x 兲 冑2
共Y 1,−1共y兲 + Y 2,−1共y兲兲 + f a共x兲Y 2,−2共y兲
+ f b共x兲共 冑3 Y 0,0共y兲 + 1
+ +
f S共 x 兲 冑2
1
1
冑2 Y 1,0共y兲
+
1
冑6 Y 2,0共y兲
兲
共Y 2,1共y兲 − Y 1,1共y兲兲 + f a共x兲共 冑13 Y 0,0共y兲 −
冑6 Y 2,0共y兲
兲 + f b共x兲Y 2,2共y兲,
1
冑2 Y 1,0共y兲
共51兲
where the fillers are functionally represented through Eq. 共42兲. 5. Representation theory
Up to this point, we gave on overview about different state space representations for complex symbolic data structures that could be regarded as formal models of mental or cognitive states. Cognitive processes, by contrast, were compared with algorithms according to the computer metaphor of the mind. Algorithms are generally sequences of instructions, or more specifically, of symbolic operations acting
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upon data structures and symbolic states. Thus, a cognitive process P = A1 ; A2 ; . . . ; An composed of n cognitive operations Ai can be seen as the concatenation product67–69 共indicated by “;”兲 of partial functions Ai : F⬁ → F⬁, where F⬁ denotes the set of filler/role bindings for symbolic expressions as introduced in Sec. II B. Two operations Ai and A j can be concatenated to another operator P = Ai ; A j if the image of A j is contained in the domain of Ai. Under this restriction, the operators Ai form an algebraic semigroup since the concatenation product is associative. Having constructed a state space representation F = 共F⬁兲 of symbolic data structures, the mapping can be formally extended to the operators acting on F⬁. Cognitive operations Ai and A j become thereby represented by mappings 共Ai兲 and 共A j兲 such that
共Ai ;A j兲 = 共Ai兲 ⴰ 共A j兲,
L
k共t兲vk共x兲 兺 k=1
λk
0.6 0.4 0.2
(a)
共53兲
for a functional representation. Here, vk共x兲 denotes the discrete cognitive state vk at time k represented by a function over feature space D and k共t兲 its corresponding timedependent amplitude. Equation 共53兲 is often referred to as an order parameter ansatz.70 The amplitudes k共t兲 are then governed by ordinary differential equations describing the continuous time dynamics of the cognitive states v共x , t兲.
0 0
10
time t
20
30
λ
3
1
共52兲
where “ⴰ” denotes the usual functional composition in representation space, defined as 共f ⴰ g兲共x兲 = f共g共x兲兲. Therefore, the mapping preserves the semigroup structure of cognitive operations and can be properly regarded as a semigroup representation in the sense of algebraic representation theory.16,17 Interestingly, Fock space representations of cognitive operations are piecewise affine linear mappings9–13,46,47,52–54,58,59 resulting in globally nonlinear maps at representation space. We present three illustrative examples in Secs. III and IV. For cognitive operators are partial functions, their representatives could be pasted together in several ways entailing nonlinear maps. The explicit construction of such maps from the well-known linear pieces establishes the inverse problem for dynamic cognitive modeling. Moreover, since many solutions of the inverse problem are, in principle, possible, their stability against parametric perturbation is of great importance. If solutions are highly unstable, the inverse problem is ill-posed. We discuss these issues in the remainder of the paper in some detail. However, we first address another, related problem, posed by Spivey and Dale.15 Symbolic operations are time discrete. When a cognitive process P = A1 ; A2 ; . . . ; An applies to a symbolic initial state s0 苸 F⬁ at time t = 0, An brings s0 into another state s1 = An共s0兲 at time t + 1 and so on. By contrast, brain dynamics is continuous in time. If 共P兲 = 共A1兲 ⴰ 共A2兲 ⴰ ¯ ⴰ 共An兲 is the representation of P and vt = 共st兲 苸 F are representations of the symbolic states at time t, we have to embed this time discrete dynamics of duration L into continuous time. A common approach is a separation ansatz v共x,t兲 =
0.8
0.5
0 1
1
0.5 0 0
λ
(b)
2
0.5 λ
1
FIG. 7. Transient amplitude dynamics of three cognitive states. 共a兲 Time course of 1共t兲 共solid兲, 2共t兲 共dashed兲, and 3共t兲 共dotted兲, obeying Eqs. 共54兲–共57兲. 共b兲 Three-dimensional phase portrait. Parameters as in Ref. 14.
In order to describe a transient dynamics of duration T = 3, where one cognitive state gradually excites its successor,14 we made the ansatz
dk共t兲 + k共t兲 = gk共0共t兲,1共t兲, . . . ,T共t兲兲 dt
with delayed couplings g0共t兲 = w · f ,
冉
共54兲
冊
1.5⌬ − t , ⌬
gl共1, . . . ,l−1兲共t兲 = w · f ,共l−1共t − ⌬兲兲,
共55兲 lⱖ1
共56兲
for t ⱖ 0. Here, the sigmoidal logistic function f with threshold and gain  is defined as f ,共z兲 =
1 1 + e−共z−兲
,
共57兲
where w, , , and are real positive constants. Figure 7 displays the dynamics of the amplitudes k共t兲 of three succeeding states v1, v2, and v3. Figure 7共a兲 shows the time courses of k共t兲, while Fig. 7共b兲 depicts a threedimensional phase portrait. Obviously, the cognitive states vk correspond to the maxima of their respective amplitudes. In phase space, they appear as saddle points, attracting states from the direction of its precursor and repelling them toward the direction of its successor. Thus, the unstable separatrices of these saddle points are connected with each other forming a heteroclinic sequence.71,72 Rabinovich and co-workers71–74 suggested stable heteroclinic sequences and stable heteroclinic channels as a universal account for transient cognitive computations. Regarding “pure” tensor product states as saddle points in continuous time dynamics implies that these symbolically
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meaningful, representational states will never be reached by the system’s evolution. Instead, the trajectory approaches these states from one direction and diverges into another direction. Thus, there is a trade-off between continuous time dynamics and the interpretation of the system’s behavior as algorithmic symbol processing. In this sense, continuous time dynamics does not implement symbolic processing, it rather approximates it. Thus, one can call dynamics and algorithmic processing incompatible with each other.75 However, this kind of incompatibility could be easily resolved assuming that tensor product states are prototypes of symbolic representations in phase space. These prototypes define equivalence classes and thereby phase space partitions. Then, symbolic states can be identified with partition cells entered by the system’s trajectory, instead of with single points in phase space. Consequently, a symbolic dynamics as discussed in Secs. II and III is obtained, providing an implementation of algorithmic processing through the change of the ontology. However, this implementation could be incompatible with the phase space dynamics unless the partition is generating.4,76–78 C. Neurodynamics
Following the top-down path of the three tier approach for dynamic cognitive modeling, the third and final steps comprise the above-mentioned implementation of a state space representation 共step two兲 constructed from the symbolic description 共step one兲. In the following, we use the term “implementation” neutrally as inspired by quantum theory, where a symmetry is implemented at the Fock space of quantum fields,17 thereby disregarding the philosophical controversy about whether connectionist models are “mere implementations” of cognitive architectures.4,11,12,36,75,79–83 A cognitive model is implemented by equipping its representation space X with a flow ⌽t : X → X solving dynamical equations in time t. If X is a finite-dimensional vector space R p, these equations are usually ordinary differential 共or difference兲 equations for the state vector u共t兲. If, conversely, X is an infinite-dimensional function space resulting from a functional representation, the states are fields u共x , t兲 obeying either partial differential equations or, more generally, integrodifferential equations. It is of crucial importance for dynamic cognitive modeling that such implementations are guided by principles from the neurosciences. Under these guiding principles, we consider ANN models or connectionist architectures in the first case of finitedimensional representations.5,6,9–12,84–86 On the other hand, neural or dynamic field models are investigated in the second case of infinite-dimensional, i.e., functional, representations.18–34
p
dui共t兲 + ui共t兲 = 兺 wij f共u j共t兲兲, dt j=1
共58兲
where ui共t兲 is the time-dependent membrane potential of the ith neuron in a network of p units. The activation function f describes the conversion of the membrane potential ui共t兲 into a spike train ri共t兲 = f共ui共t兲兲. The left-hand side of Eq. 共58兲 characterizes the intrinsic dynamics of a leaky integrator unit, i.e., an exponential decay of membrane potential with time constant ⬎ 0. The right-hand side of Eq. 共58兲 represents the net input to unit i: the weighted sum of activity delivered by all units j that are connected with unit i共j → i兲. Therefore, the synaptic weight matrix W = 共wij兲 comprises three different kinds of information: 共1兲 unit j is connected with unit i if wij ⫽ 0 共connectivity, network topology兲, 共2兲 the synapse j → i is excitatory 共wij ⬎ 0兲, or inhibitory 共wij ⬍ 0兲, and 共3兲 the strength of the synapse is given by 兩wij兩. There are essentially two important network topologies. If the synaptic weight matrix indicates a preferred direction of propagation, the network has feed-forward topology. If, on the other hand, no preferred direction is indicated, the network is recurrent. Activation functions f depend on particular modeling purposes. The most common ones are either linear 共f共z兲 = z兲 or sigmoidal as in Eq. 共57兲, which describes the stochastic all-or-nothing law of neural spike generation.19 Neural network models for cognitive processes have a long tradition.5,6,11,89–92 Siegelmann and Sontag51 used the tensor product representation 关Eq. 共24兲兴 to prove that a recurrent neural network of about 900 units implements a Turing machine. By contrast, using the Gödel representation 关Eq. 共34兲兴, Moore46,47 and Siegelmann45 demonstrated that a Turing machine can, in fact, be implemented as a lowdimensional neural network. Pollack55 and Tabor52,53 used cascaded neural networks for implementing dynamical recognizers and dynamical pushdown automata 关Eq. 共30兲兴 for syntactic language processing 共for a local representation, see Ref. 93兲. These have been recently generalized by Tabor54,94 to fractal learning neural networks. In order to model word prediction dynamics, Elman95 suggested simple recurrent neural networks. This architecture became very popular in recent years in the domain of language processing.96–99 Kawamoto100 used a Hopfield net101 with exponentially decaying activation and habituating synaptic weights to describe lexical ambiguity resolution. Similar architectures are Smolensky’s harmony machines11,12,83,102 and Haken’s synergetic computers.70,103,104 Vosse and Kempen105 described a sentence processor through local inhibition in a unification space. Neural network applications for logic and semantic processing were proposed in Refs. 49, 50, 62–65, and 106– 109. For further references, see also Refs. 39, 87, and 110.
1. Neural networks
A common choice for the dynamics of neural networks are population rate models, where the ith component ui共t兲 of the state vector u共t兲 苸 X 傺 R p describes the firing rate or the firing probability either of a single neuron or of a small neural population i in the network. The so-called leaky integrator models6,84,85,87,88 are governed by equations of the form
2. Neural field models
Analyzing and simulating large neural networks with complex topology is a very hard problem88,111–117 due to the nonlinearity of the activation function and the large number of synaptic weights. Instead of computing the sum in the right-hand side of Eq. 共58兲, a continuum limit significantly
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facilitates analytical treatment and numeric simulations. Such continuum approximations of neural networks have been proposed since the 1960s.18–31 Starting with the leaky integrator network 关Eq. 共58兲兴, the sum over all units at the right-hand side is replaced by an integral transformation of a neural field quantity u共x , t兲, where the continuous parameter x 苸 R p now indicates the position i in the network. Correspondingly, the synaptic weight matrix wij turns into a kernel function w共x , y兲. Then, Eq. 共58兲 assumes the form of the Amari equation18,30 investigated in neural field theory.
u共x,t兲 + u共x,t兲 = t
冕
w共x,y兲f共u共y,t兲兲dy,
x 苸 D,
t ⬎ 0.
D
共59兲 Interpreting the domain D 傺 R of the Amari equation 共59兲 not as a physical substrate realized by actual biological neurons, but rather as an abstract features space for computational purposes, one speaks about dynamic field theory.32–34 Neural and dynamic field theory, respectively, found several applications in dynamic cognitive modeling. Jirsa and Haken25,118 and Jirsa et al.119 modeled changes in magnetoencephalographic 共MEG兲 activity during bimanual movement coordination, while Bressloff and co-workers20,120–125 described the pattern formation and the emergence of hallucinations in primary visual cortex. Infant habituation was modeled by Schöner and colleagues32–34 through decaying dynamic fields defined over the visual field of a children performing memory tasks 共see also Ref. 3 for a review兲. p
共x,t兲 =
共x,t兲 = f共v共x,t兲兲,
x 苸 D,
t ⱖ 0,
共60兲
x 苸 D,
t ⱖ 0,
共61兲
and employing the integral operator 共W兲共x兲 =
冕
w共x,y兲共y兲dy,
x 苸 D,
共62兲
D
leads to a reformulation of the inverse problem into the equation
共x,t兲 = W共x,t兲,
x 苸 D, t ⱖ 0,
= 1, . . . ,n,
共63兲
where here the kernel w共x , y兲, x , y 苸 D of the linear integral operator W is unknown. Equation 共63兲 is linear in the kernel w. It can be rewritten as
= W
共64兲
with
= 共1, . . . , n兲,
= 共1, . . . , n兲.
For every fixed x 苸 D, we can rewrite Eq. 共63兲 as
x共t兲 =
冕
共y,t兲wx共y兲dy,
tⱖ0
共65兲
D
with wx共y兲 = w共x,y兲, x,y 苸 D, x共t兲 = 共x,t兲, x 苸 D, t ⱖ 0. We define
3. Inverse problems for neurodynamics
Up to a few examples, where neural architectures or neural fields can be explicitly designed, the top-down approach of dynamic cognitive modeling either provides discrete sequences of training patters vk共x兲 苸 X or continuous paths v共x , t兲 = 兺kkvk共x兲 关Eq. 共59兲兴 in representation space that have to be reproduced by the state evolution u共x , t兲. If the dynamics is given either by the leaky integrator 关Eq. 共58兲兴 for a neural network or by the Amari equation 共59兲 for a neural/dynamic field, the crucial task is the determination of the system parameters. This is again the inverse problem for dynamic cognitive modeling: given a prescribed trajectory v共x , t兲, then find the synaptic weight matrix W = 共wij兲 or the synaptic weight kernel w共x , y兲, respectively, such that u共x , t兲 = v共x , t兲 solves the dynamical law 关Eq. 共58兲 or 共59兲兴 for all t ⬎ 0 with initial condition u共x , 0兲 = v共x , 0兲. Here, we discuss the inverse problem in the framework of the Amari equation 共59兲. We prescribe one or several complete time-dependent patterns v共x , t兲, = 1 , . . . , n for x 苸 D, t ⱖ 0 with some domain D 傺 R p. For our further discussion, we assume that the nonlinear activation function f : R → 关0 , 1兴 is known. Then, we search for kernels w共x , y兲 for x , y 苸 D such that the solutions of the Amari equation with initial conditions u共x , 0兲 = v共x , 0兲 satisfies u共x , t兲 = v共x , t兲 for x 苸 D, t ⱖ 0, and = 1 , . . . , n. As a first step, we transform Eq. 共59兲 into a linear integral equation. Defining
v共x,t兲 + v共x,t兲, t
共Vg兲共t兲 =
冕
k共t,y兲g共y兲dy,
tⱖ0
D
with the particular choice k共t , y兲 = 共y , t兲 to write Eq. 共65兲 as
x = Vwx,
x 苸 D.
共66兲
If is continuous in y and t, then for fixed x, Eq. 共66兲 is a Fredholm integral equation 共58兲 of the first kind with continuous kernel . This equation is known to be ill-posed,126 i.e., we do not have 共a兲 existence or 共b兲 uniqueness in general and even if we have uniqueness, the solution does not depend in a 共c兲 stable way on the right-hand side. Fredholm integral equations of the form
冕
k共t,y兲g共y兲dy = f共t兲
D
with continuous kernel k共t , y兲 are well studied in mathematical literature 共cf., for example, Refs. 126–128兲. The analysis is built on the properties of compact linear operators between Hilbert spaces. Consider a compact linear operator A : X → Y with Hilbert spaces X, Y, and its adjoint Aⴱ. Then, the nonnegative square roots of the eigenvalues of the self-adjoint compact operator AⴱA : X → X are called the singular values of the operator A. The singular value decomposition126 leads to a representation of the operator as a multiplication operator within two orthonormal systems 兵gn 兩 n 苸 N其 in X and 兵f n 兩 n 苸 N其 in Y, such that
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⬁
Ag = 兺 n具g,gn典f n
共67兲
n=1
R␣Ag = 兺 q共n␣兲具Ag, f n典gn n=1 ⬁
for g 苸 X. The representation 共67兲 is also known as spectral representation of the operator A. For the orthonormal systems gn and f n we have127 A ⴱ f n = ng n .
Agn = n f n,
= 兺 q共n␣兲n具g,gn典gn n=1 N
共68兲
=兺 ¯ + n=1
If A is injective, the inverse of A is given by ⬁
A−1 f = 兺
n=1
1 具f, f n典gn . n
共69兲
If A is not injective, the inverse A−1 defined in Eq. 共69兲 projects onto the orthogonal space N共A兲⬜ = 兵g 兩 具g , ˜g典 = 0 for all ˜g 苸 N共A兲其. For compact operators A, it is well known128 that the singular values build a sequence which at most accumulates at zero. The instability of the inverse problem is then caused by behavior
冏 冏
1 → ⬁, n
⬁
R␣ f = 兺 q共n␣兲具f, f n典gn . n=1
The choice
冦
1 , n ⱕ 1/␣ = n 0, otherwise
冧
n , ␣ + 2n
n 苸 N.
共71兲
Here, the parameter ␣ is known as regularization parameter. Boundedness of q共n␣兲 can be easily obtained by elementary calculation. For both cases, we have q共n␣兲 →
1 n
for ␣ → 0
for each fixed n 苸 N. This can be used to prove 共see Ref. 126兲 that R␣Ag → g
共73兲
⬁
储
兺 n=N+1
⬁
q共n␣兲n具g,gn典gn储2 ⱕ
兩q共n␣兲n兩2兩具g,gn典兩2 → 0 兺 n=N+1 for N → ⬁.
共74兲
Further, since for every fixed n 苸 N for ␣ → 0,
for ␣ → 0,
N
兺
N
q共n␣兲具g,gn典gn
n=1
→ 兺 具g,gn典gn
for ␣ → 0.
共75兲
n=1
Combining Eqs. 共73兲–共75兲 we obtain Eq. 共72兲 by arguing that ⬁ ¯ sufficiently small we first choose N to make the tail 兺n=N+1 uniformly for all ␣ and then letting ␣ tend to zero. Clearly, if we do not have exact data f = Ag, but some right-hand side f 共␦兲 polluted by numerical or measurement error, then we cannot carry out the limit ␣ → 0. In this case we can split the total error of the reconstruction into two parts
共70兲
is known as spectral cutoff. An alternative to the abrupt cut in Eq. 共70兲 is the Tikhonov regularization q共n␣兲 =
¯.
we obtain
amplifying small errors when the inverse is applied in a naive way. The classical approach to remedy the instability is known as regularization.126 In terms of the spectral inverse 关Eq. 共69兲兴, we can easily understand the main idea of regularization by damping of the factors 1 / n for large n, i.e., by replacing 1 / n by qn which is bounded for n 苸 N. This leads to a modified inverse operator
q共n␣兲
兺
n=N+1
Since q共n␣兲n is uniformly bounded for all ␣ ⬎ 0, using the Cauchy–Schwarz inequality, we have
q共n␣兲 → 1
n → ⬁,
⬁
共72兲
i.e., in the case of exact data f = Ag, the regularized solution converges toward the true solution when the regularization parameter ␣ tends to zero. This can be seen by using 具Ag , f n典 = 具g , Aⴱ f n典 = n具g , gn典 and split of the sum into
If ␣ becomes small, the reconstruction error will decrease and tend to zero, but the data error will become large. If ␣ is large, then the data error is getting smaller, but the reconstruction error increases. Somewhere for medium size ␣, we obtain a minimum of the total error. Many methods for the automatic choice of the regularization parameter have been developed.128,129 Tikhonov regularization is a very general scheme which can be derived not only via the spectral approach but also by matrix algebra or by optimization for solving an ill-posed equation Vg = f. Clearly, the equation Vg = f is equivalent to the minimization min储Vg − f储, g苸X
共76兲
where X denotes some appropriate Hilbert space X, for example, the space L2共D兲 of square integrable functions on some domain D. The normal equations 共cf. Ref. 130兲 for the minimization problem are given by
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Inverse dynamic cognitive modeling
VⴱVg = Vⴱ f .
冕冕冋 T
The operator VⴱV does not have a bounded inverse. Stabilization is reached by adding a small multiple of the identity operator I,130 i.e., by solving 共␣I + VⴱV兲g = Vⴱ f ,
共77兲
which corresponds to adding a stabilization term ␣储g储2 to the minimization 关Eq. 共76兲兴, leading to the third form of the Tikhonov regularization
J=
D
0
g苸X
共78兲
The operator 关Eq. 共77兲兴 is usually discretized by standard procedures and then leads to a matrix equation which can be solved either directly or by iterative methods.130 The Moore–Penrose pseudoinverse is given by the limit of the Tikhonov regularization for ␣ → 0, i.e., it is V† = 共VⴱV兲−1Vⴱ = lim 共␣I + VⴱV兲−1Vⴱ . ␣→0
共79兲
However, as discussed above, this limit will lead to satisfactory reconstructions only for well-posed problems. For the above-mentioned ill-posed inverse problem for dynamic cognitive modeling, we have to employ ␣ ⬎ 0. These techniques were applied to neural pulse construction problems by Potthast and beim Graben.131,132 In particular, the authors demonstrated the feasibility of regularized construction techniques for synaptic kernel construction. Properties of the solutions were investigated and the illposedness of the problem was proven and demonstrated by particular examples. The construction of synaptic weight kernels w for the Amari equation 共59兲 is a linear problem and can be solved by linear methods. In Sec. III, we generalize the well-known linear Hebb rule to neural field models described by Eq. 共59兲. However, if the minimization problems 共76兲 and 共78兲, respectively, are more complex, also involving, besides the kernel w, space-dependent time constants = 共x兲 or driving forces p共x , t兲 added to the right-hand side of Eq. 共59兲, the inverse problem becomes highly nonlinear and different tools are required. Here, we briefly generalize the standard backpropagation algorithm6,5 to neural field models. Time-dependent recurrent backpropagation6,87,133–137 minimizes an error function for leaky integrator networks 关Eq. 共58兲兴. Its generalization aims at minimizing the error functional
冕冕 T
min w
0
D
1 共u共x,t兲 − v共x,t兲兲2dxdt 2
共80兲
between a prescribed field v共x , t兲 and the w-dependent solution of the Amari equation u共x , t兲关w兴. This problem can be tackled using field-theoretic variational calculus for the minimization problem 共80兲 under the constraint 共59兲 by introducing Lagrange multipliers 共x , t兲 and combining Eqs. 共59兲 and 共80兲 into the functional
册
+ u共x兲 − 共Wf共u兲兲共x,t兲其 dxdt.
