Dynamical Systems and Methods

Dynamical Systems and Methods

Albert C.J. Luo • Jos´e Ant´onio Tenreiro Machado Dumitru Baleanu Editors

Dynamical Systems and Methods

123

Editors Albert C.J. Luo Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1805 USA [email protected]

Jos´e Ant´onio Tenreiro Machado Department of Electrical Engineering Institute of Engineering Polytechnic Institute of Porto Rua Dr Ant´onio Bernadino Almeida 431 4200-072 Porto Portugal [email protected]

Dumitru Baleanu Mathematics and Computer Sciences Faculty of Art and Sciences Cankaya University Ankara Turkey [email protected]

ISBN 978-1-4614-0453-8 e-ISBN 978-1-4614-0454-5 DOI 10.1007/978-1-4614-0454-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011937445 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Nonlinear systems and methods for mechanical, electrical and other physical systems are present in this book along with nonlinear dynamics and mathematical methods Also covered are the nonlinear phenomena in physical systems. The aims of this edited book are to show significant achievements in nonlinear science and complexity, including nonlinear dynamics, chaos, discontinuous systems, fractional dynamics, economical, social and biological systems, and so on. Topics observed in this book are based on the those reported at the third Conference on nonlinear science and complexity (NSC), held in Ankara, Turkey from July 27–31, 2010. Due to the impact of topics on a very wide spectrum of problems in science and engineering, this conference provided a place to exchange recent developments, discoveries and progresses on nonlinear science and complexity. This conference is the continuation of the first 2006 Conference on Nonlinear Science and Complexity, held in Beijing, and the second 2008 conference on Nonlinear Science and Complexity held in Porto, Portugal. The aims of the selected papers are to present the fundamental and frontier theories and techniques for modern science and technology, and to stimulate more research interest for exploration of nonlinear science and complexity. The studies focus on fundamental theories and principles, analytical and symbolic approaches, computational techniques in nonlinear physical science and mathematics. After peer-reviews, only 20 chapters, which are divided into three parts, were selected for publication in this book. • The first part consists of six chapters about nonlinear dynamical systems. It covers parametrical excited pendulum, nonlinear dynamics in hybrid systems, dynamical system synchronization and (N+1) body dynamics. The new views different from the existing results in nonlinear dynamics will be presented in this section. • The second part is concerned with mathematical methods for dynamical systems. The conservation laws and dynamical symmetry in nonlinear differential equation will be presented as well as the invex energies in Riemannian manifolds. In addition, other mathematical methods will be presented for nonlinear dynamical systems. v

vi

Preface

• The third group discusses the nonlinear phenomena in physical problems, such as solutions, complex flows, chemical kinetics, Toda lattices and parallel manipulator. All these results provide a wide view of nonlinear dynamics existing in real worlds. The editors hope that this collection of chapters may be useful and fruitful for scholars, researchers and advanced technical members of industrial laboratory facilities, for developing new tools and products. The editors thank the Cankaya University and the Scientific and Technological Research Council of Turkey for the support needed to hold the discussions and debates and all colleagues for sharing their expertise and knowledge. for sharing their expertise and knowledge. Albert C.J. Luo Jos´e Ant´onio Tenreiro Machado Dumitru Baleanu

Contents

Part I

Nonlinear Dynamical Systems

Chaos in a Parametrically Excited Pendulum with Damping Force . . . . . . . Chunqing Lu

3

Energy and Nonlinear Dynamics of Hybrid Systems . . . .. . . . . . . . . . . . . . . . . . . . Katica R. (Stevanovi´c) Hedrih

29

Characteristics Diagnosis of Nonlinear Dynamical Systems . . . . . . . . . . . . . . . . Liming Dai and Lu Han

85

Synchronization of Two Coupled Phase Oscillators .. . . . .. . . . . . . . . . . . . . . . . . . . 105 Yongqing Wu, Changpin Li, Weigang Sun, and Yujiang Wu Chaotic Synchronization of Duffing Oscillator and Pendulum . . . . . . . . . . . . . 115 Albert C.J. Luo and Fuhong Min The Ring Problem of (N + 1) Bodies: An Overview . . . . . .. . . . . . . . . . . . . . . . . . . . 135 Tilemahos J. Kalvouridis Part II

Mathematical Methods

Symbolic Computation of Conservation Laws, Generalized Symmetries, and Recursion Operators for Nonlinear Differential–Difference Equations∗ . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 153 ¨ Unal G¨oktas¸ and Willy Hereman Approximate Polynomial Solution of a Nonlinear Differential Equation with Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 169 Constantin Bota, Bogdan C˘aruntu, and Liviu Bereteu Dynamical Symmetries of Second Order ODE . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 179 M.I. Timoshin

vii

viii

Contents

Invex Energies on Riemannian Manifolds .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 191 Constantin Udris¸te and Andreea Bejenaru Weyl’s Limit Point and Limit Circle for a Dynamic Systems . . . . . . . . . . . . . . . 215 Adil Huseynov Remarks on Suzuki (C)-Condition . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 227 Erdal Karapinar On the Eigenvalues of a Non-Hermitian Hamiltonian . . .. . . . . . . . . . . . . . . . . . . . 245 Ebru Ergun Part III