共81兲
By partial integration of the time derivative u˙ and incorporating an additional condition 共x , T兲 = 0 on , we transform Eq. 共81兲 into
冕冕冋 T
J=
D
0
min共储Vg − f储2 + ␣储g储2兲.
1 共u共x,t兲 − v共x,t兲兲2 + 共x,t兲兵共x兲u˙共x,t兲 2
1 共u共x,t兲 − v共x,t兲兲2 − ˙ 共x,t兲共x兲u共x,t兲 2
册
+ 共x,t兲u共x,t兲 − f共u兲共x,t兲共Wⴱ兲共x,t兲 dxdt +
冕
共x,0兲共x兲u共x,0兲dx,
共82兲
D
where we took the adjoint of W with respect to the scalar product over D. Since the Amari equation needs to be satisfied, the partial derivative of J with respect to should be zero at the minimum. Thus, we are free in the choice of , which we choose such that the partial derivative of J with respect to u is zero. We further remark that at time t = 0, the variation in u should vanish, i.e., the last term in Eq. 共82兲 will not contribute to the partial derivative of u. This leads to the adjoint equation for the backpropagated error signal , ˙ 共x,t兲共x兲 − 共x,t兲 = u共x,t兲 − v共x,t兲 − f ⬘共u兲共x,t兲共Wⴱ兲共x,t兲 共83兲 on 关0 , T兴 with boundary condition 共x , T兲 = 0. Alternatively, Eq. 共83兲 can also be obtained from the Euler–Lagrange equations
␦L d ␦L =0 − ␦u dt ␦u˙ applied to the Lagrange density L共x,t兲 = 21 共u共x,t兲 − v共x,t兲兲2 + 共x,t兲兵共x兲u˙共x,t兲 + u共x兲 − 共Wf共u兲兲共x,t兲其 for x 苸 D, t 苸 关0 , T兴. Equation 共83兲 is an integrodifferential equation for which the existence theory of Potthast and beim Graben131 can be applied, i.e., we obtain well-defined global solutions for every admissible choice of a kernel w for Wⴱ and forcing terms u and v. Backpropagation for neural fields is then obtained as a gradient descent algorithm in function space, where J is minimized under the constraints 共59兲 and 共83兲 with respect to w and 共and possibly p兲.138 III. METHOD
In this section, we first generalize the classical Hebbian learning rule to the Amari equation using basic results from mathematical analysis. We show that the Hebb rule is imme-
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diately entailed by the theory of orthonormal systems and is thus a natural way to solve the inverse problem in dynamic cognitive modeling. In addition, we illustrate our previous findings by means of three examples for basic cognitive processes. The first example implements the push operation equation 共1兲 in a one-dimensional neural field. The second example implements logical inference of the equivalence relation in Table I by means of traveling pulses with lateral inhibition in a layered neural field. Our third example presents a tree generator processing a simple context-free grammar. In all three examples we construct synaptic weight kernels for the Amari equation by regularized Hebbian learning such that the neural field dynamics replicate the prescribed training patterns. A. Tikhonov regularized Hebbian learning
Let us study the neural inverse problem in the form
共x,t兲 =
冕
w共x,y兲共y,t兲dy,
x 苸 D,
t ⱖ 0,
共84兲
D
where the auxiliary fields and are given by Eqs. 共60兲 and 共61兲, and we search for the kernel w. Given the images 共x , t兲 for the elements 共x , t兲 for all x 苸 D, t ⱖ 0 leads to the problem that we know a bounded linear operator W and need to construct a kernel w共x , y兲 such that 共Wg兲共x兲 =
冕
x苸D
w共x,y兲g共y兲dy,
neural networks for neural fields. This learning rule was originally suggested by Hebb139 as a psychological mechanism based on physiological arguments. Here, it appears as a basic conclusion from the Fourier theorem for orthonormal systems. The underlying physiological idea is that a synaptic connection between neurons at sites y and x, as expressed by the kernel w共x , y兲, is given by the cross correlation of their activations, reflecting the input-output pairing , . Apparently, this is a natural consequence of the representation of the kernel by orthonormal input patterns . However, training patterns are usually not orthonormal. For a general set of training patterns that are not orthogonal, the Hebb rule does not yield optimal kernels due to crosstalk. Nevertheless, as far as training patterns are still linearly independent, one can cope with cross-talk through biorthogonal bases, which lead, in turn, to the Moore–Penrose pseudoinverse for finite-dimensional neural networks.6,87,104 Potthast and beim Graben132 generalized this approach to the field-theoretic setting deploying biorthogonal bases in Hilbert spaces. If, by contrast, training patterns are neither orthogonal nor linearly independent, the inverse problem for the Amari equation is ill-posed and solutions have to be regularized to stably construct appropriate kernels. Thus, the more general theory of Sec. II C 3 is required for this aim. We propose a modified Tikhonov–Hebb rule and apply it to calculating neural fields for different cognitive dynamics. Consider a dynamical field defined by Eq. 共54兲 as
共85兲
D
for all g 苸 X with some appropriate space X. Assume that we know W. Then for any orthonormal basis 兵gn 兩 n 苸 N其, we calculate f n = Wgn. We define the kernel w as
L
v共x,t兲 =
共86兲
n=1
D
⬁
w共x,y兲gᐉ共y兲dy = 兺 f n共x兲 n=1
冕
gn共y兲gᐉ共y兲dy
n=1
If we know that W is an integral operator 共85兲 with kernel w, then we can calculate w共x,y兲gn共y兲dy,
Further, we can develop w共x , y兲 into a Fourier series with respect to the variable y and the orthonormal basis gn. We calculate w共x,y兲 = 兺
n=1
冉冕
D
w共x,y兲共y,t兲dy,
x 苸 D,
t 苸 关0,T兴
共88兲
冊
T , Nt − 1
expressing the Euler rule du共x,t兲 u共x,t + ht兲 − u共x,t兲 ⬇ , dt ht i.e., the nodes of the regular time grid are given by
n 苸 N.
D
⬁
冕
for the unknown kernel w共x , y兲. We discretize the time interval 关0 , T兴 by Nt points and step size ht =
= 兺 f n共x兲␦nᐉ = f ᐉ共x兲.
冕
共x,t兲 =
D
⬁
f n共x兲 =
t 苸 关0,T兴.
D
if the sum is convergent. Then we can calculate
冕
x 苸 D,
Introducing the auxiliary fields , according to Eqs. 共60兲 and 共61兲, again, leads to the integral equation
⬁
w共x,y兲 = 兺 f n共x兲gn共y兲
k共t兲vk共x兲, 兺 k=1
⬁
w共x,y兲gn共y兲dy gn共y兲 = 兺 f n共x兲gn共y兲. 共87兲 n=1
Equations 共86兲 and 共87兲, respectively, provide the desired generalization of the well-known Hebbian learning rule of
tk = 共k − 1兲 · ht,
k = 1, . . . ,Nt .
共89兲
On the spatial domain D we use a discretization with Nd points. For the one-dimensional case, we employ a regular grid on 关0 , 2兴; for the sphere in the second example; we chose a regular grid in polar coordinates. In both cases we write the discretization as x j, j = 1 , . . . , Nd. The time-space discretization leads to matrices
jk = f共u共x j,tk兲兲,
x 苸 D,
t ⱖ 0,
共90兲
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FIG. 8. 共Color兲 Test patterns used in the one-dimensional Amari model. 共a兲 Three “cognitive states” of a pushdown automaton 共cf. Sec. II B兲 represented by spatial waves vk共x兲 = sin共kx兲, with k = 1 共solid兲, k = 2 共dashed兲, and k = 3 共dotted兲. 共b兲 Their respective amplitudes k共t兲 as tent functions of time. 共c兲 The spatiotemporal training pattern v共x , t兲 = 兺kk共t兲vk共x兲 resulting from the separation ansatz 共53兲 for solving the inverse problem for the Amari equation 共59兲.
jk =
u共x j,tk + ht兲 − u共x j,tk兲 + u共x j,tk兲, ht
x 苸 D,
t ⱖ 0. 共91兲
Discretization of the kernel w共x , y兲 is obtained by using the above spatial grid for both x and y. We will consider the surface element dy to be a part of the discretized kernel, i.e., we approximate the operator W by the matrix W = 共w共x j,y ᐉ兲兲 j,ᐉ=1,. . .,Nd . Then, Eq. 共88兲 is transformed into the matrix equation
= W .
共92兲
With the Tikhonov inverse of given as R␣ = 共␣I + ⴱ兲−1ⴱ,
␣ ⬎ 0,
共93兲
we obtain the reconstructed synaptic weight kernel W␣ = R␣ = 共␣I + ⴱ兲−1ⴱ
共94兲
for the Tikhonov–Hebbian learning rule of the Amari equation 共59兲. Note that Oja’s rule for unsupervised Hebbian learning6 can be considered as a special case of the Tikhonov–Hebb rule by replacing the regularization constraint ming ␣储g储 in Eq. 共78兲 by ming ␣共1 − 储g储兲. B. Pushdown stack
In Eq. 共1兲 we described a stack as a list of filler symbols F = 兵a , b其 occupying slots R = 兵r1 , r2 , r3其 for the three list positions. We use the functional representations 共35兲 and 共36兲, thus obtaining the cognitive states v1共x兲 = 共␣1兲 = sin x, v2共x兲 = 共␣2兲 = sin x + sin 2x,
共95兲
v3共x兲 = 共␣3兲 = sin x + sin 2x + sin 3x,
as images of the symbolic path Eq. 共1兲 in function space where we have deliberately replaced the variable y by x since
the symbol b, represented by the function f b共x兲 = x, does not occur. Instead of describing the temporal evolution by the order parameter 关Eq. 共54兲兴, we simply use tent maps
冦
t − 共k − 1兲 − k ⱕ t ⱕ − 共k − 1兲 + 1, k共t兲 = t − 共k − 1兲 − + 1, − 共k − 1兲 ⱕ t ⱕ 共2 − k兲 ,
冧
共96兲
where indicates the maximum of amplitude k共t兲. For the first example we set = 1. Figure 8 displays the spatial patterns vk共x兲 given by Eq. 共95兲 in 共a兲, the time course of the amplitudes k共t兲 关Eq. 共96兲兴 in 共b兲, and the spatiotemporal dynamics v共x , t兲 resulting from Eq. 共53兲 in 共c兲. This prescribed trajectory in function space is trained via Tikhonov–Hebbian learning by a discretized neural field obeying the Amari equation 共59兲 with the following parameters: number of spatial discretization points Nd = 300, number of temporal discretization points Nt = 100, time constant = 0.5, synaptic gain  = 10, and activation threshold = 0.3, where we compare the Moore–Penrose pseudoinverse 共79兲 共i.e., unregularized, ␣ = 0兲 with Tikhonov regularization 关Eq. 共93兲, ␣ = 0.1兴 for the Hebb rule. C. Logic gate
One of the most persistent problems in early neural network research was the implementation of linearly nonseparable logical functions, such as xor or equiv 共see Sec. II A 1兲 by means of feed-forward architectures. These problems are not solvable with one-layered networks at all and solving them with the nonlinear backpropagation algorithm for twolayered networks is computationally very expensive 共cf. Ref. 6兲. Here, we discuss a two-dimensional neural field model where the y-dimension serves as representation space for the
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FIG. 9. 共Color兲 Training pulses for logical equivalence function equiv 共Table I兲. 共a兲 A = 0, B = 0, and C = 1; 共b兲 A = 1, B = 0, and C = 0; 共c兲 A = 0, B = 1, and C = 0; and 共d兲 A = 1, B = 1, and C = 1. The gating pulse G = 1 in all cases.
tensor product representation 关Eq. 共40兲兴, whereas the x-dimension provides a continuum of 共uncountably兲 infinite many layers. The inputs A and B for the equiv gate are given by two Gaussians centered around points x1 , x2 苸 R. Additionally, a permanent gating input G = 1 is represented by a third Gaussian centered around x3 苸 R. The inference dynamics is prescribed by Gaussian pulses traveling along linear paths along the x-axis through 2
v共x,t兲 = 1共t兲Ae−R兩x − x1共t兲兩 + 2共t兲Be−R兩x − x2共t兲兩
+ 3共t兲Ge−R兩x − x3共t兲兩
2
2
共97兲
with A , B 苸 兵0 , 1其, G = 1, x = 共x , y兲 苸 D 傺 R2, and xi共t兲 = 共0 , y i兲 + tdi for t ⱖ 0. Here, di denotes the direction of the ith neural pulse. The amplitudes i共t兲 obey a suitably chosen order parameter equation 共53兲 that reflects lateral inhibition among the three pulses. Four trajectories of neural pulses were explicitly constructed, as illustrated in Fig. 9. When A = B = 0 关Fig. 9共a兲兴, there is no inhibition at all and the gating pulse G travels into its destination, yielding output C = 1. For A = 1, B = 0 关Fig. 9共b兲兴, input A interferes with G around x = 5, leading to extinction of G. Likewise, for A = 0, B = 1 关Fig. 9共c兲兴, B and G annihilate each other around x = 5, again. Finally, if A = B = 1 关Fig. 9共d兲兴, lateral inhibition of A and B takes place much earlier around x = 2 such that the unaffected gating pulse G becomes the desired output C = 1. The four different fields of superimposed traveling pulses are used as training patterns for Tikhonov–Hebbian learning of synaptic weight kernels, as explained above. We used a discretization of Nd = 30⫻ 31 points in the spatial domain D and Nt = 80 temporal discretization points for solving
the inverse problem. As regularization parameter we set ␣ = 1. Simulations were carried out with Nt = 160 time discretization points and = 2,  = 10, and = 0.5. D. Tree generator
Our second example is related to syntactic language processing where context-free grammars are processed by pushdown automata. For the sake of simplicity, we describe the process of tree generation as used in top-down parsing approaches.13,14,37 Here we use the CFG T = 兵a其,
N = 兵S其,
P=
再
共1兲 S → S S 共2兲 S → a
冎
共98兲
,
where a is the only terminal and the start symbol S is the only nonterminal symbol. One particular path of a tree generator is shown in Fig. 10. The trees depicted in Fig. 10 are functionally represented using the following mappings:
共a兲 = f a共x兲 = 0,
共S兲 = f S共x兲 = 1
共99兲
for the fillers. Note that a is treated as the “empty word” represented by the zero function here. The tree roles are represented by spherical harmonics as outlined in Sec. IV. Using the tensor product representation deploying Clebsch–Gordan coefficients, again, yields static spatial patterns (a) S
S
(b) S
S
(c) S
S a
S S
S
(d)
S
S
S
a
S S a a
FIG. 10. Example-tree generation according to grammar 共98兲.
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FIG. 11. 共Color兲 Spherical harmonics representation of trees vk共x兲 关Eq. 共100兲兴 used in the two-dimensional Amari model 共cf. Sec. II B兲. 共a兲 Zero-level tree consisting only of root node S as in Fig. 10共a兲, 共b兲 one-level tree as in Fig. 10共b兲, and 共c兲 two-level tree as in Fig. 10共c兲. 关Note that trees from Figs. 10共c兲 and 10共d兲 are mapped onto the same spherical representation兴.
v1共x兲 = 兩Y 1,0共x兲兩,
v2共x兲 = 兩Y 1,0共x兲 + Y 1,−1共x兲 + Y 1,1共x兲兩,
共100兲
冏
v3共x兲 = Y 1,0共x兲 + Y 1,−1共x兲 +
+
1
1
1
冑2 共Y 2,1共x兲 − Y 1,1共x兲兲 1
冑3 Y 0,0共x兲 − 冑2 Y 1,0共x兲 + 冑6 Y 2,0 + Y 2,2共x兲
冏
,
which are displayed in Fig. 11. Figure 11共a兲 shows the representation v1共x兲 of the simple tree from Fig. 10共a兲 containing only the filler S bound to the root role. Note that this is actually a pz orbital of the electronic wave function in the hydrogen atom. Figure 11共b兲 shows v2共x兲 as a superposition of three spherical harmonics corresponding to Fig. 10共b兲. Accordingly, Fig. 11共c兲 displays the representation of the tree in Fig. 10共c兲 where the right branch has been further expanded. Finally, we chose the same tent map 关Eq. 共96兲兴 共although with different time scales: = 15兲 as in Sec. III B for the amplitude dynamics. The spatiotemporal patterns resulting from Eq. 共53兲 are shown in Fig. 12. Again, a discretized neural field characterized by the Amari equation 共59兲 is trained with the Tikhonov–Hebb rule 关Eq. 共93兲兴. Here, we use simulation parameters: number of spatial discretization points Nd = 6400, number of temporal discretization points Nt = 30, time constant = 0.5, synaptic gain  = 10, and activation threshold = 0.3. IV. RESULTS
We next describe the results when applying the techniques of Sec. II C 3 to construct synaptic weight kernels generating the prescribed cognitive dynamics in representation space, constructed above. A. Pushdown stack
As shown in Fig. 8共c兲, the dynamics of a pushdown stack constructed in Eqs. 共95兲 and 共96兲, appears as a onedimensional wave field 共y-axis兲 over time 共x-axis兲, realizing continuous transitions from state v1 to state v2 and from v2 to state v3.
After discretizing temporal and spatial dimensions in accordance to Eqs. 共89兲–共91兲, Fig. 13 displays the mapping of the auxiliary fields 共a兲 onto 共b兲 . This mapping is achieved by the linear integral transformation 关Eq. 共92兲兴 derived from the Amari equation 共59兲. We used Tikhonov–Hebbian inversion theory to construct a synaptic weight kernel w by Eq. 共94兲. Without regularization we do not expect to obtain a reasonably stable kernel. This is demonstrated in Fig. 14共a兲, where we carried out the reconstruction with the standard Moore–Penrose pseudoinverse 共79兲. Clearly, the classically reconstructed kernel is strongly fluctuating between −104 and +104, thus reflecting the ill-posedness of the inverse problem. On the other hand, the regularized kernel is visualized in Fig. 14共b兲. Regularization does neither require orthogonality nor even linear independence of training patterns. It is rather robust against linear dependence as resulting, e.g., from oversampling of discretized data. Note further, that the kernel depicted in Fig. 14共b兲 is not translation invariant. Hence we have explicitly constructed an inhomogeneous synaptic weight kernel from the prescribed cognitive training process. In the final step we then solved the Amari equation numerically with the reconstructed kernel. Here, it is of considerable importance to use a different discretization in time in order to avoid the inverse crime: using the same sampling for training and simulation could spuriously yield coincident patterns. Therefore, we doubled the temporal sampling rate for simulation. Simulation results are presented in Fig. 15. Image 共a兲 shows the field generated with the Moore–Penrose pseudoinverse. The field quickly explodes and moves into highorder oscillations, which correspond to the strong amplification of high modes excited by small numerical errors 关cf. Eq. 共69兲兴. Figure 15共b兲 demonstrates the reconstruction of the prescribed field dynamics. This image has been generated from the Tikhonov–Hebbian regularized kernel. The results prove that stable and quick construction of synaptic weight kernels is possible to generate prescribed dynamics constructed from representations of cognitive states.
FIG. 12. 共Color兲 Tree generation dynamics resulting from the separation ansatz 共53兲 for solving the inverse problem for the Amari equation 共59兲.
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FIG. 13. 共Color兲 Auxiliary fields 共a兲 共x , t兲 and 共b兲 共x , t兲 obtained from the training pattern v共x , t兲 by Eqs. 共60兲 and 共61兲 for parameters = 0.5,  = 10, and = 0.3.
B. Logic gate
Figure 16 shows the reconstructed traveling pulse dynamics of the logical interference neural field. Clearly, they are in good agreement with the prescribed training data shown in Fig. 9. Our results show that a hardly tractable problem for connectionist neural networks becomes linearly solvable for neural fields.
C. Tree generator
As before, Fig. 12 visualizes the prescribed cognitive dynamics constructed in Eq. 共100兲. It shows a selection of time slices for the spatiotemporal dynamics of the field representations, sampling a realization of a continuous transition over several states v1 , . . . , vn. Again, we discretized both the temporal and spatial dimensions following Eqs. 共89兲–共91兲. We then used Tikhonov–Hebbian inversion theory again to construct a synaptic weight kernel w. In turn, this kernel was used to calculate the neural field u共x , t兲 as a solution of the Amari equation 共59兲, which is shown in Fig. 17.
FIG. 15. 共Color兲 Numerical solutions u共x , t兲 of the Amari equation 共59兲 with estimated synaptic weight kernels w共x , y兲 as in Fig. 14. 共a兲 For Moore– Penrose pseudoinverse 共i.e., ␣ = 0兲 and 共b兲 for Tikhonov inverse 共93兲 with regularization parameter ␣ = 0.1 for parameters = 0.5,  = 10, and = 0.3. Compare with Fig. 8.
The results demonstrate that stable and quick construction of synaptic weight kernels is feasible over another twodimensional manifold in order to replicate neural field dynamics in cognitive representation spaces. V. DISCUSSION
We described dynamic cognitive modeling as a three tier top-down approach comprising the levels of 共1兲 cognitive processes, 共2兲 state space representations, and 共3兲 neurodynamical implementations. These levels are passed through in a top-down manner: 共1兲 cognitive processes are described as algorithms sequentially operating on complex symbolic data structures that are decomposed using filler/role bindings; 共2兲 data structures are mapped onto “points” in Fock spaces 共abstract feature spaces兲 using tensor product representations; and 共3兲 cognitive operations are implemented as dynamics of neural respective dynamic fields as continuum approximations of neural networks. The last step involves the solution of inverse problems, namely, training synaptic weights of the Amari equation to reproduce prescribed paths in representation space.
FIG. 14. 共Color兲 Synaptic weight kernels w共x , y兲 of the Amari equation 共59兲. 共a兲 For Moore–Penrose pseudoinverse 共i.e., ␣ = 0兲 and 共b兲 for Tikhonov inverse 共93兲 with regularization parameter ␣ = 0.1 for parameters = 0.5,  = 10, and = 0.3.
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FIG. 16. 共Color兲 Numerical solutions u共x , t兲 of the Amari equation 共59兲 for logical equivalence function equiv 共Table I兲. 共a兲 A = 0, B = 0, and C = 1; 共b兲 A = 1, B = 0, and C = 0; 共c兲 A = 0, B = 1, and C = 0; and 共d兲 A = 1, B = 1, and C = 1 for parameters ␣ = 1, = 2,  = 10, and = 0.5. Compare with Fig. 9.
After recasting the Amari equation into a linear integral equation that can principally be trained by Hebbian learning, we demonstrated that these problems are ill-posed and solutions have to be regularized. We suggested a modified Tikhonov–Hebb learning rule and showed its stability for particular examples of cognitive dynamics in function spaces. Tikhonov–Hebbian learning is a quick and simple 共for being linear兲 training algorithm, not requiring orthogonality or even linear independence of training patterns. In fact, the regularization is robust against linearly dependent patterns as they could result from oversampling. Regarding neural field models with traveling pulse dynamics as layered architectures,132 showed that Tikhonov–Hebbian learning also works for linearly nonseparable problems such as the persistent XOR problem for which nonlinear and computationally expensive training algorithms have to be employed for connectionist models. Moreover, Tikhonov–Hebbian learning generally leads, for given training patterns, to inhomogeneous kernels, which is a considerable progress for neural/dynamic field models of cognitive processes. Most efforts in the field so far have been done investigating either homogeneous kernels21,24,25,28,29 or by introducing inhomogeneity explicitly.20,31,122,140–143 For the sake of simplicity, we represented cognitive states as static spatial patterns in one and two dimensions by encoding symbolic fillers as constants. Using functional representations for fillers as well would further increase the dimensionality of variable domains by one. By contrast, Ref. 14 suggested a separation of time scales to represent fillers as
fast oscillations and cognitive processes as slow transients independently. This approach, however, is not feasible using Tikhonov–Hebbian learning for solving the inverse problem for the Amari equation because a synaptic weight kernel, trained on the fast time scale, would not be able to replicate the slow cognitive dynamics and vice versa. As the neural field is a deterministic dynamical system, the same fast cycle would be repeated after training the first one. Nevertheless, separation of time scales could still be possible using more sophisticated solutions of the inverse problem involving time-dependent input to the neural field. We outlined one possible solution of the nonlinear inverse problem by functional backpropagation for learning weight kernels, time constants and possibly input forces simultaneously. In the present state, dynamic cognitive modeling deals with an abstract, theoretical task, namely, solving the inverse problem of finding a neurodynamical implementation for a given algorithmic symbol processor in a top-down fashion. It does currently not address another inverse problem prevalent in cognitive neuroscience, namely the bottom-up reconstruction of neurodynamics from observed physiological time series, such as electroencephalographic 共EEG兲,144–149 MEG,150 optical diffusion tomographic,151 or functional magnetic resonance tomographic152 data. In particular, it is not related to dynamic causal modeling153,154 or similar 155,156 where the inverse problem of finding neural approaches generators from physiological data is addressed. Certainly, dynamic causal modeling and dynamic cognitive modeling ought to be connected at the intermediate level of neurody-
FIG. 17. 共Color兲 Numerical solutions u共x,t兲 of the Amari equation 共59兲 for tree generation dynamics. Compare with Fig. 12.
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namics in the long run, one coming from the bottom level of physiological data, the other one coming from the top level of cognitive architecture. We leave this ambitious project for future research. ACKNOWLEDGMENTS
This work has been supported by an EPSCR Bridging the Gaps grant on Cognitive Systems Sciences. Inspiring discussions with and helpful comments from Sabrina Gerth, Whitney Tabor, Paul Smolensky, and Johannes Haack are gratefully acknowledged. We would like to thank Tito Arecchi and Jürgen Kurths for their kind invitation to contribute to this focus issue on Nonlinear Dynamics in Cognitive and Neural Systems. 1
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Lectures in Supercomputational Neuroscience: Dynamics in Complex Brain Networks, edited by P. beim Graben, C. Zhou, M. Thiel, and J. Kurths 共Springer, Berlin, 2008兲, pp. 195–223. 88 P. beim Graben and J. Kurths, Neurocomputing 71, 999 共2008兲. 89 Neurocomputing: Foundations of Research, edited by J. A. Anderson and E. Rosenfeld 共MIT Press, Cambridge, MA, 1988兲, Vol. 1. 90 Neurocomputing: Directions for Research, edited by J. A. Anderson, A. Pellionisz, and E. Rosenfeld 共MIT Press, Cambridge, MA, 1990兲, Vol. 2. 91 P. S. Churchland and T. J. Sejnowski, The Computational Brain 共MIT Press, Cambridge, MA, 1994兲. 92 The Handbook of Brain Theory and Neural Networks, edited by M. A. Arbib 共MIT Press, Cambridge, MA, 1998兲. 93 C.-H. Chen and V. Honavar, IEEE Trans. Neural Netw. 10, 1239 共1999兲. 94 W. Tabor, IEEE Trans. Neural Netw. 14, 444 共2003兲. 95 J. L. Elman, in Mind as Motion: Explorations in the Dynamics of Cognition, edited by R. F. Port and T. van Gelder 共MIT Press, Cambridge, MA, 1995兲, pp. 195–223. 96 W. Tabor, C. Juliano, and M. K. Tanenhaus, Lang. Cognit. Processes 12, 211 共1997兲. 97 W. Tabor and M. K. Tanenhaus, Cogn. Sci. 23, 491 共1999兲. 98 S. Lawrence, C. L. Giles, and S. Fong, IEEE Trans. Knowl. Data Eng. 12, 126 共2000兲. 99 I. Farkas and M. W. Crocker, Neurocomputing 71, 1172 共2008兲. 100 A. H. Kawamoto, J. Mem. Lang. 32, 474 共1993兲. 101 J. J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 79, 2554 共1982兲. 102 P. Smolensky, in Parallel Distributed Processing: Explorations in the Microstructure of Cognition, edited by D. E. Rumelhart, J. L. McClelland, and the PDP Research Group 共MIT Press, Cambridge, MA, 1986兲, Vol. 1, pp. 194–281. 103 H. Haken, Synergetic Computers and Cognition: A Top-Down Approach to Neural Nets 共Springer, Berlin, 1991兲. 104 H. Haken, Principles of Brain Functioning 共Springer, Berlin, 1996兲. 105 T. Vosse and G. Kempen, Cognition 75, 105 共2000兲. 106 C. Balkenius and P. Gärdenfors, in Principles of Knowledge Representation and Reasoning, edited by J. A. Allan, R. Fikes, and E. Sandewall 共Morgan Kaufmann, San Mateo, CA, 1991兲, pp. 32–39. 107 P. Gärdenfors, Foundations of Computation, edited by J. v. Eijck and A. Visser 共MIT-Press, Cambridge, MA, 1994兲, pp. 49–77. 108 R. Blutner, Synthese 141 共2004兲. 109 E. Mizraji and J. Lin, Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, 155 共2001兲. 110 M. H. Christiansen and N. Chater, Cogn. Sci. 23, 417 共1999兲. 111 S.-I. Amari, Kybernetik 14, 201 共1974兲. 112 R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 共2002兲. 113 L. Zemanová, C. Zhou, and J. Kurths, Physica D 224, 202 共2006兲. 114 C. Zhou, L. Zemanová, G. Zamora, C. C. Hilgetag, and J. Kurths, Phys. Rev. Lett. 97, 238103 共2006兲. 115 M. Kaiser and C. C. Hilgetag, Neurocomputing 58–60, 297 共2004兲. 116 O. Sporns, D. R. Chialvo, M. Kaiser, and C. C. Hilgetag, Trends Cogn. Sci. 8, 418 共2004兲. 117 W. Maass, T. Natschläger, and H. Markram, Neural Comput. 14, 2531 共2002兲. 118 V. K. Jirsa and H. Haken, Physica D 99, 503 共1997兲. 119 V. K. Jirsa, A. Fuchs, and J. A. S. Kelso, Neural Comput. 10, 2019 共1998兲. 120 P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas, and M. C. Wiener, Philos. Trans. R. Soc. London, Ser. B 356, 299 共2001兲.
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P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas, and M. C. Wiener, Neural Comput. 14, 473 共2002兲. 122 P. C. Bressloff, Phys. Rev. Lett. 89, 088101 共2002兲. 123 P. C. Bressloff and J. D. Cowan, J. Physiol. 共Paris兲 97, 221 共2003兲. 124 P. C. Bressloff, Biol. Cybern. 93, 256 共2005兲. 125 A. M. Oster and P. C. Bressloff, Bull. Math. Biol. 68, 73 共2006兲. 126 R. Kress, Linear Integral Equations 共Springer-Verlag, Berlin, 1989兲. 127 D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory 共Springer-Verlag, Berlin, 1998兲. 128 H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems 共Kluwer Academic, Dordrecht, 1996兲. 129 A. Neumaier, SIAM Rev. 40, 636 共1998兲. 130 R. Kress, Numerical Analysis, Graduate Texts in Mathematics 共SpringerVerlag, New York, 1999兲. 131 R. Potthast and P. beim Graben, “Existence and properties of solutions for neural field equations,” Math. Models Meth. Appl. Sci. 共to be published兲. 132 R. Potthast and P. beim Graben, “Inverse problems in neural field theory,” SIAM J. Appl. Dyn. Syst. 共unpublished兲. 133 H. Bersini, M. Saerens, and L. G. Sotelino, IEEE Trans. Neural Netw. 5, 945 共1994兲. 134 B. A. Pearlmutter, Neural Comput. 1, 263 共1989兲. 135 B. A. Pearlmutter, IEEE Trans. Neural Netw. 6, 1212 共1995兲. 136 L. G. Sotelino, M. Saerens, and H. Bersini, Neural Netw. 7, 767 共1994兲. 137 G.-Z. Sun, H.-H. Chen, and Y.-C. Le, Proceedings International Joint Conference on Neural Networks 共IJCNN 91兲, 1991 共unpublished兲, Vol. 2, pp. 13–18. 138 C. Igel, W. Erlhagen, and D. Jancke, Neurocomputing 36, 225 共2001兲. 139 D. O. Hebb, The Organization of Behavior 共Wiley, New York, NY, 1949兲. 140 M. Breakspear, J. A. Roberts, J. R. Terry, S. Rodrigues, N. Mahant, and P. A. Robinson, Cereb. Cortex 16, 1296 共2006兲. 141 Z. P. Kilpatrick, S. E. Folias, and P. C. Bressloff, SIAM J. Appl. Dyn. Syst. 7, 161 共2008兲. 142 V. K. Jirsa and J. A. S. Kelso, Phys. Rev. E 62, 8462 共2000兲. 143 C. J. Rennie, P. A. Robinson, and J. J. Wright, Biol. Cybern. 86, 457 共2002兲. 144 D. Lehmann, Electroencephalogr. Clin. Neurophysiol. 31, 439 共1971兲. 145 D. Lehmann, H. Ozaki, and I. Pal, Electroencephalogr. Clin. Neurophysiol. 67, 271 共1987兲. 146 R. Friedrich, A. Fuchs, and H. Haken, in Rhythms in Physiological Systems, edited by H. Haken and H. P. Koepchen 共Springer, Berlin, 1991兲, pp. 315–338. 147 C. Allefeld and J. Kurths, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 417 共2004兲. 148 M. Paluš, I. Dvořák, and I. David, Physica A 185, 433 共1992兲. 149 P. beim Graben, S. Frisch, A. Fink, D. Saddy, and J. Kurths, Phys. Rev. E 72, 051916 共2005兲. 150 V. K. Jirsa, R. Friedrich, and H. Haken, Physica D 89, 100 共1995兲. 151 H. Obrig and A. Villringer, J. Cereb. Blood Flow Metab. 23, 1 共2003兲. 152 K. J. Friston, P. Fletcher, O. Josephs, A. Holmes, M. D. Rugg, and R. Turner, Neuroimage 7, 30 共1998兲. 153 O. David and K. J. Friston, Neuroimage 20, 1743 共2003兲. 154 O. David, S. J. Kiebel, L. Harrison, J. Mattout, J. Kilner, and K. J. Friston, Neuroimage 30, 1255 共2006兲. 155 A. Galka, O. Yamashita, T. Ozaki, R. Biscay, and P. Valdeś-Sosa, Neuroimage 23, 435 共2004兲. 156 A. Galka, T. Ozaki, H. Muhle, U. Stephani, and M. Siniatchkin, Cognit. Neurodynamics 2, 101 共2008兲.
CHAOS 19, 015104 共2009兲
Control of transient synchronization with external stimuli Marzena Ciszak,1 Alberto Montina,2 and F. Tito Arecchi1,2 1
C.N.R.-Istituto Nazionale di Ottica Applicata, Largo E. Fermi 6, 50125 Firenze, Italy Dipartimento di Fisica, Universitá di Firenze, 50019 Sesto Fiorentino, Italy
2
共Received 20 January 2009; accepted 21 January 2009; published online 31 March 2009兲 A network of coupled chaotic oscillators can switch spontaneously to a state of collective synchronization at some critical coupling strength. We show that for a locally coupled network of units with coexisting quiescence and chaotic spiking states, set slightly below the critical coupling value, the collective excitable or bistable states of synchronization arise in response to a stimulus applied to a single node. We provide an explanation of this behavior and show that it is due to a combination of the dynamical properties of a single node and the coupling topology. By the use of entropy as a collective indicator, we present a new method for controlling the transient synchronization. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3080195兴 Current views see the brain as a self-organizing dynamical system operating close to instability points which allow flexible switching between different states.1 It has been experimentally demonstrated that an external stimulus can induce oscillatory and synchronous neuronal activity with the predominant frequency in the 40 Hz range (gamma wave activity). In sensory systems, phase synchronization of oscillation links neurons functionally together to groups that respond to identical stimulus features.2 Through synchronization, the neurons which perceive the stimulus characteristics amplify their activity and contribute to a temporal increase in a local field potential (firing rates)3 which gives rise to the attention and conscious perception processes. The principal properties of such a collective oscillatory activity is the simultaneity and very short onset latency.4 Exploring a wider frequency band reveals an intrinsic dynamics of long lasting activity patterns of spatial extent, initiated by a sensory event. Essential results of numerous studies show that brain activity undergoes a sudden transition from one stable state to another at some critical stimulus parameter. This phenomenon suggests that the universal property of complex physical systems operating in a metastable dynamics can be attributed to the brain.5 The global neuronal workspace hypothesis6 considers a phase transition in the metastable dynamics of the brain. This implies that only one single global representation can be sustained at each time. Under this hypothesis the occurrence of conscious states would be a sudden transition in brain space activity which has been linked to bifurcations and phase transitions.7 I. INTRODUCTION
Synchronization processes were intensively studied by Wiener8 who first argued that frequency adjustment was a universal mechanism for self-organization operating everywhere in nature. Later studies revealed that many processes—specially the self-organized ones—work, thanks to the mutual cooperation of many constituents. In biology, at the biochemical level, such cooperation is a way to accomplish the sophisticated tasks in living organisms. Synchronization is of great importance not only in biology9 but also in 1054-1500/2009/19共1兲/015104/5/$25.00
physics,1 chemistry,10 neuroscience,11 and medicine.12,13 The problem of controlled synchronization processes has attracted great attention recently due to the new discoveries of a possible role of synchronization in conscious perception processes.4 Numerous physical processes are caused by pulsatile stimuli and are transient,14 e.g., transient short-term brain responses evoked by sensory stimuli play a key role in the study of cerebral information processing. In particular, various experiments have confirmed that transient synchronization is used by neural systems for encoding the olfactory stimuli15 and for spatiotemporal integration in visual system.16 In view of the experimental observations, in this paper we present possible mechanism for the control of stimulusdependent transient synchronization in an array of oscillators in the presence of internal noise 共chaos兲. The synchronization, as well as the global complex dynamics in neural networks, has been widely studied, e.g., see Ref. 17 for review. However, there are no reports concerning the transient synchronization mechanism and its dependence on sensory stimuli. In view of the actual debate concerning the brain functionality, consideration of how sensory stimuli induce synchronization is of crucial importance. Our model is made of interacting nodes, each one exhibiting chaotic spiking dynamics. This dynamics can be produced by single neurons or by columns of neurons, each of them being described by a simple equation like that of FitzHugh–Nagumo 共FHN兲. The use of a column as a node is due to a recent results on chaotic spiking dynamics which can arise through an interaction of simple FHN units.18 Moreover, it has been proposed in Ref. 19 that the network of coupled FHN neurons can lead to an appearance of heteroclinic connections between saddle regions 共fixed points or limit cycles兲, giving a network the possibility for efficient encoding of inputs. Our model provides a novel dynamical mechanism of transient synchronization. Application of an external stimulus switches the system to a metastable synchronization state which lasts for a few interspike intervals 共ISIs兲. Such a transient synchronization looks as a collective excitability inter-
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val in an order parameter. As order parameter we introduce entropy, which describes the degree of order between interacting dynamical systems. If the spike timings are correlated, then the entropy reaches a low value, otherwise it remains large. We show that, depending on the values of the coupling strength between neurons and on the amplitude of the external stimulus, the overall system exhibits entropy oscillations as well as bistable behavior. We demonstrate the existence of an excitability threshold in the system since at a fixed coupling the transient synchronization appears at a critical value of the perturbation amplitude. After the transient synchronization interval during which the system is insensitive to another stimulus 共refractory period兲, the dynamics switches again to a disordered state. We explain this phenomenon in terms of the internal dynamical properties of the coupled nodes. An external stimulus applied at one site can change the dynamics of that node; that is, it can either be in chaotic autonomous spiking regime or in a steady fixed point state. As the input node is “switched off” into the steady state for a short time or permanently, the global collective properties of the array change. II. MODEL
The model consists of N mutually coupled units xi, x˙ i = F共xi兲 + P共t兲 + ⑀共xi−1 + xi+1 − 2xi兲.
共1兲
The coupling consists of mutual excitatory and inhibitory interactions. Since the neural system is an open system 共with many inputs兲, we consider no-flux boundary conditions; thus the last term in the equation reduces to ⑀共xi−1 − xi兲 and ⑀共xi+1 − xi兲 at i = N and i = 1, respectively. P共t兲 is an external stimulus, being nonzero only at one site. The nodes xi are bistable, that is, either in chaotic spiking or a fixed point. The effect of having bistable nodes in an array is such that in the case of large coupling strengths all sites will finally settle in a fixed point state. Our model however operates far away from such a high value of the coupling strength. We fix the network topology and concentrate on the dependencies of the synchronization on the internal dynamics of the perturbed site. The nodes x˙ = F共x兲 describe the dynamics of single neurons or a column of neurons. In our paper we concentrate on irregular spiking produced by a column of neurons for which global dynamics can be represented by a set of equations exhibiting irregular spiking 共see, e.g., Ref. 20兲, x˙i1 = k0xi1共xi2 − 1 − k1 sin2 xi6兲, x˙i2 = − ␥1xi2 − 2k0xi1xi2 + gxi3 + xi4 + p, x˙i3 = − ␥1xi3 + gxi2 + xi5 + p, x˙i4 = − ␥2xi4 + zxi2 + gxi5 + zp, x˙i5 = − ␥2xi5 + zxi3 + gxi4 + zp, i+1 i x˙i6 = − 兵xi6 − b0 + r关f共xi1兲 + ⑀共xi−1 1 + x1 − 2 共t兲兲兴其,
共2兲
where f共xi1兲 = xi1 / 共1 + ␣xi1兲 and i共t兲 is a variable obeying the filter equation ˙ i = −d共i − xi1兲 with d = 10−3. The index i denotes the ith site position for i = 1 , . . . , M. The values of parameters are k0 = 28.5714, k1 = 4.5556, ␥1 = 10.0643, ␥2 = 1.0643, g = 0.05, p = 0.016, z = 10,  = 0.4286, ␣ = 32.8767, r = 160, and b0 = 0.1032. Chaos in these equations is due to the homoclinic return to a saddle focus, thus it implies a high sensitivity to an external perturbation in the neighborhood of the saddle. There is also a heteroclinic connection between a saddle focus and a fixed point which gives a contribution to a coexistence of two states: chaotic spiking and quiescence 共bistability兲. At each pseudoperiod or ISI, equations yield the alternation of a regular large spike and a small chaotic background. The chaotic background is the sensitive region where the activation from the neighbors occurs, while the spike provides a suitable signal to activate the coupling. Coupling of N HC 共homoclinic chaos兲 equations corresponds to a network containing heteroclinic connections between N saddle focus points. Despite the oscillating regular or irregular regimes, also bistability is a common feature found experimentally in neural networks.21 We show that introducing the bistable features into a network, i.e., coexistence of a saddle focus with a fixed point in a system, adds a new dynamical characteristics to a network and allows the efficient control of synchronization. The degree of order of the system is measured by the entropy S of the generation times Tg. The generation time Tg is defined as the shortest time difference between spike occurrence at neighboring sites starting from the first. Then the entropy S is, S共t兲 = −兺Tg p共Tg , t兲ln p共Tg , t兲. The probability function p共Tg , t兲 is a discrete and normalized probability distribution of a continuous variable Tg evaluated at time t and is defined as the ratio of the number of elements with given Tg to the total number of elements. We choose entropy to characterize synchronization because, at variance with other statistical quantities, it allows us to measure also nonisochronous coherence in a precise way. The synchronization transition for the systems under consideration is sharp and occurs spontaneously at the critical value ⑀c of the coupling strength.22 As the coupling strength is set slightly below ⑀c, a coupled array displays low intersite correlation. An external stimulus of amplitude A in the form of the modulation of some system parameter b0, e.g., b0 → b0⬘ = b0 + A for a time ⌬t, shorter than, or equal to, 具ISI典 共the average ISI兲 is applied at the first site i = 1. Depending on the stimulus amplitudes, the array responds in various manners. After the stimulus application, it can either not respond at all, remaining in the uncorrelated 共high entropy兲 state, or it can switch to the synchronized state remaining in it permanently. Such a behavior indicates the existence of a bistability. Moreover, for a limited range of the stimulus amplitudes, it can synchronize for a short time and then come back again to the uncorrelated state. This behavior, shown in Fig. 1, is reminiscent of excitable dynamics. The collective dynamical regimes mentioned above are shown in the upper panel of Fig. 2, where by changing the coupling strength ⑀ and with fixed stimuli features, we report the fixed point 关Fig. 2共a兲兴, excitable 关Fig. 2共b兲兴, oscillatory 关Fig. 2共c兲兴, and bistable 关Fig. 2共d兲兴 regimes. The use of en-
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FIG. 1. 共Color online兲 Time evolution of entropy S for an array of N = 30 coupled HC systems perturbed at the first site at time t = 40 ISI by a pulse of amplitudes of 共a兲 A = 0.09 and 共b兲 A = 0.104. In both cases the pulse duration is ⌬t = 具ISI典 and the coupling strength ⑀ = 0.104. 共c兲 Histograms of generation times during time evolution. At time t = 40 the external stimulus is applied which causes the sharpening of the generation times distribution 共with a shift to the right of their maxima兲. This sharpening corresponds to the temporal lowering of entropy.
tropy as the indicator of the collective properties of the array establishes an analogy between the collective response of an array and a dynamical system. Excitable and oscillatory responses follow cycles in a phase space of the system variables. In our case we have only one variable, but we can reconstruct a phase space by an embedding method23 共lower panel兲. A standard definition of excitable system says that the form of the system response, beyond a particular excitability threshold, is determined by the system parameters and not by the external signal features. There is however a crucial difference between excitability in dynamical systems and col-
lective excitable behavior of an array. In the latter case, the excitability exists for some range of the stimulus amplitude, whereas the classical excitability refers to the fixed response for all stimuli amplitudes. In Figs. 3共a兲 and 3共b兲 we plot the average entropy values in the parameter space of the coupling strength ⑀ and the stimulus amplitude A. The average entropy is calculated from the time series far beyond the stimulus was applied. All collective dynamical regimes are seen: excitability, oscillations, and bistability. An interesting phenomenon can be noticed from the figures: For different initial conditions, different configurations of the excitability regions are formed. It is a kind of a memory of the initial state.
insensitive
excitable
oscillatory
bistable
1.14
A
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1.02
FIG. 2. Upper panel: entropy S vs time for N = 30 weakly coupled HC units with different coupling strength values: 共a兲 ⑀ = 0.1015, 共b兲 ⑀ = 0.1045, 共c兲 ⑀ = 0.105 795, and 共d兲 ⑀ = 0.106, perturbed with a stimulus of amplitude A = 1.04 and duration ⌬t = 具ISI典 applied at time t = 40. Lower panels 共e兲–共h兲 show phase spaces for the embedded variable S corresponding to figures 共a兲–共d兲, respectively.