Nonlinear Physics

Perturbation Methods for Solitons and Their Behavior as Particles .. . . . . . 257 L.A. Ostrovsky Complex Holomorphic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 269 Constantin Udris¸te and Romeo Bercia Unsteady MHD Flow Past a Stretching Sheet Due to a Heat Source/Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 291 A.K. Banerjee, A. Vanav Kumar, and V. Kumaran Effect of Chemical Kinetics on Permeability of a Porous Rock Scaling by Concentration of Active Fluid . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 301 Tapati Dutta, Supti Sadhukhan, and Sujata Tarafdar Exciton–Phonon Dynamics with Long-Range Interaction . . . . . . . . . . . . . . . . . . 311 Nick Laskin Time Evolution of the Spectral Data Associated with the Finite Complex Toda Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 323 Aydin Huseynov and Gusein Sh. Guseinov Efficient Dynamic Modeling of a Hexa-Type Parallel Manipulator .. . . . . . . 335 Ant´onio M. Lopes

Contributors

Dumitru Baleanu Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, Ankara, Turkey, [email protected] A.K. Banerjee Department of Mathematics, National Institute of Technology, Tiruchirappalli - 620015, India, [email protected] Andreea Bejenaru Faculty of Applied Sciences, Department of MathematicsInformatics I, University Politehnica of Bucharest, Splaiul Independentei 313, Bucharest, 060042, Romania, [email protected] Romeo Bercia University Politehnica of Bucharest, Faculty of Applied Sciences, 313 Splaiul Independentei, 060042, Bucharest, Romania, r [email protected] Liviu Bereteu Faculty of Mechanics, “Politehnica” University of Timis¸oara, Blv. Mihai Viteazu, Timis¸oara, 300222, Romania, [email protected] Constantin Bota Department of Mathematics, “Politehnica” University of Timis¸oara, P-t¸a Victoriei, 2, Timis¸oara, 300006, Romania, [email protected] Bogdan C˘aruntu Department of Mathematics, “Politehnica” University of Timis¸oara, P-t¸a Victoriei, 2, Timis¸oara, 300006, Romania, [email protected] Liming Dai University of Regina, Regina, SK, S4S 0A2, Canada, [email protected] Tapati Dutta Physics Department, St. Xavier’s College, 30, Mother Teresa Sarani, Kolkata 700032, India, tapati [email protected] Ebru Ergun Department of Physics, Ankara University, 06100 Tandogan, Ankara, Turkey, [email protected] ¨ ¨ Unal G¨oktas¸ Department of Computer Engineering, Turgut Ozal University, Kec¸i¨oren, Ankara 06010, Turkey, [email protected]

ix

x

Contributors

Gusein Sh. Guseinov Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey, [email protected] Lu Han University of Regina, Regina, SK, S4S 0A2, Canada, [email protected] Katica R. (Stevanovi´c) Hedrih Head of Department for Mechanics and Faculty of Mechanical Engineering, Mathematical Institute SANU Belgrade, University of Niˇs, Knez Mihailova 36/III, 11009 - Belgrade, Serbia, [email protected] Willy Hereman Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401-1887, USA, [email protected] Adil Huseynov Department of Mathematics, Ankara University, 06100 Tandogan, Ankara, Turkey, [email protected] Aydin Huseynov Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, AZ1141 Baku, Azerbaijan, [email protected] Tilemahos J. Kalvouridis Department of Mechanics, National Technical University of Athens, School of Applied Mathematics and Physical Sciences, 5, Heroes of Politechnion Ave., Zografou Campus, 157 73, Athens, Greece, [email protected] Erdal Karapinar Atılım University, Department of Mathematics, 06836, Incek, Ankara, Turkey, [email protected]; [email protected] V. Kumaran Department of Mathematics, National Institute of Technology, Tiruchirappalli - 620015, India, [email protected] Nick Laskin TopQuark Inc., Toronto, ON, M6P 2P2, Canada, [email protected] Changpin Li Department of Mathematics, Shanghai University, Shanghai 200444, China, [email protected] Ant´onio M. Lopes Unidade de Integrac¸a˜ o de Sistemas e Processos Automatizados, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465, Porto, Portugal, [email protected] Chunqing Lu Department of Mathematics and Statistics, College of Arts and Sciences, Southern Illinois University at Edwardsville, Edwardsville, Illinois 62026, USA, [email protected] Albert C.J. Luo Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA, [email protected] Jos´e Ant´onio Tenreiro Machado Department of Electrical Engineering, Institute of Engineering, Polytechnic Institute of Porto, Rua Dr. Ant´onio Bernadino Almeida 431, 4200-072 Porto Portugal, [email protected]