(a)
0.1025 0.105 0.1075 0.11 ε
(b)
0.1025 0.105 0.1075 0.11 ε
FIG. 3. 共Color online兲 The stimulus of duration ⌬t = 具ISI典 and amplitude A applied to the first of N = 30 weakly coupled systems induces excitable, bistable, and oscillatory responses. 共a兲 and 共b兲 are produced for two different sets of the initial conditions.
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0.1
A
0.08 0.06 0.04 0.02 0
FIG. 4. 共a兲 Spontaneous synchronization transitions in N = 40 interacting systems for different boundary conditions at site i = 1 determined by the following coupling forms: ⑀共xi+1兲 共solid line兲, ⑀共xi+1 − 2xi兲 共dashed line兲, and ⑀共xi+1 − 3xi兲 共dotted line兲. 共b兲 Raster plots for N = 20 coupled HC systems with ⑀ = 0.136 far beyond the initial transient. Various final synchronization states arise from different sets of initial conditions.
III. DYNAMICAL MECHANISM OF COLLECTIVE BEHAVIORS
The mechanism underlying collective excitability and bistability is related to the internal characteristics of the single nodes. As the stimulus is applied to one site, for a particular value of the stimulus amplitude, the node collapses to a fixed point. Before the stimulus is applied, the coupling at i = 1 is ⑀共xi+1 − xi兲 for the first site, whereas for the second site it is ⑀共xi+1 + xi−1 − 2xi兲. Now, if the dynamics of the first site is switched off, then the new coupling for the second site is ⑀共xi+1 − 2xi兲, that is different from the previous one. In order to elaborate the above observations, let us look closer on the synchronization features in the case of various types of boundary conditions. For the nearest neighbor and no-flux boundary conditions, we notice that as the nodes at the boundaries are more strongly coupled to the array, the spontaneous transition to synchronization occurs for smaller coupling strengths 共decreasing of ⑀c兲. In Fig. 4共a兲 the synchronization transition for three different cases are shown. Entropy is calculated from the distribution of the generation times Tg for the following coupling configurations at site i = 1: ⑀共xi+1兲, ⑀共xi+1 − 2xi兲, and ⑀共xi+1 − 3xi兲. Thus, besides the single node dynamics, also the boundary conditions are responsible for setting the collective synchronization. The typical patterns of spontaneous synchronization which arise above ⑀c are shown in Fig. 4共b兲. Synchronization patterns are formed by various clusters of oscillators enslaved in a unison motion which are determined by the initial conditions and by the leading sites at boundaries. The number of such states increases linearly with the number of sites N in an array. The synchronization states can exist because above ⑀c, nodes in the array enter a regular oscillatory regime. As was demonstrated in Ref. 22, the value of ⑀c remains the same for any system size N. However, as shown in Fig. 4共a兲, the synchronization transition occurs for a smaller value of the coupling strength as the ending sites are tied more strongly to the array. Thus, if the temporal change in boundary conditions is such that the array synchronizes for ⑀c⬘ ⬍ ⑀c, then the excitability and bistability can be observed. When the spiking stops at the first site, the array’s boundary conditions change and the value of ⑀c for the synchronization decreases until
2000
4000 time
6000
8000
FIG. 5. Response characteristics for a single HC system to external stimuli of various amplitudes A and fixed duration ⌬t = 具ISI典. Dots represent spiking events. External stimulus is applied at time t = 2000. The dashed line marks a threshold for A at which the system is pushed to a steady fixed point state 共bistability兲.
the first site recovers its chaotic dynamics again. The characterization of the responses of the single node to external stimuli is presented in Fig. 5. A single system equation is integrated in time starting from different initial conditions. At time t = 2000, an external stimulus is applied, causing the delayed reappearance of spiking. As the amplitude of the stimulus increases, the delay time becomes longer and finally becomes infinite 共spiking does not reappear兲. The critical value of A at which the spiking is suppressed marks the bistability regime for an array of coupled systems. On the other side, values of A, which cause the delayed spiking 共temporal switching off of the system dynamics兲, allow excitable regimes in an array. Finally, when no delay in spiking is present, the array is insensitive to an external stimulus. IV. CONCLUSIONS
In conclusion, we have shown that below the critical point of the coupling strength at which synchronization arises spontaneously, a coupled array responds collectively to a stimulus. The collective behavior arises even though the stimulus is applied at a single site. We have shown that in the response to a single perturbation, an array exhibits bistable-, oscillatory- and excitablelike behavior. That is, the array switches from a disordered state into the perfectly synchronized one permanently or for a short time. Such a transition appears at a critical value Ac of the perturbation amplitude. In the excitablelike regime, after a transient synchronization, the system goes back to its previous disordered state. As we decrease the amplitude A, the system does not respond at all to the external stimuli, thus there exists an excitability threshold. We have here introduced the sufficient general conditions for the control of collective behavior which may be summarized as follows: population of coupled bistable units (coexistence of fixed point and chaotic attractor) satisfying the condition for a sharp transition to synchronization22 with no-flux boundary conditions. The collective phenomena appear when an external stimulus applied at one site changes the dynamics of that bistable node. As the dynamics of that node is temporally or permanently switched off into the steady state, the global collective properties of the array change. This is due to the boundary effects on the critical value of the coupling strength needed to induce array synchronization. Thus the collective behavior of an array can be
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controlled by application of an external stimulus at one site inducing transient or permanent synchronization. The behavior here discussed persists for the systems which have a bistable dynamics with the coexistence of autonomous spiking 共not necessary chaotic兲 and fixed point state.24,25 Since in animals excitable,26 as well as bistable,21 cells are well known to constitute the brain columns and networks, the model presented here may play an important role during the interactions of the neural assemblies in response to sensory stimuli. Moreover the mechanism presented here may be applied in other areas of science to control systems, e.g., temporal or permanent global switching on-off of large networks by a one-point input. In the absence of definite experimental evidence on the role of collective synchronization as outlined here, in brain processes, we ask the question: Where in the cognitive ladder starting with sensorial detectors and ending in a decision 共motor or linguistic兲27 collective synchronization might be crucial? The first step of the ladder, called “encoding” is the buildup of sequences of spikes in coupled neuron arrays that code the input information. Two recent strategies have been discussed at length, namely, “chaotic itinerancy” by Tsuda et al.28 and “stable heteroclinic chains” by Rabinovich et al.19 The next crucial question is how the brain utilizes the encoded information in selecting specific motor decisions. We draw a tentative boundary between two relevant time scales, a short one within which decisions are almost automatic and based on probabilistic 共Bayes兲 inference29 and a long one, where there is enough time to introduce top-down perturbations from memorized learning to the bottom-up signals resulting from encoding.30 A comparison of the cognitive role of different time windows has been done by Poppel31 and we refer to those papers for further details. It is sound to hypothesize that decisions are based on the recruitment of large brain areas acting coherently for a sizable time span. In view of this, we are exploring how the collective synchronization outlined here might be a workable strategy to interface encoding to decisions. ACKNOWLEDGMENTS
M.C. acknowledges a Marie Curie Intra-European Program within the 6th European Community Framework Programme. A.M. acknowledges Ente Cassa di Risparmio di Firenze under the Project Dinamiche cerebrali caotiche.
H. Haken, Advanced Synergetics 共Springer, Berlin, 1983兲. M. G. Gray and W. Singer, Proc. Natl. Acad. Sci. U.S.A. 86, 1698 共1989兲; R. Eckhorn, R. Bauer, W. Jordan, M. Broisch, M. Kruse, M. Munk, and H. J. Reitboeck, Biol. Cybern. 60, 121 共1988兲. 3 F. Varela, J. P. Lachaux, E. Rodriguez, and J. Martineri, Nat. Rev. Neurosci. 2, 229 共2001兲. 4 P. R. Roelfsema, A. K. Engel, P. Koening, and W. Singer, Nature 共London兲 385, 157 共1997兲. 5 S. Bressler and J. A. S. Kelso, Trends Cogn. Sci. 5, 26 共2001兲; W. J. Freeman and M. D. Holmes, Neural Networks 18, 497 共2005兲. 6 J.-P. Changeux and S. Dehaene, Cognition 33, 63 共1989兲. 7 C. Sergent and S. Dehaene, Psychol. Sci. 15, 720 共2004兲. 8 N. Wiener, Cybernetics, 2nd ed. 共MIT, Cambridge, 1961兲. 9 A. T. Winfree, The Geometry of Biological Time 共Springer, Berlin, 1980兲. 10 Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence 共Springer, Berlin, 1984兲. 11 H. Steriade, E. G. Jones, and R. R. Llinas, Thalamic Oscillations and Signaling 共Wiley, New York, 1990兲. 12 P. A. Tass, Phase Resetting in Medicine and Biology. Stochastic Modelling and Data Analysis 共Springer, Berlin, 1999兲; P. A. Tass, Europhys. Lett. 57, 164 共2002兲. 13 K. H. Chiappa, Evoked Potentials in Clinical Medicine 共Raven, New York, 1983兲. 14 P. A. Tass, Europhys. Lett. 59, 199 共2002兲. 15 R. W. Friedrich and G. Laurent, Science 291, 889 共2001兲. 16 J. J. Hopfield and C. D. Brody, Proc. Natl. Acad. Sci. U.S.A. 98, 1282 共2001兲. 17 M. V. Ivanchenko, G. V. Osipov, V. D. Shalfeev, and J. Kurths, Phys. Rev. Lett. 93, 134101 共2004兲; D. Battaglia, N. Brunel, and D. Hansel, ibid. 99, 238106 共2007兲. 18 T. Yanagita, T. Ichinomiya, and Y. Oyama, Phys. Rev. E 72, 056218 共2005兲. 19 M. Rabinovich, A. Volkovskii, P. Lecanda, R. Huerta, H. D. I. Abarbanel, and G. Laurent, Phys. Rev. Lett. 87, 068102 共2001兲. 20 A. N. Pisarchik, R. Meucci, and F. T. Arecchi, Eur. Phys. J. D 13, 385 共2001兲; F. T. Arecchi, Physica A 338, 218 共2004兲. 21 C. Monteiro, D. Lima, and V. Galhardo, Neurosci. Lett. 398, 258 共2006兲. 22 M. Ciszak, A. Montina, and F. T. Arecchi, Phys. Rev. E 78, 016202 共2008兲. 23 N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, Phys. Rev. Lett. 45, 712 共1980兲. 24 H. R. Wilson, Spikes, Decisions, and Actions: The Dynamical Foundations of Neuroscience 共Oxford University Press, New York, 1999兲. 25 E. M. Izhikevich, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 1171 共2000兲. 26 M. A. Segraves and M. E. Goldberg, J. Neurophysiol. 58, 1387 共1987兲. 27 W. Freeman, How brains make up their minds 共Columbia University Press, New York, 2000兲. 28 I. Tsuda, Chaos 19, 015113 共2009兲. 29 J. M. Beck, W. J. Ma, R. Kiani, T. Hanks, A. K. Churchland, J. Roitman, M. N. Shadlen, P. E. Latham, and A. Pouget, Neuron 60, 1142 共2008兲. 30 S. Grossberg, Behav. Brain Sci. 21, 473 共1998兲. 31 E. Poppel, Trends Cogn. Sci. 1, 56 共1997兲; Acta Neurobiol. Exp. 共Warsz兲 64, 295 共2004兲. 1 2
CHAOS 19, 015105 共2009兲
A combined method to estimate parameters of neuron from a heavily noise-corrupted time series of active potential Bin Deng, Jiang Wang,a兲 and Yenqiu Che School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People’s Republic of China
共Received 10 December 2008; accepted 11 February 2009; published online 31 March 2009兲 A method that combines the means of unscented Kalman filter 共UKF兲 with the technique of synchronization-based parameter estimation is introduced for estimating unknown parameters of neuron when only a heavily noise-corrupted time series of active potential is given. Compared with other synchronization-based methods, this approach uses the state variables estimated by UKF instead of the measured data to drive the auxiliary system. The synchronization-based approach supplies a systematic and analytical procedure for estimating parameters from time series; however, it is only robust against weak noise of measurement, so the UKF is employed to estimate state variables which are used by the synchronization-based method to estimate all unknown parameters of neuron model. It is found out that the estimation accuracy of this combined method is much higher than only using UKF or synchronization-based method when the data of measurement were heavily noise corrupted. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3092907兴 For a quantitative understanding of time varying phenomena from nature and technology, it is often desired and quite informative to fit coefficients of nonlinear models to time series of observations. These models may contain quantities that cannot be measured directly. Instead, only a rather small portion of noise-corrupted observations is available. Reliable reconstruction of all model components and parameters using such “real-world data” is one of the most challenging research topics in nonlinear data analysis. Several algorithms have been proposed and successfully applied for considerably restricted model classes for the system and observation process. The restrictions demand only weak nonlinearities or small amounts of noise, for example. Unfortunately, most real-world systems do not possess these properties. The general approach is to treat the problem of estimating parametrized models from incomplete time series amounts in a state space description. For linear state space models with Gaussian process and observation noise, the well-known Kalman filter is the method of choice for the consistent estimation of the indirectly observed or unobserved states. However for the estimation of parameters, even for linear models this inevitably leads to nonlinear state space equations, which prevent the direct use of the Kalman filter. Very recently, Julier and Uhlmann developed a substantial extension of the Kalman filter for nonlinear models, the unscented Kalman filter (UKF). Compared with the widely used extended Kalman filter, nonlinearities are handled in a more superior way in the sense that a better quality of estimates is achieved with less computational expense. In this paper, the UKF and the well-known synchronization-based apa兲
Author to whom correspondence should be addressed. Electronic addresses: [email protected] and [email protected].
1054-1500/2009/19共1兲/015105/9/$25.00
proach are both employed to estimate all the unknown parameters of neuron model from heavily noisecorrupted time series of active potential. I. INTRODUCTION
In neuroscience research, certain conductance-based neuron models, such as the well-known Hodgki–Huxley 共HH兲, Morris–Lecar, FitzHugh–Nagumo 共FHN兲, and Hidmarsh–Rose 共HR兲 models,1–4 are highly nonlinear and involve many electrophysiological variables and parameters. However, only some of the variables, e.g., the membrane potential, are easily measured by voltage-clamp experiments, while other parameters are difficult to experimentally determine. Moreover, it is known that the effect of noise in the neural research is not avoidable. There are various sources of fluctuations which include the probabilistic open and closure of ion channels, the release of neurotransmitters from chemical synapses, and the random synaptic inputs from other neurons which impact on a neuron. Further more, the signal of active potential is very weak, so the data measured in neural experiment may be heavily noise corrupted. Therefore, in order to explore more precisely the dynamics of neuronal models, it is crucial to estimate the parameters accurately from heavily noise-corrupted time series of active potential. To study the detailed structure of the equations of the underlying dynamical system that govern its temporal evolution is one of the objectives of time series analysis. This includes the number of independent variables, the form of the flow function, the nonlinearities in them, and parameters of the system.5 In recent years, some dynamic methods6–11 of parameter estimation based on the synchronization of chaotic system and adaptive control have been proposed to solve the subject of estimating partial or all unknown model param-
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eters of nonlinear dynamical systems from observed time series when the number of independent variables and the structure of underlying dynamical equation for experimental systems are in advance available. The restrictions of those methods mentioned above demand small amount of noise, but most real-world systems do not possess these properties. Especially in the experimental research on neurons, the heavy noises of process and observation are not avoidable. A general solution to this problem, in the linear case, is provided by the Kalman filter 共KF兲.12 In the case of nonlinear systems, the KF is inadequate. The way to overcome such a shortcoming is to linearize the system equation based on a Taylor-series expansion at each iteration and simply run the standard KF. This procedure is known as the extended KF 共EKF兲.13 Very recently, Julier and Uhlmann developed a substantial extension of the KF for nonlinear models, the unscented KF 共UKF兲 共see Ref. 14 and references therein兲. Compared with EKF, nonlinearities are handled in a more superior way in the sense that a better quality of estimates is achieved with less computational expense, so it has already attracted attention in the context of nonlinear dynamics.15–18 In this paper the state variables of neuron model are estimated from noise-corrupted data by employing the UKF, and then they are used by the synchronization-based method proposed in Refs. 10 and 11 to estimate the unknown parameters of neuron model so that the high accuracy of parameter estimation can be guaranteed. The rest of this paper is organized as follows. The UKF and synchronization-based method of parameter estimation are briefly reviewed separately and the combined method is given in Sec. II The combined method is employed to estimate parameters of the FHN and HR model from heavily noise-corrupted time series of active potential in Sec. III Finally, Sec. IV discusses the main points of the paper. II. STATE AND PARAMETER ESTIMATION A. The UKF
The state estimation problem for the system x共k兲 = Ax共k − 1兲 + Bu共k − 1兲 + w共k − 1兲, 共1兲 y共k兲 = Hx共k兲 + r共k兲, where A 苸 Rn⫻n, B 苸 Rn⫻p, and H 苸 Rm⫻n are constant matrices, can be described as follows. Suppose that only known data are the initial conditions x共0兲 苸 Rn, the measurement y共k兲 苸 Rm, and the control inputs u共k兲 苸 R p. Process noise w共k兲 苸 Rn and measurement noise r共k兲 苸 Rm are assumed white Gaussian and are mutually independent of covariance matrices Q and R, respectively. If there are no exogenous signals, then simply let u共k兲 = 0 ∀ k. It is desired to obtain estimation for the state vector x共k兲, k = 1 , 2 , . . .. The standard solution to this problem is the classical KF 共Ref. 12兲 xˆ 共k兩k − 1兲 = Axˆ 共k − 1兩k − 1兲 + Bu共k − 1兲, yˆ 共k兩k − 1兲 = Hxˆ 共k兩k − 1兲,
P共k兩k − 1兲 = AP共k − 1兩k − 1兲AT + Q共k兲, Pyy共k兩k − 1兲 = HP共k兩k − 1兲HT + R共k兲,
共2兲
Pxy共k兩k − 1兲 = P共k兩k − 1兲HT , K共k兲 = Pxy共k兩k − 1兲P−1 yy 共k兩k − 1兲, xˆ 共k兩k兲 = xˆ 共k兩k − 1兲 + K共k兲关y共k兲 − Hxˆ 共k兩k − 1兲兴, P共k兩k兲 = P共k兩k − 1兲 − K共k兲Pyy共k兩k − 1兲KT共k兲,
共3兲
where K is the Kalman gain matrix, the carets indicate the mean of the corresponding density function, and P is the covariance matrix of the vector of estimation errors. Moreover, the notation z共k 兩 k − 1兲 indicates the value of the quantity z at time k using information taken up to time k − 1. Likewise, z共k 兩 k兲 indicates the value of z computed at time k using the information available up to and including time k. The first equation in Eq. 共2兲 shows how the state at time k − 1 is propagated to time k and the second equation shows how the propagated state maps onto the output. Similarly, the third equation shows how the estimation error vector covariance matrix propagates from time k − 1 to time k. A key remark here is to notice that such propagations are made using the linear dynamical model available for the system represented by matrices 共A , B , H兲. The two equations in Eq. 共3兲 show how the current values, that is, at time k, of the state vector and its covariance matrix can be updated after new information becomes available. The equations that describe the propagation of P, Pxy, and Pyy, together with the linear state propagation equations, as stated in Eq. 共2兲, are valid only for linear system. It is interesting to notice that the other equations remain valid even in nonlinear case. The traditional approach to propagate the covariance matrices in nonlinear case is to linearize Eq. 共1兲 at each step and then apply the KF equations. This result is the well-known EKF.19 The problem about the EKF is that if nonlinearities cannot be approximated well by linearized terms, most EKF estimates are biased and inconsistent. A novel procedure for dealing with estimation in strong nonlinear state space models has been proposed recently by Julier and Uhlman.14 This procedure belongs to the class of statistical linearization schemes20 in which densities are truncated instead of the models f and h which are the nonlinear function of the state and the observation function, respectively. A sample set with same mean and covariance is generated and propagated through the full 共not approximated as with EKF兲 state space model. This sample set 兵i其2n 0 is given by the so-called sigma points X0共t − ⌬t兩t − ⌬t兲 = xˆ 共t − ⌬t兩t − ⌬t兲, Xi共t − ⌬t兩t − ⌬t兲 = xˆ 共t − ⌬t兩t − ⌬t兲 + 关冑共n + 兲P共t − ⌬t兩t − ⌬t兲兴i ,
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Estimate parameters
Xi+n共t − ⌬t兩t − ⌬t兲 = xˆ 共t − ⌬t兩t − ⌬t兲
deal with nonlinear dynamical evolution, it is easy to consider a parameter as a “virtual” state which has to be estimated. Often it is expected that the system parameters do not vary or, if they do, the variation is much slower than that of the system state.
− 关冑共n + 兲P共t − ⌬t兩t − ⌬t兲兴i with associated weights given by W0 =
, n+
Wi =
1 , 2共n + 兲
B. Synchronization-based estimation method
where i = 1 , 2 . . . , 2n, n is the dimension of Eq. 共3兲 and 关冑共·兲兴i is either the ith row or column of the matrix square root. Then the UKF equations can be expressed as21 Xi共t兩t − ⌬t兲 = f共Xi共t − ⌬t兩t − ⌬t兲,,u兲, Yi共t兩t − ⌬t兲 = h共Xi共t兩t − ⌬t兲兲. 2n The resulting sample sets 兵i共t 兩 t − ⌬t兲其2n 0 and 兵␥i共t 兩 t − ⌬t兲其0 may represent a density with higher order moments but due to linearization only mean 2n
xˆ 共t兩t − ⌬t兲 = 兺 WiXi共t兩t − ⌬t兲,
Assuming that the number of independent variables and the structure of underlying dynamical equations for a chaotic system are available, in Refs. 10 and 11 the method of parameter estimation based on adaptive-feedback control is proposed. Considering the noise free dynamic system 共5兲
x˙ = F共x,,t兲, where x = 共x1 , x2 , . . . , xn兲 苸 R is the state vector, n
F共x,,t兲 = 共F1共x,,t兲,F2共x,,t兲, . . . Fn共x,,t兲兲 is a nonlinear vector function with
i=0
m
Fi共x,,t兲 = gi共x,t兲 + 兺 ij f ij共x,t兲,
2n
yˆ 共t兩t − ⌬t兲 = 兺 WiYi共t兩t − ⌬t兲,
Here gi共x , t兲 and f ij共x , t兲 are nonlinear functions, and = ij 苸 U 傺 Rnm are unknown parameters to be estimated. Assuming the vector field satisfies the uniform Lipschitz condition, i.e., for any, 苸 U, x = 共x1 , x2 , . . . , xn兲, x0 = 共x01 , x02 , . . . , x0n兲 there exists a constant l ⬎ 0 so that
i=0
and covariance 2n
P共t兩t − ⌬t兲 = 兺 Wi关Xi共t兩t − ⌬t兲 − xˆ i共t兩t − ⌬t兲兴 i=0
兩Fi共x,,t兲 − Fi共x0,,t兲兩 ⱕ l max 兩x j − x0j 兩,
⫻关Xi共t兩t − ⌬t兲 − xˆ i共t兩t − ⌬t兲兴T ,
i = 1,2, . . . ,n.
1ⱕjⱕn
共6兲
2n
Pyy共t兩t − ⌬t兲 = 兺 Wi关Yi共t兩t − ⌬t兲 − yˆ i共t兩t − ⌬t兲兴 i=0
⫻关Yi共t兩t − ⌬t兲 − yˆ i共t兩t − ⌬t兲兴T , 2n
Pxy共t兩t − ⌬t兲 = 兺 Wi关Xi共t兩t − ⌬t兲 − xˆ i共t兩t − ⌬t兲兴 i=0
⫻关Yi共t兩t − ⌬t兲 − yˆ i共t兩t − ⌬t兲兴T are used. The updating equations are K共t兲 = Pxy共t兩t − ⌬t兲P−1 yy 共t兩t − ⌬t兲, xˆ 共t兩t兲 = xˆ 共t兩t − ⌬t兲 + K共t兲关y共t兲 − h共xˆ 共t兩t − ⌬t兲兲兴,
i = 1,2, . . . ,n.
j=1
共4兲
P共t兩t兲 = P共t兩t − ⌬t兲 − K共t兲Pyy共t兩t − ⌬t兲KT共t兲. In closing this section, it is important to notice that the updating equations of the three filter algorithms are the same. On the other hand, the covariance matrix propagation is carried out using the linear model for the KF, using the Jacobian, that is, it is locally approximate only for the EKF, and using the full nonlinear model in the case of UKF. An important characteristic of the EKF and UKF algorithms is that they can be readily used to jointly estimate the system state and parameters. Once the algorithms are able to
The above condition is the so called uniform Lipschitz condition and l refers to the uniform Lipschitz constant. Note this condition is very loose. One way is to check easily that the class of Eqs. 共5兲 and 共6兲 includes almost all well-known chaotic system such as Lorenz system, Chua circuit, and neuron model such as FHN model and HR model. We assume that time series for all variables of system 共5兲, as the experimental output of the system, are available. To estimate all parameters from these time series, an auxiliary system of variables y = 共y 1 , y 2 , . . . y n兲 whose evolution equations have identical form to that of x, is introduced. However, the corresponding parameters are not some, which will be set to arbitrary initial values, say, q = qij, i = 1 , 2 , . . . , n and j = 1 , 2 , . . . , m. In contrast to the experimental system 共5兲, the auxiliary system can be controlled in practice, which is also called the receiver system is given by the following equation: 共7兲
y˙ = F共y,q,t兲 + u,
where u is a simple linear feedback control and can be described by the following equation: ui = i共y i − xi兲,
i = 1,2, . . . ,n.
共8兲
Instead of the usual linear feedback, the feedback strength = 共1 , 2 , . . . , n兲 here will be adapted duly according to the following update law:
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˙ i = − ␣ie2i ,
共9兲
i = 1,2, . . . ,n,
where e = 共e1 , e2 , . . . , en兲, ei = y i − xi i = 1 , 2 , . . . , n denoting the synchronization error of systems 共5兲 and 共7兲 and ␣i ⬎ 0, i = 1 , 2 , . . . , n are arbitrary constants. The equations governing the evolution of the parameters q are chosen as follows: q˙ij = − ijei f ij共y,t兲,
i = 1,2, . . . ,n,
j = 1,2, . . . ,m,
共10兲
where ij ⬎ 0, i = 1 , 2 , . . . , n, j = 1 , 2 , . . . , m, are arbitrary constants. Next the rigorous proof of the main results will be given. System 共4兲 can be rewritten as x˙ = F共x,,t兲,
˙ = 0.
共11兲
For the system consisting of the error equation between Eqs. 共7兲 and 共10兲 and Eqs. 共11兲 and 共6兲, which is formally called the augment system, we introduce a scalar Lyapunov function V=
1 2
冋兺 n
i=1
n
m
e2i + 兺 兺 i=1 j=1
册
n
1 1 共qij − ij兲2 + 兺 共i + L兲2 , 共12兲 ij i=1 ␣i
where L ⬎ nl. By differentiating the function V along the trajectories of the augment system, we obtain n
n
m
n
1 1 共qij − ij兲q˙ij + 兺 共i + L兲˙ i V˙ = 兺 ei共y˙ i − x˙i兲 + 兺 兺 i=1 i=1 i=1 ij i=1 ␣i n
= 兺 ei关Fi共y,q,t兲 − Fi共x,,t兲 + iei兴 i=1
n
m
n
− 兺 兺 共qij − ij兲ei f ij共y兲 − 兺 共i + L兲e2i i=1 j=1
termine any additive parameters; 共ii兲 systematic because the control technique in the form of Eqs. 共7兲–共10兲 can be applied to all chaotic systems satisfying the uniform Lipschitz condition.
C. The combined method
In order to use the synchronization-based parameter estimation proposed above, we need to know all the state variables of the system. The method does nothing to deal with the noise of measurement, so it is only robust against weak observation noise. As the UKF can deal well with heavily noise-corrupted data, a combined method that uses the state variables estimated by UKF to estimate all unknown parameters of nonlinear system will be proposed. Namely, the state variables of the master system used in Eq. 共8兲 are replaced by the state variables of system 共1兲 estimated by UKF in Eq. 共4兲, so Eq. 共8兲 can be rewritten as follows: ui = i共y i − xˆi共t兩t兲兲,
共13兲
i = 1,2, . . . ,n,
where xˆi共t 兩 t兲, i = 1 , 2 , . . . , n, is given by the second equation of the UKF updating Eq. 共4兲. The combined method can be described as follow: first, the UKF is employed to estimate all the state variables of the experimental system 共1兲 from noise-corrupted data of measurement. Then, the state variables estimated in Eq. 共4兲 are used as the driven signals to drive the receiver systems 共7兲–共10兲 to synchronize to system 共5兲. Thus, the unknown parameters may be dynamically estimated from q in the receiver system with high accuracy.
i=1
n
III. ESTIMATING PARAMETERS OF NEURON MODEL
= 兺 ei关Fi共y,q,t兲 − F共x,,t兲兴 i=1
A. FHN model
n
n
− 兺 ei关Fi共y,q,t兲 − Fi共y,,t兲兴 − L 兺
e2i
i=1
i=1
n
n
= 兺 ei关Fi共y,p,t兲 − Fi共x,,t兲兴 − L 兺 e2i i=1
i=1
n
n
ⱕ 兺 eil max 兩e j兩 − L 兺 i=1
1ⱕjⱕn
i=1
n
e2i
ⱕ 共nl − L兲 兺 e2i ⱕ 0. i=1
It is obvious that V˙ = 0 if and only if ei = 0, i = 1 , 2 , . . . , n. Therefore the set E = 兵e = 0 , − p = 0 , = 0 苸 Rn其 is the largest invariant set contained in V˙ = 0 for the augment system. Then according to the well-known invariance principle of differential equations, the solution starting from arbitrary initial values of Eqs. 共7兲–共10兲 possesses asymptotic behavior: y → x, q → as t → ⬁. Namely, the unknown parameters may be dynamically estimated from q in the receiver system. In comparison with previous methods for synchronization-based parameter estimation,6–11 the distinguished characteristic of this method is 共i兲 analytical and rigorous because it does not require one to numerically de-
As a simple but representative example of excitable systems, the FHN neuron model, was originally suggested for the description of firing behaviors of sensory neurons;3 it was also widely used in the modeling of a spiral wave in a twodimensional excitable medium. A simplified form of it which is in the bistable regime can be written as follows: x˙1 = x1共x1 − 1兲共1 − 1x1兲 − x2 + I共t兲,
x˙2 = 2x1 .
共14兲
Here x stands for the membrane voltage variable, while y is the so-called recovery variable. 1 and 2 are the parameters to be estimated and I共t兲 is the injected stimulus which is written as I共t兲 = A cos共2 ft兲. Let 1 = 10, 2 = 1, A = 0.125, and f = 0.1271 to generate a chaotic time series of active potential which will be used to estimate the parameters by the combined method proposed above. The stationary solution is the trajectory of the system in the state space as shown in Fig. 1共a兲. Assuming that we know the functional form of the model 共14兲, we can consider the unknown parameters as virtual states that have to be estimated. The joint system can be written as follows:
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FIG. 1. 共Color online兲 共a兲 Clean and 共b兲 estimated attractor for the neuron model. Clean and noisy observation 共dots兲 for this limit cycle can be seen in 共c兲. Clean and estimated states are shown in 共d兲.
x共t兲 = f共x共t − ⌬t兲, 共t − ⌬t兲兲
共⌬t ⬎ 0兲,
共t兲 = 共t − ⌬t兲,y共t兲 = h共x共t兲兲 + r共t兲.
共15兲
Here x = 共x1 , x2兲, = 共1 , 2兲 ⌬t = 0.01, r共t兲 is the measurement noise, and f denotes the integral equation for Eq. 共14兲 which is solved using the four-order Runge–Kutta method. As only the active potential can be measured, let h = 关1 0兴. The observations are generated by corrupting the active potential with white Gaussian noise whose strength is D = 0.1. Here signal-noise-ratio is about 7 dB or, equivalently, we deal with 50% measurement noise. The noisecorrupted signal and the noise free signal are plotted in Fig. 1共c兲. Under the assumption that the underlying dynamical model and the statistic of the measurement noise are known, the UKF is employed to estimate the state of the joint system 共15兲. The initial guess for the parameter is chosen to be ˆ = 关5 0兴, and the initial state estimation is set to xˆ = 关0.2 0.2兴. The differences between the estimated and the clean states are negligible small as shown in Fig. 1共d兲 if related to the difference between clean states and noisy data in Fig. 1共c兲. The estimated trajectory is plotted in Fig. 1共b兲. The estimated states are used by the combined method described in Eqs. 共4兲, 共7兲, 共9兲, 共10兲, and 共13兲 to estimate the unknown parameters of FHN model, the unknown param-
eters may be dynamically estimated from q in the receiver system as shown in Figs. 2共a兲 and 3共a兲, respectively. It is noted that the unknown parameters are the states of the joint system 共15兲, so the UKF can also be used to estimate the parameters but with low accuracy. The evolutions of the parameters estimated by UKF only are plotted in Figs. 2共b兲 and 3共b兲. Figures 2共c兲 and 3共c兲 show the convergence behavior of the estimated parameters by the synchronization-based method only from the noisy observation data directly. As shown in Figs. 2 and 3, the accuracy of the parameters estimated by the combined method is higher than only by the UKF or synchronization-based method. Moreover it almost failed in employing only the synchronization-based method to estimate the unknown parameters from the observation data which is corrupted by the noise with the strength D = 0.1. For the purpose of studying the effect of the observation noise on the accuracy in the estimation of parameters, we define the estimation accuracy of as A = 兩 − q兩 / , = 关1 2兴 q = 关q1 q2兴, i.e., the normalized absolute error of parameter estimation, and then analyze its development as a function of noise strength D. The observation noise with strength D still takes additive uniformly distributed values from −D to D. Figure 4 depicts the asymptotic value A against the noise strength D. It can be seen from Fig. 4共a兲 that the absolute error of the parameter 1 estimated by the
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FIG. 2. 共Color online兲 The parameter 1 estimated by combined method 共a兲, UKF only 共b兲, and the synchronization-based method only 共c兲 from the noise-corrupted time series.
FIG. 3. 共Color online兲 The parameter 2 estimated by combined method 共a兲, UKF only 共b兲, and the synchronization-based method only 共c兲 from the noise-corrupted time series.
B. HR model
combined method is always about 0.1%, which means the high estimation accuracy. The absolute error by UKF is always less than 2%, and it is noted that the synchronizationbased method is only against weak noise, it can fail while the strength of observation noise increases from 0.07 to 0.1. As shown in Fig. 4共b兲, the absolute error of the parameter 2 estimated by the combined method is less than 2% and the synchronization-based method also fails, while the observation noise strength increases to 0.01.
The HR model is a simplified model that uses a polynomial approximation to the right-hand side of a HH model. It can be written as follows: x˙1 = x2 − 1x31 + 2x21 − x3 + I, x˙2 = c − dx21 − x2,
x˙3 = r关s共x1 − x0兲 − x3兴,
共16兲
where x1 is the membrane potential, x2 describes fast currents, x3 describes slow current, and I is an external current. Here, c = 1, d = 5, s = 4, r = 0.006, x0 = −1.6, and 1 and 2 are the parameters to be estimated. Let 1 = 1 and 2 = 3 to gen-
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FIG. 4. 共Color online兲 The normalized absolute error of parameter estimation of the three methods for 共a兲 1 and 共b兲 2 vs the observation noise strength D.
FIG. 6. 共Color online兲 The parameter 1 estimated by combined method 共a兲, UKF only, 共b兲 and the synchronization-based method only 共c兲 from the noise-corrupted time series.
FIG. 5. 共Color online兲 Clean 共red line兲 and noisy observation 共blue dots兲 can be seen in 共a兲. Clean 共red line兲 and estimated states 共blue line兲 are shown in 共b兲.
erate a chaotic time series of active potential which will be used to estimate the parameters by the combined method proposed above. The observations are generated by corrupting the active potential with white Gaussian noise whose strength is D = 0.1. Here, the signal-noise-ratio is about 7 dB or, equivalently, we deal with 50% measurement noise. The noisecorrupted signal and the noise free signal are plotted in Fig. 5共a兲. To use the UKF to estimate the states of system 共16兲, the initial guess of parameter is chosen to be ˆ = 关0 0兴, and the
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FIG. 8. 共Color online兲 The normalized absolute error of parameter estimation of the three methods for 共a兲 1 and 共b兲 2 vs the observation noise strength D.
vergence behavior of the estimated parameters by the synchronization-based method only from the noisy observation data directly. Figure 8 depicts the asymptotic value A against the noise strength D.
IV. CONCLUSIONS
FIG. 7. 共Color online兲 The parameter 2 estimated by combined method 共a兲, UKF only, 共b兲 and the synchronization-based method only 共c兲 from the noise-corrupted time series.
initial state estimation is set to xˆ = 关0.1 0.1 0.1兴. The differences between the estimated and the clean states are negligible small as shown in Fig. 5共b兲 if related to the difference between clean states and noisy data in Fig. 5共a兲. The estimated states are used by the combined method described in Eqs. 共4兲, 共7兲, 共9兲, 共10兲, and 共13兲 to estimate the unknown parameters of HR model, the unknown parameters may be dynamically estimated from q in the receiver system as shown in Figs. 6共a兲 and 7共a兲, respectively. The evolutions of the parameters estimated by UKF only are plotted in Figs. 6共b兲 and 7共b兲. Figures 6共c兲 and 7共c兲 show the con-
To summarize, we have shown that a combination of UKF with the synchronization-based adaptive feedback parameter estimation method enables the high accuracy estimation of the unknown parameters of neuron model from heavily noise-corrupted time series of active potential. The feedback comes from the states estimated by UKF from the noise-corrupted data instead of the noise-corrupted data directly. The technique of UKF enables simultaneous state and parameter estimation from data with relatively large amounts of measurement noise. Contrary to methods that yield similar estimation results for deterministic system, the UKF can be used with stochastic systems as well. Unlike in many other approaches, the model nonlinearities are taken as what they are and not approximated by Taylor series expansion. Further more, time expensive stochastic simulations are not necessary. This makes UKF very flexible, and it can also be applied to systems where the explicit forms of the nonlinearities are not known or the derivatives are difficult to compute. It must be noted that UKF estimates unknown parameters by considering the unknown parameters as virtual states of a
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joint system, so the accuracy of the parameter estimation will not be very high. The parameter estimation method of synchronization based on adaptive feedback can obtain accurate estimation of parameters but it is only robust against weak noise, so when the observed data was heavily noise corrupted, the feedback can be replaced by the state estimated by UKF, then accurate parameter estimation can also be obtained. In the examples studied here, the functional form of the state space model must be known beforehand. To our knowledge, there are no methods of state estimation which would also allow for the estimation of functional form of the model. Our future research will concentrate on a combination of parametric modeling and nonparametric methods.22 ACKNOWLEDGMENTS
This work was supported by the Key Grant of National Natural Science Foundation of China 共Grant No. 50537030兲, the Grant of National Natural Science Foundation of China 共Grant No. 50707020兲, the Grant of Postdoctoral Science Foundation of China 共Grant No. 20070410756兲, and the Special Grant of Postdoctoral Science Foundation of China 共Grant No. 200801212兲.
A. L. Hodgkin and A. F. Huxley, J. Physiol. 共London兲 117, 500 共1952兲. C. Morris and H. Lecar, Biophys. J. 35, 193 共1981兲. 3 R. FitzHugh, in Biological Engineering, edited by H. P. Schwann 共McGraw-Hill, New York, 1969兲. 4 J. L. Hindmarsh and R. M. Rose, Nature 共London兲 296, 162 共1982兲. 5 H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. Sh. Tsimring, Rev. Mod. Phys. 65, 1331 共1993兲. 6 U. Parlitz, Phys. Rev. Lett. 76, 1232 共1996兲. 7 U. Parlitz, L. Junge, and L. Kocarev, Phys. Rev. E 54, 2115 共1996兲. 8 A. Maybhate and R. E. Amritkar, Phys. Rev. E 61, 6461 共2000兲. 9 R. Konnur, Phys. Rev. E 67, 027204 共2003兲. 10 D. Huang, Phys. Rev. E 73, 066204 共2006兲. 11 D. Huang, Phys. Rev. E 69, 067204 共2004兲. 12 R. E. Kalman, J. Basic Eng. 82, 35 共1960兲. 13 L. Ljung, IEEE Trans. Autom. Control 24, 36 共1979兲. 14 S. J. Julier and J. K. Uhlmann, Proc. IEEE 92, 401 共2004兲. 15 A. Sitz, U. Schwarz, J. Kurths, and H. U. Voss, Phys. Rev. E 66, 016210 共2002兲. 16 A. Sitz, U. Schwarz, and J. Kurths, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 2093 共2004兲. 17 L. A. Aguirre, B. O. S. Teixeira, and L. A. B. Torres, Phys. Rev. E 72, 026226 共2005兲. 18 H. U. Voss, J. Timmer, and J. Kurths, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1905 共2004兲. 19 J. Honerkamp, Stochastic Dynamical Systems 共VCH, New York, 1993兲. 20 A. Gelb, Applied Optimal Estimation 共MIT Press, Cambridge, MA, 1974兲. 21 S. Haykin, Kalman Filtering and Neural Networks 共Wiley, New York, 2002兲. 22 J. Timmer, H. Rust, W. Horbelt, and H. U. Voss, Phys. Lett. A 274, 123 共2000兲. 1 2
CHAOS 19, 015106 共2009兲
Automated synchrogram analysis applied to heartbeat and reconstructed respiration Claudia Hamann,1 Ronny P. Bartsch,2 Aicko Y. Schumann,3 Thomas Penzel,4 Shlomo Havlin,2 and Jan W. Kantelhardt3 1
Institut für Physik, Technische Universität Ilmenau, 98684 Ilmenau, Germany Department of Physics and Minerva Center, Bar-Ilan University, Ramat Gan 52900, Israel 3 Institute of Physics, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany 4 Schlafmedizinisches Zentrum der Charité Berlin, 10117 Berlin, Germany 2
共Received 4 December 2008; accepted 17 February 2009; published online 31 March 2009兲 Phase synchronization between two weakly coupled oscillators has been studied in chaotic systems for a long time. However, it is difficult to unambiguously detect such synchronization in experimental data from complex physiological systems. In this paper we review our study of phase synchronization between heartbeat and respiration in 150 healthy subjects during sleep using an automated procedure for screening the synchrograms. We found that this synchronization is significantly enhanced during non-rapid-eye-movement 共non-REM兲 sleep 共deep sleep and light sleep兲 and is reduced during REM sleep. In addition, we show that the respiration signal can be reconstructed from the heartbeat recordings in many subjects. Our reconstruction procedure, which works particularly well during non-REM sleep, allows the detection of cardiorespiratory synchronization even if only heartbeat intervals were recorded. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3096415兴 About one-third of our life we spend sleeping. Since sleep is required for recreation, a disturbed sleep negatively affects our daily physical and mental fitness. However, at least 10% of the human population in the industrialized world suffer from sleep related disorders or sleep-wake dysfunctions. Investigating human physiology during sleep is of high interest not only for identifying sleep related disorders but also for detecting and understanding changes in sleep patterns related to other diseases such as, e.g., Alzheimer or Parkinson disease. The standard procedure in a hospitals’ sleep laboratory includes full night polysomnography, where many sensors and electrodes are attached to the patient’s body in order to measure heartbeat, respiration, muscle activity, brain waves, and eye movements. Although the major aim of these measurements is to monitor natural sleeping behavior, sleep is often disturbed by the unfamiliar environment and the distempering measuring devices. In contrast, the electrocardiogram (ECG) can be recorded with portable, rather inexpensive and comfortable Holter recorders. In this paper we explore the possibility of extracting respiration signals from ECG recordings by taking advantage of the variation in heart rate during a respiratory cycle (“respiratory sinus arrhythmia”). Based on therewith extracted respiration signals, we study the phase relationship between respiration and heartbeat (“cardiorespiratory synchronization”) utilizing an automated procedure for screening cardiorespiratory synchrograms, and find that cardiorespiratory synchronization differs significantly between the main sleep stages. Reconstructed respiration from ECG together with the automated synchrogram analysis might pave the way for assessing sleep and sleep disorders by simply analyzing Holter recordings.