Contributors

xi

Fuhong Min Nanjing Normal University, Nanjing, Jiangsu, 210042, China, [email protected] L.A. Ostrovsky Zel Technologies and University of Colorado, Boulder, Colorado, USA, [email protected] Supti Sadhukhan Physics Department, Jogesh Chandra Choudhuri College, Kolkata 700033, India, [email protected] Weigang Sun School of Science, Hangzhou Dianzi University, Hangzhou 310018, China, [email protected] Sujata Tarafdar Condensed Matter Physics Research Centre, Physics Department, Jadavpur University, Kolkata 700032, India, sujata [email protected] M.I. Timoshin Ulyanovsk State Technical University, Ulyanovsk, Russian Federation, [email protected] Constantin Udris¸te Faculty of Applied Sciences, Department of MathematicsInformatics I, University Politehnica of Bucharest, Splaiul Independentei 313, Bucharest, 060042, Romania, [email protected] A. Vanav Kumar Department of Mathematics, Vellore Institute of Technology, Chennai - 600 048, India, vanav [email protected] Yongqing Wu School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China Basic Teaching Department, Liaoning Technical University, 125105, Huludao, China, [email protected] Yujiang Wu School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China, [email protected]

Part I

Nonlinear Dynamical Systems

Chaos in a Parametrically Excited Pendulum with Damping Force Chunqing Lu

Abstract The shooting method is applied to prove that a damped pendulum with oscillatory forcing makes chaotic motions for certain parameters. The method is more intuitive than the Poincar´e map method and provides more information about when the chaos occurs.

1 Introduction Consider a simple pendulum with a mass m and a rod of length l. If its support is subjected to a vertical excitation, and there is no damping force, the equation of the motion in nondimensional form is y + δ y + (1 + γ sin ε t) sin y = 0

(1)

where γ , δ , and ε are constants with |γ | < 1. In this chapter, we always assume that all the parameters δ , ε , and γ are positive. A similar equation y + εδ y + (1 + γ sin ε t) sin y = 0

(2)

has been studied by many researchers [1,4,5,8,9]. Wiggins [9] applied the Melnikov method to obtain the chaos by showing the system has a “homoclinic tangle.” He used this equation as an example to show the equivalence of the system (2) and the Smale Horseshoe map. Hastings and McLeod [4] used topological shooting technique to prove the pendulum can have the following motion if the parameter ε is sufficiently small: for any sequence of positive integers m1 , m2 , · · · there is

C. Lu () Department of Mathematics and Statistics, College of Arts and Sciences, Southern Illinois University at Edwardsville, Edwardsville, Illinois 62026, USA e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 1, © Springer Science+Business Media, LLC 2012

3

4

C. Lu

a solution of (1) corresponding to a motion consists of exactly m1 full clockwise rotations, followed by exactly m2 full counter-clockwise rotations, then m3 full clockwise rotations, and so forth. If the sequence is finite, then eventually the pendulum stops making full rotations. In [4], Hastings and McLeod pointed out the chaotic behavior may occur for larger value of ε and for δ = 0. But, they did not provide a proof. This author generalizes their works and provided a quantitative estimate about ε and γ in the case δ = 0. This chapter studies the case δ > 0 and presents a proof of the existence of the chaotic solutions, although it still requires δ and γ to be sufficiently small. It proves that the there a set of solutions corresponding to each sequence of positive integers m1 , m2 , .... The proof applies the shooting method without considering the Poincare´e mapping to get the Smale Horseshoe. It should be mentioned that the proofs in [6,7] rely on the symmetry of the solutions of (1) when δ = 0. In the case δ = 0, the odd and even symmetry are no longer held. However, the method applied in this chapter can still be used for the undamped case δ = 0.

2 Main Result The main result of this chapter is the following theorem. Theorem. For any given ε > 0, there is a γ0 = γ0 (ε ) and δ0 = δ0 (ε ) such that for any γ < γ0 , δ < δ0 and for any sequence of positive integers m1 , m2 , · · ·

(3)

there exists at least a countably infinite set of solutions of (1) corresponding to a motion consists of exactly m1 full clockwise rotations, followed by exactly m2 full counterclockwise rotations, then m3 full clockwise rotations, and so forth. If the sequence is finite, then eventually the pendulum stops making full rotations.