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I. INTRODUCTION
Transitions in the synchronization behavior of two coupled oscillators have been shown to be important characteristics of model systems.1 It has been found that noise, when applied identically to different nonlinear oscillators, can induce, enhance, or destroy synchronization among them.2–4 However, phase synchronization is difficult to study in experimental data, which are very often inherently nonstationary and thus contain only quasiperiodic oscillations. Among the few recent experimental studies are coupled electrochemical oscillators,3 laser systems,4 and climate variables.5 Moreover, the question of detecting synchronization between several interacting processes with different time scales in univariate signals has been addressed6 and methods for characterizing two different types of phase locking, softand hard phase locking, and its detection by analyzing univariate data were suggested.7 In physiology, the study of phase synchronization focuses on cardiorespiratory data and encephalographic data.8 First approaches for the study of cardiorespiratory synchronization have been undertaken by the analysis of the relative position of inspiration within the corresponding cardiac cycle.9 More recently, phase synchronization between heartbeat and breathing has been studied during wakefulness using the synchrogram method.10–13 While long synchronization episodes were observed in athletes and heart transplant patients 共several hundreds of seconds兲,11,12 shorter episodes were detected in normal subjects 共typical duration less than hundred seconds兲.12–14 For two recent models of cardiorespiratory synchronization, see Ref. 15. The cardiorespiratory system consisting of the heart, the blood vessels as well as the lungs plays a major role in hu-
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man physiology. Therefore, it has been studied extensively during the past decades. Most approaches covered monovariate analysis of either heartbeat or breathing signals, e.g., employing spectral methods to investigate heart rate variability16,17 or exploring correlation behavior and fluctuations by applying detrended fluctuation analysis,18–20 or recently phase rectified signal averaging.21 Cross modulations and cross interactions between the components of the cardiorespiratory system have been studied by means of crosscorrelation analysis, transfer function analysis, 共phase兲 synchronization analysis, and more recently by bivariate phase rectified signal averaging 共see Ref. 22 and references therein兲. In this work, we review our automated synchrogram based procedure23 using the concept of phase synchronization10–13 and study interactions between cardiac and respiratory oscillations under different physiological conditions. Usually, the cardiorespiratory system is continuously influenced and altered by its environment, e.g., we respond to external stimuli such as sonic or visual input and are sensitive to mental stress. However, during sleep the cardiorespiratory system is self-sustained, reflecting the intrinsic characteristics of the autonomous nervous system and the subjects’ physiology. In order to obtain reliable experimental evidences of transitions in phase synchronization behavior, we have thus considered cardiorespiratory synchronization in humans during sleep.23 It is well known that healthy sleep consists of approximately five cycles of roughly 1–2 h duration. Each cycle usually evolves from non-rapid-eyemovement 共non-REM兲 sleep, consisting of light and deep sleep, to REM sleep.24 Homogeneous long-term data for well-defined conditions of a complex physiological system are thus available from sleep laboratories.25 Investigating human physiology during sleep is of diagnostic value not only for sleep related diseases such as sleep apnea. It can also enhance the understanding and identification of nonsleep related diseases. We have found the intriguing result that during REM sleep cardiorespiratory synchronization is suppressed by approximately a factor of 3 compared with wakefulness.23 On the other hand, during non-REM sleep, it is enhanced by a factor of 2.4, again compared with wakefulness. In addition, we have found that these significant differences between synchronization in REM and non-REM sleep are very stable and occur in the same way for males and females, independent of age and independent of the body mass index 共BMI兲. Our results regarding synchronization efficiency suggest that the synchronization is mainly due to a weak influence of the breathing oscillator upon the heartbeat oscillator, which is disturbed in the presence of long-term correlated noise, superimposed by the activity of higher brain regions during REM sleep. While heartbeat data can be measured easily and conveniently by employing portable devices, the recording of brain waves 共electroencephalogram, EEG兲 and respiration is technically much more difficult. Usually, respiration is measured by either one of the following rather uncomfortable and encumbering and/or invasive and thus ambulatory ill suited methods: 共i兲 by using stretch sensors embedded in a belt and
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FIG. 1. 共Color online兲 共a兲 Heartbeat interval time series RRi from a healthy young subject during sleep. 共b兲 Power spectrum of the signal shown in 共a兲. The LF band 共0.04–0.15 Hz兲 is assumed to reflect blood pressure oscillations, whereas the HF band 共0.15–0.4 Hz兲 reflects respiration via the RSA effect. It can thus be used to reconstruct the respiratory signal.
attached to the chest and/or abdomen surveying excursions and motions of the body surface, 共ii兲 by means of a thermistor and/or a spirometer 共flow meter兲 incorporated in a mouthpiece, a nose clamp or a breathing mask covering the whole nose-mouth area, 共iii兲 by expensive inductive plethysmographs or respiratory magnetometers, or 共iv兲 by impedance pneumography based on impedance cardiographic signals.26 The major aim of polysomnographic measurements27 is to monitor natural sleeping behavior of the patient. However, sleep can be disturbed significantly in a sleep laboratory because of the unfamiliar environment and the distempering measuring devices, both resulting in additional physical and mental stress. We thus explore in this paper a possibility for the reconstruction of respiratory signals from heartbeat intervals since heartbeat can be recorded at home with portable devices. Our reconstruction algorithm is based on the physiological phenomenon that respiration influences the sympathovagal autonomous nervous system. While inspiration enhances sympathetic components followed by an increase in heart rate, expiration suppresses sympathetic and activates vagal components resulting in a heart rate decrease. This variation in the heart rate during the respiratory cycle is generally known as respiratory sinus arrhythmia 共RSA兲. It is important to note that RSA is not equivalent to cardiorespiratory synchronization. While RSA only leads to a cyclical variation of heart rate, cardiorespiratory synchronization is only observed when a constant number of heartbeats occur at the same instantaneous phases within the breathing cycle for a period of several consecutive breaths. Both phenomena can occur independent of each other, although an increased RSA might reduce cardiorespiratory synchronization.11 In spectral analysis of heartbeat interval data 共see Fig. 1兲, two prominent peaks are often observed corresponding to characteristic frequency components in the low frequency 共LF兲 共0.04–0.15 Hz兲 and high frequency 共HF兲 共0.15–0.4 Hz兲 bands.16 The LF band has been associated with sympathetic activation; the corresponding peak might be related with blood pressure oscillations 共Mayer waves兲. The HF band is related with vagal components, and it has been shown that HF spectral power is significantly influenced by breathing volume and breathing rate, i.e., changing the breathing pat-
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II. DATA AND DATA PREPARATION
Our studies are based on data collected in the framework of the EU project SIESTA from seven European sleep laboratories.25 The data consist of full night polysomnographic recordings27,29 from 150 healthy subjects 共age: 50.3⫾ 19.4 y兲 recorded during one regular night 共an adaption night was preceding each recording night兲. Note that for this manuscript we have increased the number of subjects to 150 compared with our original publication23 where we only considered 112 subjects, and which are a subset of the 150 subjects analyzed here. The average duration of the recordings is 7.9⫾ 0.4 h. Based on visual inspection of the recordings and following international standards,24 sleep stages were identified and assigned in intervals of 30 s. Sleep stage classifications include REM sleep 共stage 5兲 and non-REM sleep, i.e., light sleep 共stages 1 and 2兲 and deep sleep 共stages 3 and 4兲 with some additional shorter wake periods. In this paper we study a one channel electrocardiogram 共ECG兲 signal as well as the oronasal airflow signal measured by a thermistor. Depending on the laboratory, the ECG signal was sampled at 100, 200, or 256 Hz, while airflow was sampled at 16, 20, 100, or 200 Hz. From the raw ECGs we detected the time positions of heartbeats 共R peaks兲 applying either a wavelet based peak detector or a peak detector designed at the German Heart Center and Klinikum Rechts der Isar Munich, Germany.30 We have validated the comparability of both detectors by statistical tests, considering 20 randomly chosen subjects. For an extensive review on ECG beat detection, we refer to Ref. 31. In the next step we derived the series of time intervals between each two successive heartbeats 共RR intervals兲. These series usually contain some artifacts caused by incorrectly detected beat positions, measurement errors, or subject movements. To eliminate these artifacts, we removed beat-to-beat intervals smaller than 300 ms, larger than 2000 ms, or differing by more than 1000 ms from the preceding or the following beat-to-beat interval. Removed data points were linearly interpolated in order to preserve the timing. Regarding respiration, we studied two types of signals: 共i兲 respiration obtained directly from oronasal airflow and 共ii兲 respiration reconstructed from heartbeat intervals. Noise in oronasal airflow data consists mainly of spikes 共outliers兲 in the quite sinusoidal signal. A simple threshold filter is thus sufficient. All data points exceeding a threshold of 95% of
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tern alters the HF spectral components.28 Therefore, respiratory components can be extracted from a heartbeat interval time series by Fourier filtering, i.e., applying a band-pass filter adjusted to HF components. This way, we use the RSA effect for the reconstruction of the respiration signal from heartbeat intervals. The paper is structured as follows. Section II describes our data recordings and the data preprocessing for studying phase synchronization. This includes the reconstruction of respiration signals from heartbeat signals. In Sec. III we describe our phase synchronization analysis for real and reconstructed respiration as well as our automated procedure for the detection of phase synchronization. In Sec. IV we apply this procedure to study cardiorespiratory synchronization.
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FIG. 2. 共Color online兲 Analysis of respiration signals. 共a兲 and 共b兲 Recorded oronasal airflow 共black solid line兲 is compared with the respiratory signals reconstructed from heartbeat 共red dashed lines兲 for two segments of a recording from a healthy subject. 共c兲 and 共d兲 Instantaneous phases calculated from the oronasal airflow 关1共t兲, black solid line兴 and from the reconstructed respiration 关2共t兲, red dashed line兴 are compared for the same segments. 共e兲 and 共f兲 Histogram of the phase differences 1共t兲 − 2共t兲 between real and reconstructed respiration signals. The peak in 共e兲 indicates that the reconstructed signal resembles the original signal with an unimportant systematic phase shift of approximately / 2. The uniform distribution in 共f兲 indicates that the respiratory signal could not be reconstructed probably due to a diminished influence of the breathing upon the heartbeat signal, i.e., very weak RSA.
the maximum value or dropping below 95% of the 共negative兲 minimum value within a moving time window are clipped to the corresponding threshold value. After filtering, respiration was resampled at 4 Hz, preparing the signal for phase detection via Hilbert transform. The reason for the selection of this sampling rate lies in the optimal number of 10–20 data points per oscillation for the Hilbert transform of noisy periodic signals. The resampling is done by dividing the original sampling rate by 4 Hz and averaging the corresponding number of data points. To reconstruct an alternative respiration signal from heartbeat, we began with resampling the beat-to-beat interval series in equidistant time steps. This means we linearly interpolated between the RRi values occurring at times tk = 兺ki=1RRi. We chose a time resolution of 0.25 s corresponding to the 4 Hz of the original respiration signal for comparability. Then, respiration is reconstructed by band-pass Fourier filtering, i.e., calculating the fast Fourier transform 共FFT兲 of the heartbeat interval signal, setting all Fourier coefficients outside the desired HF band 共0.15–0.4 Hz兲 to zero, and calculating the inverse FFT. Two illustrative examples of the reconstructed respiration and the corresponding filtered oronasal airflow signals are shown in Figs. 2共a兲 and 2共b兲. In order to investigate and quantify the reconstruction quality of the respiratory signal, we subdivided all signals into segments corresponding to different sleep stages and employed methods of phase-synchronization analysis 共see below兲. III. ANALYSIS METHODS
In this section we review our automated synchrogram method to study phase synchronization between a continuous
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signal 共respiration兲 and a point-process 共heartbeats兲, which we briefly introduced in Ref. 23. In addition, we describe the method we applied to investigate synchronization between measured oronasal airflow and ECG based reconstructed respiration. A. Calculating and testing instantaneous respiratory phases
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where PV denotes the Cauchy principal value. In practice, for homogeneously sampled signals, the Hilbert transform can easily be calculated by the following steps: 共i兲 applying FFT to the original signal x共t兲, 共ii兲 multiplying the spectral coefficients by −i sgn共f兲 with frequency f, and 共iii兲 transforming the manipulated signal back into time domain by applying inverse FFT. Finally, instantaneous phases can be defined from real and imaginary parts of the analytical sig˜ 共t兲 / x共t兲兴. Additionally, a sign logic has to nal, 共t兲 = arctan关x be implemented to obtain values − ⬍ ⱕ rather than just − / 2 ⬍ ⱕ / 2 共note that this is available as atan2 in most computer libraries兲. Note further that a Hilbert transform requires a rather narrow-banded input signal,10,33 and thus a suitable band-pass filter might have to be applied before the transform. Our resampling of the respiration signals at 4 Hz represents an easy implementation of a low pass filter eliminating HFs that would disturb the phase calculations. Figures 2共a兲 and 2共b兲 show two exemplary parts of both, recorded respiration and ECG based reconstructed respiration. The corresponding instantaneous phase signals obtained from Hilbert transforms are shown in Figs. 2共c兲 and 2共d兲. It is clearly seen that the reconstruction works well in the section shown in parts 共a兲 and 共c兲, while it completely fails in the section shown in 共b兲 and 共d兲. In order to quantify the quality of the reconstruction, we have plotted in Figs. 2共e兲 and 2共f兲 the histograms of the corresponding phase differences. If recorded and reconstructed respiration signals resemble each other, they exhibit 1:1 phase synchronization and the histogram is strongly peaked. For not synchronized signals, on the other hand, all phase differences have identical probability and the histogram is flat. This can be computationally checked either by calculating standard synchronization indices10 or by a direct study of the histograms. However, we find that the standard indices strongly depend on the noise level in the signal, sudden phase jumps, and artifacts. In addition, they do not fully vanish for unsynchronized surrogate data, e.g., for reconstructed respiration from one subject and the measured oronasal airflow from another. We thus decided to classify the histograms exemplified in
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In order to study phase synchronization between two signals, it is necessary to obtain instantaneous phases at least for the slower oscillating signal, i.e., for respiration in our case. For a real valued continuous signal x共t兲, this can be done within an analytic signal approach, adding a corresponding ˜ 共t兲 to the signal. ˜x共t兲 is calculated by emimaginary part ix ploying Hilbert transform,32,33 ˜x共t兲 =
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tk [sec] FIG. 3. Examples illustrating the automated synchrogram method for real共left兲 and reconstructed 共right兲 respiration signals. Symbols in 共a兲 and 共b兲 show the instantaneous respiratory phases at the time of the heartbeats. 共c兲 and 共d兲 means and standard deviations of the phases, calculated in time intervals of length = 30 s around each point in the horizontal lines. 共e兲 and 共f兲 Phase points with a standard deviation larger than the threshold were deleted and then sequences shorter than the threshold T were also deleted. Note that T must be slightly smaller for reconstructed breathing 共right兲 since the continuous segments are shorter.
Figs. 2共e兲 and 2共f兲 by calculating the mean phase differences ⌬ = 具1共t兲 − 2共t兲典 and the corresponding standard deviations within windows with length of 30 s, i.e., 120 data points. We define both signals as synchronized if the standard deviation is below 0.5 rad. In this way, we obtain for each subject the overall percentage of synchronized windows for the whole night as well as separately for wake, REM sleep, and non-REM sleep 共cf. Table I兲. B. Automated phase-synchrogram analysis
In order to study phase synchronization between heartbeat intervals and either the recorded oronasal airflow or the reconstructed respiration signal, we employ the method of phase-synchrogram analysis. This method is ideal for studying phase synchronization between a point process 共here, heartbeat兲 and a continuous signal 共here, respiration兲. While cardiorespiratory synchrograms are traditionally analyzed by visual inspection, we recently suggested a completely automated approach.23 First, instantaneous phases 1共t兲 and 2共t兲 are calculated for both respiratory time series 共see Sec. III A兲. Second, we calculate cumulative respiratory phases ⌽ j共t兲 = j共t兲 + 2n, j = 1 , 2, starting with n = 0 and incrementing n if the instantaneous phase j共t兲 drops by a value larger than . Ideally, a new breathing cycle would start with a drop in phase by 2. However, smaller values occur in practice due to limited time resolution, noise, etc. In rare cases, when the instantaneous phase increases by more than , n is decremented. The cardiorespiratory synchrogram is then obtained by mapping the times tk = 兺ki=1RRi of the heartbeats onto the continuous cumulative phases ⌽ j共t兲. Figures 3共a兲 and 3共b兲 illustrate two representative parts of the corresponding synchrograms for measured oronasal airflow 共a兲 and reconstructed breathing 共b兲. In these synchrograms for studying synchronization of n heartbeats within one breathing cycle 共n : 1 synchronization兲, ⌿1,j共tk兲 = ⌽ j共tk兲mod 2 is plotted versus tk. In areas with n : 1 phase synchronization, n parallel horizontal lines appear. The lines vanish if synchronization breaks
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down. Figures 3共a兲 and 3共b兲 show events of 4:1 phase synchronization. The similarities between 共a兲 and 共b兲 indicate a good reconstruction quality in the considered time window. Arbitrary synchronization ratios n : m, i.e., the occurrence of n heartbeats during m breathing cycles, can be studied easily considering synchrograms of ⌿m,j共tk兲 = ⌽ j共tk兲mod 2m versus tk and again looking for n parallel horizontal lines. To study data of many subjects and nights, it is necessary to automatically detect and distinguish synchronized and unsynchronized areas in the synchrograms.23 We therefore applied a centered moving average filter of window length separately for every r = 1 , . . . , n heartbeats observed within the m considered breathing cycles in the following sense: 共i兲 m breathing cycles, which are assumed to be at the center of the averaging window, are taken and the number n of heartbeats occurring within these m breathing cycles is counted. The times of these heartbeats are denoted as t共r兲 c , 共ii兲 a regularly spaced phase interval associated with each single heart共n兲 = 2m / n is calculated, beat event at the center position ⌬m 共iii兲 all phases m共tk兲 belonging to neighboring breathing 共r兲 cycles within the time interval Tr = 关t共r兲 c − / 2 , tc + / 2兴 are averaged with respect to their dedicated phase range Rr 共n兲 共n兲 = 关共r − 1兲⌬m , r⌬m 兲 , r = 1 , . . . , n, 共r兲 具⌿m 典共t共r兲 c 兲=
1 兺 ⌿共r兲共tk兲. NRr tk苸Tr m
共2兲
Here, NRr denotes the number of phases occurring in the time window Tr and the phase range Rr as obtained from the synchrogram. Note that even when n heartbeats occur during m breathing cycles at the center position, there might be a different number of heartbeats during other m breathing cycles within the same considered moving average window of width . In addition to the average, we calculate for each of the r heartbeats a standard deviation ˆ r. 共r兲 In the next step, every value ⌿m 共tk = t共r兲 c 兲 during the centered m breathing cycles is replaced by the corresponding 共r兲 共tk兲典, as illustrated for n = 4 and m = 1 in mean value 具⌿m Figs. 3共c兲 and 3共d兲. In addition, the four different ˆ r are shown as error bars. In the final step shown in Figs. 3共e兲 and 3共f兲 we remove all points in the synchrogram where the condition ˆ r ⬍ 2m / n␦ is violated and keep only episodes of constant n that are longer than a minimum period T. From the remaining synchronized episodes, we can determine the percentage of synchronized episodes compared with the total sleep duration.
C. Optimizing synchrogram evaluation parameters
For our automated analysis of phase synchrograms, three parameters needed to be optimized:23 共i兲 the 共time兲 width of the moving average filter, 共ii兲 the standard deviation limit parameter ␦, and 共iii兲 the minimum episode duration T. To perform the optimization, we studied the influence of the parameters on the overall synchronization for total nights, comparing the results for the real data 共heartbeat and oronasal airflow兲 with those for 共unsynchronized兲 surrogate data.
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BMI [kg/m ] FIG. 4. 共Color online兲 Medians, upper and lower quartiles 共bars兲, and means 共filled symbols兲 for the detected synchronization rates 共a兲 vs T for all original data 共black dotted bars and circles兲 and surrogate data 共magenta striped bars and triangles兲, left of dotted line ␦ = 6, right of dotted line ␦ = 5. 共b兲 The results for optimized parameters 共for ␦ = 5 and T = 30 s兲 are shown vs BMIs and gender for wakefulness 共blue dotted bars and circles兲, REM sleep 共red blank bars and triangles兲, and non-REM sleep 共green striped bars and diamonds兲. Note the similar synchronization behavior in all subgroups. This figure is based on our original data set of 112 subjects and is adapted from Ref. 23.
The surrogate data were obtained by randomly combining heartbeat data from one subject with breathing data from another subject. Figure 4共a兲 shows the total night synchronization percentage for different ␦ and T. As expected, the largest ratio of synchronized episodes was found for small T and small ␦ 共i.e., a large limit for the standard deviations兲. However, in this case, a rather large number of synchronized episodes are also reported for the unsynchronized surrogate data. The ratio of the mean percentage of synchronization in real data over the mean percentage in surrogate data increases from 1.6 for T = 20 s to 3.4 for T = 40 s. However, for T = 40 s only very few synchronization episodes were detected. We therefore suggested choosing ␦ = 5 and T = 30 s to optimize the ratio between correctly detected real synchronization episodes and falsely detected synchronization episodes in surrogate data. Together with = 30 s, these parameter values provide a good separation and, furthermore, the time parameters coincide with the time interval of 30 s used in sleep stage classification. Note that ␦ has a similar influence on the results as T 共not shown in detail兲, while just weakly effects the results. When comparing synchronized episodes for real and reconstructed respiration 关see Figs. 3共e兲 and 3共f兲兴, one observes, in general, shorter synchronized episodes for the reconstructed respiration. This is due to instabilities in the reconstruction process. We thus adjusted Trec = 24 s for reconstructed breathing, keeping both and ␦ at the same values. This led to similar durations of the synchronized episodes in all subjects compared with recorded respiration and T = 30 s. IV. RESULTS AND DISCUSSION A. Phase synchronization with recorded respiration
In this subsection we review our results obtained from 112 subjects from the SIESTA database,23 which we origi-
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nally used to study cardiorespiratory synchronization by applying our automated synchrogram analysis 共see Sec. III兲 to heartbeat and recorded respiratory data. In Secs. IV B and IV C, when studying cardiorespiratory synchronization based on reconstructed respiratory data, we have increased the number of subjects to 150 including all 112 subjects of the previous study. Calculating the mean, median, and quartiles for the time ratio of phase synchronized episodes separately for wakefulness, REM sleep, and non-REM sleep, we found highly significant differences. During non-REM sleep, we observed phase synchronization during 3.8% of the total time compared with just 0.6% during REM sleep—a difference by a factor of 6.3. Wakefulness during the night—excluding times before the initial sleep onset and after final awakening—was clearly intermediate, since we found 1.6% of it to show cardiorespiratory phase synchronization. Similar differences were observed for other values of T and ␦. We also studied synchronization separately for several groups with different BMIs, and men and women. Figure 4共b兲 shows that the results are very similar in these subcategories of subjects. The same holds for different age groups,23 although both heart rate and breathing rate are known to depend on BMI and age. However, when comparing very young and old subjects, one observes a shift to lower values in synchronized episodes during non-REM sleep for older subjects. Altogether, these results prove that our finding of significant differences between the cardiorespiratory synchronization in REM and non-REM sleep is very stable. Evidence of stable but pronounced differences between REM and non-REM sleep has been reported in the fluctuations of both heartbeat19 and respiration.20 Influences of the central nervous system 共with its sleep stage regulation in higher brain regions兲 on the autonomous nervous system was suggested to be responsible for these differences. The similarity led us to hypothesize that the diminished synchronization during REM sleep is also caused by influences of the central nervous system. We note that there is very little 共⬇14%兲 increase in the amplitude of heartbeat or breathing fluctuations during REM sleep when compared with nonREM sleep. The changes in the synchronization behavior can thus not be due to variations in the strength of the influences from the brain. Rather they must be due to the correlation structure imposed by these influences. Long-term correlations are nearly absent in both heartbeat and breathing during non-REM sleep. We suggested the following physiological mechanism to explain our findings of the sleep stage differences in cardiorespiratory phase synchronization. As long as the heartbeat oscillator and the breathing oscillator 共as parts of the autonomous nervous system兲 are only affected by uncorrelated noise from higher brain regions, they run like two weakly coupled oscillators, and they clearly show synchronization as expected, possibly enhanced by the noise.2 However, if the higher brain regions are more active and impose long-term correlated noise on the two oscillators, as is the case during REM sleep, the noise disturbs the emergence of synchronized patterns, leading to a drastic reduction in synchroniza-
TABLE I. Average synchronization between recorded oronasal airflow and ECG based reconstructed breathing within different sleep stages in percent. Results are shown for all subjects and for subject characterized by different relative fluctuations of the reconstructed breathing. Numbers in brackets denote the number of considered individuals in the respective group.
All ⱕ 0.2 ⱕ 0.25 ⱕ 0.3
Whole night
Wake
REM
Non-REM
30.5共150兲 47.9 共74兲 39.5共110兲 33.2共136兲
17.9共150兲 33.3 共33兲 25.1 共95兲 20.6共129兲
17.8共149兲 44.0 共33兲 25.5 共95兲 19.2共137兲
35.9共150兲 52.3 共88兲 44.3共117兲 37.4共143兲
tion episodes. Hence, we suggest from the experimental data that correlated noise rather suppresses synchronization, while uncorrelated noise might increase it. Our results and interpretations are in agreement with the finding of enhanced cardiorespiratory synchronization in heart transplanted patients, where correlated signals from the brain can hardly affect the heartbeat oscillator.12 They further affirm the recently reported theory that synchronization pattern can only indirectly be related to cardiac impairments.34 Reduced long-term correlated regulation activity could possibly explain the increase in synchronization in well-trained athletes,11 where fluctuations of heartbeat and breathing might be avoided to optimize the cardiovascular system for optimal performance. For a discussion of the phase synchronization ratios n : m between oronasal airflow and heartbeat data, we refer to our original paper.23 There we have shown that mainly n : 1 synchronization is effective in the cardiorespiratory system. B. Quality of reconstructed respiration
Now we want to study cardiorespiratory synchronization based solely on heartbeat data. The first step of this task — the reconstruction of respiration from heartbeat time series — has been described in Sec. III A. After the reconstruction, we should make sure that the reconstructed respiration is reliable as in the example shown in Figs. 2共a兲, 2共c兲, and 2共e兲 rather than quite arbitrary as in the example shown in Figs. 2共b兲, 2共d兲, and 2共f兲. We have thus checked for all subjects the reliability of the respiration reconstruction, calculating for each of them the overall percentage of synchronized 30 s windows of real and reconstructed respiration. The procedure described at the end of Sec. III A is performed for the whole nights as well as separately for wake, REM sleep, and non-REM sleep. Average values can be found in Table I. Figure 5共a兲 shows the percentage of whole-night synchronization between both respiratory signals as a function of the relative fluctuations of the reconstructed respiration. Considering the time intervals between phase increases in 2 in the cumulative reconstructed respiratory phases ⌽2共t兲, is defined as the quotient of the standard deviation of these breathing intervals over the mean breathing interval. Since the respiratory data is sometimes nonstationary, we calculate for windows of T = 300 s and then average these values over the whole night.
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FIG. 5. 共Color online兲 共a兲 Percentage of synchronized time between realand reconstructed respiration signals vs relative fluctuations of the reconstructed respiration for whole nights. Values of between 0.2 and 0.3 seem appropriate to exclude subjects with failing reconstruction of respiration. 共b兲 The percentage of synchronized time between real- and reconstructed respiration signals is calculated separately during REM sleep and non-REM sleep, and the pairs of values are plotted in a scatter plot. Since the points for most subjects are above the diagonal 共dashed line兲, the reconstruction is more reliable during non-REM sleep.
Each point in Fig. 5共a兲 represents one subject from our database. It is obvious that there are subjects where the reconstruction is rather successful 共large percentage of correctly, i.e., synchronous reconstructed respiration兲, while for others it more or less fails. Obviously, the value of is usually larger for subjects with failing reconstruction. The relative fluctuations of the reconstructed respiration, i.e., , can thus be used as an approximate parameter for the quality of the reconstruction. Consequently, we compare results regarding reconstructed cardiorespiratory synchronization taking into account just subjects with values of below given thresholds 共see also Table I兲. We note that we tried to use different parameters characterizing the height of the peak in the HF band of the power spectrum 关see Fig. 1共b兲兴 as replacements for in classifying good and bad reconstruction of respiration. However, turned out to be superior to all these parameters. Figure 5共b兲 shows the percentage of synchronization between both respiratory signals during REM sleep versus the corresponding percentage during non-REM sleep in the same subject. Again, each point represents one subject. One clearly observes that well reconstructed respiration is found rather during non-REM sleep than during REM sleep since more subjects are found above the diagonal in Fig. 5共b兲. This observation is stable for different thresholds for 共see Table I兲. Except for ⱕ 0.2, results for wakefulness and REM sleep are basically the same, also reflecting the well-known statistical resemblance of REM and wake stages. C. Phase synchronization with reconstructed respiration
Figure 6共a兲 shows the whole-night percentages of cardiorespiratory synchronization for the real respiration 共left subpanel兲 and the reconstructed respiration 共right subpanel兲 considering subsets of the 150 subjects with below and above the indicated thresholds and slightly reduced Trec = 24 s. We have employed the automated synchrogram analysis algorithm described in Sect. III to calculate these percentages of phase synchronization. The close similarity of the results for real and reconstructed respiration prove that car-
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FIG. 6. 共Color online兲 共a兲 Median, upper and lower quartiles, and mean 共dot兲 of percentage of phase synchronized time regarding heartbeat and real breathing signal 共left兲. The three sets on the right show the results for heartbeat and reconstructed respiration for different . The violet striped bars represent the values for subjects with below the threshold value and the dark yellow blank bars for breathing signals with above the threshold. 共b兲 Cardiorespiratory synchronization percentages for real respiration signals 共left set兲 and reconstructed respiration signals 共right set兲 during different sleep stages 共wake= blue, REM= red, non-REM= green兲.
diorespiratory synchronization can be calculated based solely on heartbeat data. Comparing the values with those in the left subpanel, we think that the limit ⬍ 0.25 is most appropriate. For Fig. 6共b兲 we have split the data into parts of wakefulness, non-REM sleep, and REM sleep. Again there is a close similarity for the results based on real respiration and reconstructed respiration. The main finding of drastically reduced cardiorespiratory synchronization during REM sleep and enhanced cardiorespiratory synchronization during nonREM sleep compared with wakefulness is fully confirmed. V. CONCLUSION
We studied cardiorespiratory phase synchronization for a large database of healthy subjects, further increasing the number of subjects considered in our original paper.23 For this purpose, we developed and thoroughly described an algorithm detecting epochs of synchronization automatically and systematically in synchrogram plots. Comparing the synchronization behavior during different well-defined physiological stages, we observed clearly reduced synchronization during REM sleep and enhanced synchronization during non-REM sleep compared with wakefulness. Since REM and non-REM sleep differ mainly in the type of activity of higher brain centers, it seems probable that the differences in cardiorespiratory synchronization are caused by the more and less long-term correlated regulation actions of the brain during REM and non-REM sleep, respectively. In addition, we developed and tested a method for the reconstruction of respiration signals from interheartbeat time series based on the RSA effect and the HF spectral component of heartbeat. We have shown that the reliability of the reconstruction can be checked for each subject by calculating the relative standard deviation of the reconstructed breathing intervals. In general, the reconstruction is more reliable dur-
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ing non-REM sleep compared with REM sleep. The respiration reconstruction works well in most subjects and yields very similar results for the cardiorespiratory phase synchronization as the recorded respiration data. Hence, a simple Holter recording will be sufficient for the study of cardiorespiratory synchronization in many subjects. The findings should be helpful in the discrimination of sleep stages based only on Holter recordings. Alternatively, the reconstruction of respiration might be based on the heights of the R peaks in multiple lead ECG recordings if such records are available. Possibly, a different reconstruction method can improve the reconstruction quality particularly during REM sleep. Such a method might be combined with the approach studied in this paper. This possibility should be explored in future work. In addition, the consistency of the reconstruction with respiration recorded with stretch sensors could be checked. ACKNOWLEDGMENTS
We would like to acknowledge financial support from the European Union 共project DAPHNet, Grant No. 0184742兲, from Deutsche Forschungsgemeinschaft 共DFG兲 共Grant Nos. KA 1676/3 and Pe628/3兲, and the Adar Foundation for advancing heart research at Bar-Ilan University. C.H. is particularly grateful to the Minerva Foundation for funding her visit to Bar-Ilan University. 1
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Numerical studies of slow rhythms emergence in neural microcircuits: Bifurcations and stability M. A. Komarov,1,2 G. V. Osipov,1,2 J. A. K. Suykens,2 and M. I. Rabinovich3 1
Department of Control Theory, Nizhny Novgorod University, 23 Gagarin Avenue, 603950 Nizhny Novgorod, Russia 2 ESAT-SCD/SISTA, K.U. Leuven, Kasteelpark Arenberg 10, B-3001 Leuven (Heverlee), Belgium 3 Institute for Nonlinear Science, University of California, San Diego, 9500 Gilman Drive 0402, La Jolla, California 92093-0402, USA
共Received 2 December 2008; accepted 9 February 2009; published online 31 March 2009兲 There is a growing body of evidence that slow brain rhythms are generated by simple inhibitory neural networks. Sequential switching of tonic spiking activity is a widespread phenomenon underlying such rhythms. A realistic generative model explaining such reproducible switching is a dynamical system that employs a closed stable heteroclinic channel 共SHC兲 in its phase space. Despite strong evidence on the existence of SHC, the conditions on its emergence in a spiking network are unclear. In this paper, we analyze a minimal, reciprocally connected circuit of three spiking units and explore all possible dynamical regimes and transitions between them. We show that the SHC arises due to a Neimark–Sacker bifurcation of an unstable cycle. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3096412兴 The validity of dynamical models can be confirmed only when their solutions explaining the investigated phenomenon are structurally stable. An exhaustive sweep over the control parameters enlightens both regions of such solutions in the parameter space and, more interestingly, the evolution of the behavioral qualities along a particular change in parameters. Motivated by recent experimental observations that brain rhythms are products of local inhibitory networks, we have analyzed the dynamics of a minimal inhibitory circuit of three neurons. The considered microcircuit is capable of generating a global rhythm that does not depend on the details of the spiking activity in individual units. We have shown that the mathematical image of this behavior is a closed heteroclinic channel enclosing contour of saddle limit cycles and the heteroclinic orbits connecting them. Our bifurcation analysis yields the conditions on the emergence and the structural stability of this regime.
I. INTRODUCTION
The appearance and timing relationship of oscillatory activities with strongly different frequencies in complex neural systems and in the brain is one of the key problems of neuroscience. Many experiments indicate that spiking and bursting dynamics are involved in different ways in neuronal microcircuit functions and in brain rhythm generation.1–4 In particular, spiking 共temporal兲 and bursting 共rate兲 activity can be independent and code for different entities or sensory variables.5 What is the dynamical origin of the slow rhythm generation? We analyzed the minimal inhibitory neural circuit of spiking neurons that is modeled by Bonhoeffer–Van der Pol equations. We showed here that subcritical Neimark– Sacker bifurcation leads to the appearance of structurally stable heteroclinic channel 共SHC兲. The skeleton of this channel is a heteroclinic contour that consists of saddle limit 1054-1500/2009/19共1兲/015107/8/$25.00
cycles and the heteroclinic orbits connecting them. It happens when the degree of the nonsymmetry of the network connectivity exceeds a critical level. Similar problems have been investigated in Ref. 6. The authors of that work compared the bifurcation sequence from tonic spiking activity to burst generation in an inhibitory network of Hodgkin–Huxley neurons with the sequence of qualitative transformations of the phase portrait that leads to the appearance of a heteroclinic cycle in the framework of a time-averaged 共rate兲 model of the same network and found that these sequences are the same. In this paper we directly calculate the Floquet multipliers of limit cycles and determine the critical parameter values when two complex conjugate multipliers reach unit modulus. The observed bifurcation leads to the appearance of a structurally stable regime of the sequential switching of the activity of spiking neurons. Due to structural stability, slow rhythm generation practically does not depend on the neural model. It only depends on the connectivity parameters. However, to investigate the bifurcations in detail, we need to use a rather representative model on the one hand, but also rather convenient for analysis on the other hand. II. NETWORK MODEL
We consider the network of three spiking neurons 共shown in Fig. 1兲, modeled by the Bonhoeffer–Van der Pol equations,
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FIG. 1. Neuronal network: motif of three reciprocally inhibitory coupled neurons.
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Here xi共t兲 denotes the membrane potential of the ith neuron, y i共t兲 the variable corresponding to the action of all ionic currents, Si the external stimuli to each neuron, v the reversal potential, gij the coupling coefficients between the ith and jth neuron and F共x j兲 = 1 / 共1 + exp共共0.5− x j兲 / 20兲兲. The values of the parameters are fixed in all simulations to a = 0.7, b = 0.8, 1 = 0.08, 2 = 3.1, and v = −1.5, and we chose the parameter Si ⬎ 0.35 that corresponds to tonic spiking regime of individual uncoupled neurons. Depending on the level of nonsymmetry of inhibitory coupling, this simple network demonstrates the variety of dynamical regimes. • One neuron is active 共spiking oscillations兲 and two other neurons are suppressed 共subthreshold oscillations兲. Time series is shown in Fig. 2共a兲. • Two neurons are active 共spiking oscillations兲 and one neuron is suppressed 共subthreshold oscillations兲. Time series is shown in Fig. 2共b兲. • The regime of synchronous in-phase spiking oscillations of all three neurons 共x1 = x2 = x3兲. Time series is shown in Fig. 2共c兲. • Various regimes of sequential activation of the neurons. Time series are shown in Figs. 2共d兲–2共f兲. III. DISTRIBUTION OF THE CONTROL PARAMETERS SPACE
Figures 3共a兲 and 3共b兲 present the bifurcation diagram in the plane 共g1 , g2兲 of the regimes in systems 共1兲 and 共2兲. For better representation of the main results we assume g1 = g12 = g23 = g31 for a clockwise coupling, and g2 = g13 = g32 = g21 for a counterclockwise coupling. Because of identical neurons in the ensemble, the diagram is symmetric with respect to the diagonal line, which is characterized by g1 = g2. Figure 3共b兲 is the detailed area in Fig. 3共a兲. At sufficiently large and symmetric 共g1 ⬇ g2兲 couplings 共region A兲, six limit cycles can be observed. First, three limit cycles 共we denote them L11,2,3兲 correspond to the dynamics when one of the three neurons produces periodic spikes and suppresses the spiking activity of the two other neurons 关Fig. 2共a兲兴. Second, three limit cycles 共we denote them L21,2,3兲 correspond to the dynamics when one of the three neurons is suppressed 共subthreshold oscillations兲 by the other two active 共in-phase spiking oscillations兲 neurons 关Fig. 2共b兲兴. At
transition from region A to region B, the limit cycles L21,2,3 on the curve h2 disappear through the saddle-node limit cycle bifurcation 共one of real multipliers reaches a value of +1兲. In region B, systems 共1兲 and 共2兲 have only three limit cycles L11,2,3. In region C, in systems 共1兲 and 共2兲 there exists only one regime: periodical sequential activation of all neurons 关Fig. 2共d兲兴. The transition from region B to region C 共boundary line h1兲 is very important because it corresponds to the most realistic values of the parameters: one coupling is sufficiently strong 共strong inhibition兲 and the other coupling is rather small or absent. For this reason a detailed description of the bifurcation on line h1 will be given now. Region D is the region of coexistence of seven limit cycles: three limit cycles L11,2,3, three limit cycles L21,2,3, and one limit cycle corresponding to synchronous in-phase spiking oscillations of all three neurons 关we denote it by L3; the time series are shown in Fig. 2共c兲. In region E, four stable limit cycles can be observed: the three cycles L21,2,3 and limit cycle L3. The transition from D to E is accompanied by a subcritical Neimark–Sacker bifurcation of the limit cycles L11,2,3. Region F has a complex structure. There are areas there that the three limit cycles L21,2,3 coexist with the limit cycle L3. In the rest there is the coexistence of the sequential dynamics and the stable limit cycle L3. The transition from region E to region F leads to the disappearance of the three limit cycles L21,2,3 through the saddle-node limit cycle bifurcation and the appearance of sequential dynamics. A more detailed description of such a transition will be given below. In region G, only the limit cycle L3 corresponding to identical behavior 共x1 = x2 = x3兲 is stable. On the boundary of curve h3 共transition from F to G兲, the limit cycle corresponding to the sequential dynamics disappears. One of the multipliers of this cycle reaches the value of ⫺1, i.e., the stable limit cycle merges with the saddle limit cycle of doubled period. However, on the top boundary of region F 共curve h4兲, limit cycle L3 loses stability via a subcritical Neimark–Sacker bifurcation. Finally on the curve h2, between regions E and G, three limit cycles L21,2,3 disappear through the saddle-node bifurcation. Hence, region G is the region where only one limit cycle L3 exists. Systems 共1兲 and 共2兲 also have areas in the diagram 共Fig. 3兲 where there exist three limit cycles which correspond to the antiphase synchronous spiking activity of two neurons and the subthreshold oscillation of one neuron. IV. NEIMARK–SACKER BIFURCATION: EMERGENCE OF HETEROCLINIC SEQUENCE
In order to study the bifurcation leading to the appearance of the regime of sequential switching on line h1 关Fig. 3共a兲兴, we calculated the dependencies of the multipliers of the limit cycles on the relation of coupling coefficients ␣i = gij / g ji, where gij are the coefficients of the clockwise coupling and g ji are the coefficients of the counterclockwise coupling 共in experiments the coefficients of the counterclockwise coupling remain constant and are equal to 0.5兲. It was found that the pair of complex multipliers 1,2 of the limit cycle L11 共further denoted by L1兲 reaches the unit modulus with a decrease in the relation of synaptic couplings ␣1 = g12 / g21 共Fig. 4兲.
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FIG. 2. 共Color online兲 共a兲 One neuron is active and suppresses the activity of the two other neurons. Parameters: gij = g ji = 0.5 and i , j = 1 , . . . , 3. 共b兲 Two neurons are active and suppress the activity of the other neuron Parameters: gij = g ji = 0.3 and i , j = 1 , . . . , 3. 共c兲 Synchronous in-phase 共x1 = x2 = x3兲 spiking mode. Parameters:gij = g ji = 0.03 and i , j = 1 , . . . , 3.; 共d兲 Periodical sequential activation of the neurons. Parameters: g12 = g23 = g31 = 0.5, g21 = g13 = g32 = 0.05, and 共␣1 = ␣2 = ␣3 = 0.1兲. 关共e兲 and 共f兲兴 Transient sequential activation of the neurons. Parameters: 共e兲 g12 = 0.06, g23 = 0.07, g31 = 0.0732, g21 = g32 = g13 = 0.5共␣1 = 0.12, ␣2 = 0.1404, ␣3 = 0.1464兲. 共f兲 g12 = g23 = 0.06, g31 = 0.0732, g21 = g32 = g13 = 0.5共␣1 = ␣2 = 0.12, ␣3 = 0.1464兲.
The eigenvectors of the monodromy matrix corresponding to the multipliers 1,2 depend only on x2 and y 2. Therefore we are able to introduce Poincaré section which allows us to study the possible bifurcation in systems 共1兲 and 共2兲 in detail. Mapping of the plane ⌸1 = 兵x1 = 0 , y 1 = −0.18, z1 = 0 , z2 = 0.01, x3 = −0.9, y 3 = −0.32, z3 = 0.018其 to itself 共all values of the variables were chosen on the limit cycle L1 except the variables x2 , y 2兲 allows us to detect the existence of the saddle torus T1. Figure 5 shows the sections of the saddle torus T1 with the plane ⌸1. From region A 关Fig. 5共a兲兴, all trajectories go to the stable limit cycle L1 共infinite spiking oscillation of the first element, i.e., fixed point in mapping of plan ⌸1 to itself兲. From region B, all trajectories go to the stable limit cycle L2 共infinite spiking oscillations of the second element兲. When decreasing
␣1, the saddle torus T1 goes to the stable limit cycle L1, and at the bifurcation value of ␣1, it merges with the limit cycle and passes its instability 关Fig. 5共b兲兴. Thus, a subcritical Neimark–Sacker bifurcation takes place.7 The numerical investigation of different initial conditions shows that before the bifurcation all trajectories from the vicinity of the unstable saddle torus go to the stable limit cycle L1 or to the stable limit cycle L2 关Figs. 5共a兲, 5共F兲, 5共i兲, 5共g兲, and 7兴. Figure 6 shows the unstable torus T1 and the stable limit cycle L1 in the subspace of the transformed coordinates 1 = x1 + x2 cos共兲, 2⫽y1⫹x2 sin共兲, 3⫽y2⫹10z1, ⫽arctg共y˙1/x˙1兲. Figure 7 shows a few trajectories that go from the vicinity of the unstable torus T1 to the stable limit cycle L2. For a better representation, the trajectories were plotted in two sub-
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FIG. 3. 共Color online兲 Bifurcation diagram of regimes in an ensemble of three inhibitory coupled neurons. Region A: coexistence of three limit cycles L11,2,3 关Fig. 2共a兲兴 and three limit cycles L21,2,3 关Fig. 2共b兲兴. Region B: coexistence of three limit cycles L11,2,3. Region C: periodic sequential switching of activity between all neurons 关Fig. 2共c兲兴. Region D: coexistence of three limit cycles L11,2,3, three limit cycles L21,2,3, and limit cycle L3 关Fig. 2共d兲兴. Region E: coexistence of three limit cycles L11,2,3 and limit cycle L3. Region F: region with complex structure. The black areas in the inserted figure correspond to the coexistence of the three limit cycles L21,2,3 with the limit cycle L3. The white regions are the areas of the coexistence of the sequential dynamics and the stable limit cycle L3. Region G: the existence of limit cycle L3.
spaces: the trajectories situated near T1 关subspace 共1 , 2 , 3兲兴 and phase points going to the stable limit cycle L2 关subspace 共x2 + x1 , y 2 , 10z1兲兴. It is necessary to notice that such bifurcation is typical for the other limit cycles L2 and L3. With decreasing g23 and g31, the subcritical Neimark–Sacker bifurcation takes place for L2 and L3 correspondingly. The behavior of the trajecto-
FIG. 4. 共Color online兲 Real and imaginary parts of multipliers 1,2 of the limit cycle L1. At ␣1 ⬇ 0.1362, the absolute values of the multipliers are equal to 1.
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FIG. 5. 共Color online兲 共a兲 Mapping of the plane ⌸1 to itself. Red line t1: intersection of the saddle torus T1 with plane ⌸1. From region A, all trajectories go to the stable limit cycle L1 共infinite spiking oscillation of the first element, i.e., fixed point in mapping of the plane ⌸1 to itself兲. From region B, all trajectories go the stable limit cycle L2 共infinite spiking oscillations of the second element兲. ␣1 = ␣2 = ␣3 = 0.1384. 共b兲 Intersections of the torus T1 with plane ⌸1 at different values of the coupling strength ␣1 = 0.1464, 0.1404, 0.1384, and 0.1366. When decreasing ␣1, the saddle torus T1 becomes closer to the stable limit cycle L1, and at a bifurcation value of ␣1, it merges with the limit cycle and passes it its instability.
ries at ␣1 = 0.1384, ␣2 = 0.1404, and ␣3 = 0.1464 共before bifurcations of each limit cycle L1,2,3兲 is illustrated in Fig. 8. The red curves t2 and t3 are the intersections of the saddle tori T2 and T3 with planes ⌸2 and ⌸3 correspondingly 共planes ⌸2 and ⌸3 were chosen in analogous way as ⌸1兲. The black line with the arrow, which goes from T1 to L2, represents the set of trajectories that go from the vicinity of saddle torus T1 to the stable limit cycle L2 共Fig. 7兲. In Refs. 8 and 9 it was shown that heteroclinic orbits and sequences of heteroclinic orbits between saddle points in the phase space of dynamical system are the mathematical image
FIG. 6. 共Color online兲 Illustration of the saddle torus T1 and the stable limit cycle L1 共green curve兲 in the subspace 共1 , 2 , 3兲. The intersections 共closed blue curves兲 of torus T1 with different planes are shown.
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FIG. 7. 共Color兲 共a兲 Saddle torus T1 and a few trajectories going from the vicinity of T1 to the stable limit cycle L2. 共b兲 Continuing of the trajectories plotted in 共a兲: the trajectories go to the stable limit cycle L2. Here ␣1 = ␣2 = ␣3 = 0.1464.
of sequential activity in the networks modeled by a modified Lotka–Volterra model 共rate model兲. In our case the arising of stable heteroclinic orbits is also at the origin of the sequential firing. Let us consider the case when ␣1 ⬇ 0.1362 is approximately equal to the critical value when the subcritical Neimark–Sacker bifurcation takes place and the saddle torus T1 merges with the stable limit cycle L1. Before bifurcation, the set of trajectories goes from the vicinity of T3 to L1. Hence, at the moment of the bifurcation between the saddle torus T3 and the unstable limit cycle L1, the set of heteroclinic orbits appears. Such a set of heteroclinic orbits between saddle modes is the mathematical image in the phase space of the sequential switching of activity. The schematic illustrations of the trajectories and time series are shown in Figs. 9 and 2共e兲 correspondingly. The initial conditions in the vicinity of saddle torus T3 关in region B in Fig. 5共a兲兴 provide finite oscillations of the third element 关Figs. 9 and 2共e兲兴. Next, due to the instability of the torus T3, the phase point leaves the vicinity of T3 near the heteroclinic orbit and goes to the unstable L1 共black line with arrow in Fig. 9兲. Then, due to the instability of L1, the phase point remains located at the vicinity of L1 for a certain time. This fact provides finite oscillations of the first element and suppression of spiking oscillations in the other elements. Finally, the phase point leaves the vicinity of L1 and goes to the stable limit cycle L2 共infinite oscillations of the second element and suppression of the other elements兲. When further decreasing ␣1, a heteroclinic orbit between T3 and L1 disappears but the switching behavior remains.8,9 So we can claim the existence of a SHC.10
FIG. 8. 共Color online兲 Schematic illustration of the trajectories at ␣1 = 0.1384, ␣2 = 0.1404, and ␣3 = 0.1464. The black lines with the arrows illustrate the set of trajectories going from the vicinity of the saddle tori to the stable limit cycles 共Fig. 7兲.
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FIG. 9. 共Color online兲 Illustration of the trajectories for ␣1 = 0.1344, ␣2 = 0.1404, and ␣3 = 0.1464. Sequential switching of the activity arising from a heteroclinic orbit formation 关time series presented in Fig. 2共e兲兴.
When simultaneously decreasing ␣1 and ␣2 to the bifurcation value, a sequence of heteroclinic orbits between saddle modes arises: 共i兲 the heteroclinic orbit between saddle torus T3 and the saddle limit cycle L1 and 共ii兲 the heteroclinic orbit between the saddle limit cycle L1 and the saddle limit cycle L2. Such a sequence also collapses when further decreasing ␣1 and ␣2, but the heteroclinic channel remains stable up to zero values of the coupling strength. Schematic presentations of the trajectories and the time series are shown in Figs. 10 and 2共f兲, respectively. The generation of sequential activity is also finite in time, but now it covers all neurons. Let us notice that heteroclinic channels are constructions in the phase space capable at describing transient generation of bursting waves in neuronal ensembles, unlike stable limit cycles, which are images of periodic activity. Finally, simultaneous decrease in all three coupling coefficients g12, g23, and g31 leads to the formation of heteroclinic orbits between the saddle limit cycles L1, L2, and L3. When further decreasing the conductances, the heteroclinic contour collapses and in its vicinity stable limit cycle appears. It is the image of a periodical sequential activity occurring in the ensemble 关Figs. 11 and 2共d兲兴. Note that a similar bifurcation of the formation of a stable limit cycle, appearing as a destruction of the heteroclinical contour, is analytically studied in Refs. 6 and 8. On the boundary h2 between regions E and F, a transition from the periodic dynamics to sequential switching activity was also observed. Remember that in region E three stable limit cycles L21,2,3 coexist. Each limit cycle corresponds to the periodic spiking activity of two neurons and a subthreshold
oscillation of the third neuron 关the time series are shown in Fig. 2共b兲兴. It was found that in the phase space other three limit cycles exist. However they are saddle cycles. The stable manifolds of these cycles separate the basins of attractions of stable cycles L21,2,3. When a decrease in the ratio g1 / g2 on the boundary h2, one of the multipliers of stable limit cycles reaches the value of +1. It means that each stable limit cycle merges with the saddle cycle. At this moment the heteroclinic contour arises.6 When further decreasing g1 / g2, this contour collapses and irregular behavior corresponding to sequential bursting activity is set, as shown in Fig. 12. V. CONCLUSION
The modulation instability that is related to subcritical Neimark–Sacker bifurcation is a general mechanism of slow oscillation generation. In an inhibitory network with nonsymmetrical connections, the period of such slow oscillations is determined by cycling inhibition and does not depend on the details of the neuronal spiking activity. No one neuron can be pointed as the leader of the rhythm. This is a winnerless competition principle that has been suggested in Ref. 11. The results of this paper showed that the closed heteroclinic contour 共mathematical image of sequential switching in the network of inhibitory coupled spiking neurons兲 is a structurally stable object. The structural stability of the discussed slow oscillation is an important point for understanding the possible mechanisms of information processing in the brain. In particular, the interaction between spiking and bursting dynamics is de-
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FIG. 10. 共Color online兲 Trajectories corresponding to ␣1 = ␣2 = 0.1344 and ␣3 = 0.1464. Sequential switching of the activity arising from a heteroclinic orbit formation 关time series presented in Fig. 2共f兲兴.
FIG. 11. 共Color online兲 Trajectories corresponding to ␣1 = ␣2 = ␣3 = 0.1344. The periodic generation of sequential activity is shown 关time series presented in Fig. 2共d兲兴. A limit cycle arises in the vicinity of the heteroclinic sequence between the saddle limit cycles.
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RFBR, under Project Nos. 06-02-16596, 08-02-97049, and 08-02-92004. J. A. K. Suykens acknowledges support from K. U. Leuven, the Flemish government, FWO, and the Belgian federal science policy office 共FWO Contract No. G.0226.06, CoE Contract No. EF/05/006, GOA AMBioRICS, IUAP DYSCO, and BIL/05/43兲. M. I. Rabinovich acknowledges support from ONR under Grant No. N00014-071-0741.
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FIG. 12. 共Color online兲 Irregular sequential activation of neurons in region F.
termined to be the mechanism of working memory, different coding strategies in the sensory systems, the mechanisms of motor command generation, and neural microcircuits coordination.2 Synchronization of the ensemble of microcircuits, which generates the rhythm in different parts of the brain, can be the origin of different brain rhythms.1 While we have investigated a minimal inhibitory network in this paper, the phenomenon of sequential switching activity of spiking neurons between quasistationary states, which is typical for a heteroclinic contour, has been observed in vivo in the gustatory cortex12 and other systems.13 ACKNOWLEDGMENTS
We thank V. Afraimovich and V. Belykh for fruitful discussions. This work is done with financial support from
G. Buzsáki, Rhythms of the Brain 共Oxford University Press, Oxford, 2006兲. 2 M. I. Rabinovich, P. Varona, A. I. Selverston, and H. D. I. Abarbanel, Rev. Mod. Phys. 78, 1213 共2006兲. 3 R. H. R. Hahnloser, A. A. Kozhevnikov, and M. S. Fee, Nature 共London兲 419, 65 共2002兲. 4 O. Mazor and G. Laurent, Neuron 48, 661 共2005兲. 5 J. Huxter, N. Burgess, and J. O’Keefe, Nature 共London兲 425, 828 共2003兲. 6 T. Nowotny and M. I. Rabinovich, Phys. Rev. Lett. 98, 128106 共2007兲. 7 E. Ott, Chaos in Dynamical Systems 共Cambridge University Press, Cambridge, 1993兲. 8 V. S. Afraimovich, M. I. Rabinovich, and P. Varona, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1195 共2004兲. 9 V. S. Afraimovich, V. P. Zhigulin, and M. I. Rabinovich, Chaos 14, 1123 共2004兲. 10 M. I. Rabinovich, R. Huerta, P. Varona, and V. S. Afraimovich, PLOS Comput. Biol. 4, e1000072 共2008兲. 11 M. Rabinovich, A. Volkovskii, P. Lecandra, R. Huerta, H. D. I. Abarbanel, and G. Laurent, Phys. Rev. Lett. 87, 068102 共2001兲. 12 L. M. Jones, A. Fontanini, B. F. Sadacca, P. Miller, and D. B. Katz, Proc. Natl. Acad. Sci. U.S.A. 104, 18772 共2007兲. 13 M. Rabinovich, R. Huerta, and G. Laurent, Science 321, 48 共2008兲. 1
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Hypothesis test for synchronization: Twin surrogates revisited M. Carmen Romano,1,2,a兲 Marco Thiel,1 Jürgen Kurths,3 Konstantin Mergenthaler,4 and Ralf Engbert4 1
Department of Physics, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom Institute of Medical Sciences, University of Aberdeen, Aberdeen AB25 2ZD, United Kingdom 3 Potsdam Institute for Climate Impact Research, 14412 Potsdam, Germany and Institute of Physics, Humboldt University Berlin, 12489 Berlin, Germany 4 Computational Neuroscience, Department of Psychology, University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany 2
共Received 15 December 2008; accepted 29 December 2008; published online 31 March 2009兲 The method of twin surrogates has been introduced to test for phase synchronization of complex systems in the case of passive experiments. In this paper we derive new analytical expressions for the number of twins depending on the size of the neighborhood, as well as on the length of the trajectory. This allows us to determine the optimal parameters for the generation of twin surrogates. Furthermore, we determine the quality of the twin surrogates with respect to several linear and nonlinear statistics depending on the parameters of the method. In the second part of the paper we perform a hypothesis test for phase synchronization in the case of experimental data from fixational eye movements. These miniature eye movements have been shown to play a central role in neural information processing underlying the perception of static visual scenes. The high number of data sets 共21 subjects and 30 trials per person兲 allows us to compare the generated twin surrogates with the “natural” surrogates that correspond to the different trials. We show that the generated twin surrogates reproduce very well all linear and nonlinear characteristics of the underlying experimental system. The synchronization analysis of fixational eye movements by means of twin surrogates reveals that the synchronization between the left and right eye is significant, indicating that either the centers in the brain stem generating fixational eye movements are closely linked, or, alternatively that there is only one center controlling both eyes. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3072784兴 In a typical laboratory experiment, in which phase synchronization of two systems is studied, the coupling strength between both systems is systematically increased, until both systems adapt their rhythms, and hence, become phase synchronized. In the case of passive experiments, it is not possible to systematically vary the coupling strength. This is the case in many natural systems, such as, the synchronization among the electrical activity of different brain areas. There, we have only access to one single value of the coupling strength. Computing the phase synchronization index in these cases is not enough to assess the statistical significance of the result. The method of twin surrogates has been proposed to overcome this problem, allowing the performance of a hypothesis test that assess the significance of the result. In this paper, we revisit the method of twin surrogates and derive new analytical expressions for the number of twins depending on the size of the recurrence neighborhood and the number of points of the trajectory. These results allow us to determine the optimal parameters for the generation of twin surrogates, which is a very relevant problem in the case of experimental data. Moreover, we validate the method of twin surrogates comparing the generated surrogates to “natural” surrogates in an experimental system consisting of fixational eye movements, a兲
Electronic mail: m [email protected].
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and show that the phase synchronization of the left and right fixational eye movements is statistically significant.
I. INTRODUCTION
Synchronization of complex systems has been intensively studied in the last decade. This nonlinear phenomenon has been found in numerous technical and natural systems.1 Recently, the conditions for synchronizability in complex networks has become a main focus of research.1 In spite of the large number of papers about this topic, the problem of synchronization analysis of experimental data in passive experiments remains an open problem. Passive experiments are those in which it is practically impossible to systematically change the main parameters responsible for synchronization: the coupling strength or the frequencies of the two or more interacting systems. This is the case in many natural systems, such as, in geophysical and neurophysical ones. For example, synchronization is often analyzed between the electrical activity from different brain areas. In such cases, we obtain just one value for the synchronization index, and then, it is difficult to statistically judge whether the result is significant or not. In one of the standard textbooks for synchronization,1 the authors state: “The general problem is, what kind of information can be obtained from a passive experiment. In particular, a natural question appears, whether one can detect synchronization by analyzing bivariate data.
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Generally, the answer to the formulated question is negative.” In Ref. 2 the authors encountered the same problem when analyzing phase synchronization 共PS兲 between the heartbeats of pregnant women and their fetuses. To assess the significance of their results, they computed the synchronization index between a mother and the fetus of another pregnant surrogate woman. By means of this analysis, they obtained statistically significant results. However, the different women were shown to have rather different distributions of times between consecutive heartbeats, and therefore, it was not clear whether the synchronization results were significant just due to the physiological differences between the women. In order to overcome this problem, we have recently proposed a mathematical algorithm to generate trajectories which correspond to the same underlying system but starting at different initial conditions.3 This method, called twin surrogates 共TS兲, generates an independent copy of the whole system. If we now compute the synchronization index between one subsystem from the original, and the other subsystem from the surrogate system 共just as in the case of the pregnant women兲, we are able to assess the significance of the results. By means of this algorithm we avoid the difficulties with different statistical properties in different subjects or nonstationarities in the case of comparison between different realizations of the same subject. Therefore, the method of twin surrogates allows testing for synchronization of complex systems even in the case of passive experiments, which is a prevalent problem in synchronization research. In this paper, we first review the method of twin surrogates 共Sec. II兲 and then thoroughly analyze the influence of the parameters of the method on the quality of the generated surrogates. We derive analytical expressions for the average number of twins in the underlying trajectory depending on the parameters of the method and compare the theoretical expressions with numerical simulations 共Sec. III兲. Moreover, we compute several linear and nonlinear statistics for the surrogates and compare them with the ones obtained from trajectories starting at different initial conditions generated from the equations of the underlying system 共Sec. IV兲. Having done all these computations, allows us to choose optimal parameters of the method to generate twin surrogates in the case of experimental data. Hence, we exemplify in Sec. V how to apply the method of twin surrogates to a passive experiment: phase synchronization of fixational eye movements. This experiment is very appropriate to validate the twin surrogate technique in the case of experimental data, because measurements from 21 subjects with 30 trials per person are available. Then, we can compare the generated twin surrogates to the different trials performed by the same subject. But first of all, we review the twin surrogates algorithm in the next section. II. ALGORITHM FOR THE GENERATION OF TWIN SURROGATES
The algorithm to generate twin surrogates is based on the recurrence matrix Ri,j = ⌰共␦ − 储xជ 共i兲 − xជ 共j兲储兲,
i, j = 1, . . . ,N,
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FIG. 1. Recurrence plot of a trajectory from the Lorenz system 关Eq. 共A2兲兴.
where ⌰共·兲 denotes the Heaviside function, 储 · 储 a norm 共e.g., Euclidean or maximum norm兲, and ␦ is a predefined threshold. xជ 共i兲 denotes the vector of the trajectory of the system in phase space at time t = i⌬t, with ⌬t being the sampling time of the trajectory and i = 1 , . . . , N. In the case that only a scalar time series has been observed, the trajectory of the system has to be reconstructed using some embedding technique, such as, the delay coordinates.4,5 Coding the “1’s” in the matrix as black dots and the “0’s” as white ones, we obtain the recurrence plot 共RP兲 of the trajectory. The method of RPs was introduced in Ref. 6 to visualize the trajectories of dynamical systems in phase space. This method and the related “Recurrence Quantification Analysis” have proven to be very useful for the analysis of data, as can be shown by the numerous publications in many different fields of research.7 In Fig. 1 the RP of a trajectory of the Lorenz system in the chaotic regime 关Eq. 共A2兲兴 is represented for illustration. Note that the RP consists mainly of diagonal lines of different lengths. A diagonal line indicates that the trajectory recurs to the neighborhood of a former visited point of phase space, and that the trajectory evolves similarly to the past during a certain time interval, which is given by the length of the diagonal line. Since the system is chaotic, after some time interval two segments of the trajectory starting at slightly different initial conditions diverge, and therefore, the diagonal lines in the RP are interrupted. There are different statistics based on the distribution of the diagonal lines in RPs, which are the basis for the RQA. Furthermore, it has been shown that it is possible to estimate several invariants of the dynamics using the recurrence matrix,8 and even the rank order of a univariate time series can be reconstructed from its recurrence matrix.9,10 These facts suggest that the recurrence matrix contains the topological information about the underlying system. Hence, a first idea for the generation of surrogates is to change the structures in a RP consistently with the ones produced by the underlying dynamical system and then reconstruct the trajectory from the modified RP. Furthermore, we use the fact that in a RP there are identical columns, i.e., Rk,i = Rk,j ∀ k. This is because two different points of the trajectory can have exactly the same set of neighbors with respect to the threshold ␦. Thus, there are points which are not only neighbors 关i.e., 储xជ 共i兲 − xជ 共j兲储 ⬍ ␦兴, but which also share the same neighborhood. These points are called twins. Twins are special points of the time series as they are dynamically
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indistinguishable considering their neighborhoods but in general different and hence, have different pasts and, more important, different futures. The key idea of how to introduce the randomness needed for the generation of surrogates of a deterministic system is that one can jump randoml