3 Proof of Main Result For convenience we rescale (1) by introducing τ = ε t and y( τε ) = u(τ ) to get 1 + γ sin τ δ sin u = 0. u + u + ε ε2

(4)

We will consider the initial conditions u(0) = λ , u (0) = 0,

(5)

Chaos of a Pendulum

5

where λ ∈ (−π , 0). In the remainder of the chapter, u always denotes the solution of the initial value problem, (4)–(5). We also need to consider other initial conditions u(0) = −π , u (0) = σ

(6)

To emphasize the dependence on the initial value and parameters, we will write u = u(t, λ ) or u(t; λ , γ , δ ) depending on the context.

3.1 Lemmas and Theorems Lemma 1. Let u be a solution of (4)–(5). Then there exists a number λ = λr such that u(r, λr ) = 0 and u (r, λr ) is positive and bounded as r → ∞. Proof. It is obvious that for any λ ∈ (−π , 0) the corresponding solution u must reach the τ axis, since u ≥ 0 as long as −π < u < 0. Assume that the first zero point of u is T depending on λ . From (4) u = −

1 + γ sin τ δ sin u − u . ε2 ε

(7)

For u ∈ (−π , 0), we have 1+γ δ δ 1−γ (− sin u)u − (u )2 ≤ u u ≤ 2 (− sin u)u − (u )2 . 2 ε ε ε ε

(8)

Integrating (8) with respect to τ from 0 to T (correspondingly, u from λ to 0) yields 1−γ (cos u − cos λ ) − ε2 ≤

0

ε

(u )2 dt

u2 1+γ ≤ 2 (cos u − cos λ ) − 2 ε

Hence, 0 ≤ u ≤ Thus,

 T δ

 u λ

(9)

 T δ 0

ε

(u )2 dt.

1 2(1 + γ )(cosu − cos λ ). ε

dr √ ≤ cos r − cos λ

 2(1 + γ ) τ. ε

(10)

(11)

(12)

It follows from (11) that u (T ) ≤

1 2(1 + γ )(1 − cos λ ) ε

(13)

6

C. Lu

which implies that u (T ) is bounded for all λ ∈ (−π , 0). Also, from (12) 

ε 2(1 + γ )

 0 λ

√

dr ≤T cos r − cos λ

(14)

Make a change of variable

λ u = sin sin φ . 2 2 The integral appeared above can be written in the form (see [2]) sin

 0 λ

Denote

√

√ du = 2 cos u − cos λ



π 2

0

 0

π 2

dφ  1 − sin2 λ2 sin2 φ

  2λ  = K sin 2 1 − sin2 λ2 sin2 φ dφ

(15)

(16)

(17)

which is the complete elliptic integral of the first kind. Then the function T satisfies the following inequality   ε 2λ √ ≤T (18) K sin 1+γ 2 It is also seen that u > 0 at the first zero of u (for otherwise, u = u = 0 at the same point which gives the unique trivial solution u ≡ 0, a contradiction). This means that u (T, λ ) > 0 while u(T, λ ) = 0. Thus, by the implicit function theorem, T = T (λ ) is a continuous function of λ . Noting from (18) that K(sin2 λ2 ) → ∞ as λ → −π and K(sin2 λ2 ) → π2 as λ → 0,we see that T is a continuous mapping from (−π , 0) to an open interval containing (A1 , ∞), where A1 is a constant depending on δ , ε , and γ , by the theorem that continuous functions preserve the connectedness. This proves the lemma. Similarly, we see that

ε  2(1 + γ )

 −π 2 λ

dr √ ≤s cos r − cos λ

(19)

u (s) is also bounded where u(s) = − π2 . Also, there exists at least a sequence λn of λ such that lim u (Tλn ) exists as λn → −π . Especially, we can choose λn such that T (λn ) = mn π where mn is odd and assume mn+1 − mn ≥ 4. As λ → −π , both T (λ ) and s(λ ) diverge to infinity. This can also be proved by the theorem that the solution is continuous in its initial values. So, limn→∞ s(λn ) = ∞ and u (s(λn )) → β > 0 or at least for some subsequence. Definition. α (ε , γ , δ ) = limλn →−π u (mn π ).

Chaos of a Pendulum

7

Lemma 2. For given ε and γ , there exists a value δ1 such that for δ ∈ [0, δ1 ) if u(s, λ ) = − π2 and u(T, λ ) = 0 at the first time, then T (λ ) − s(λ ) is bounded as λ → −π . Proof. From (8), we see that from u = − π2 to u ∈ (− π2 , 0]. 1−γ cos u − ε2 ≤

 T δ s

ε

(u )2 dt

u2 u (s)2 1+γ − ≤ 2 cos u − 2 2 ε

 T δ s

ε

(u )2 dt

where u(s) = − π2 . Since u is bounded by (11), we may choose δ small enough, say δ < δ1 so that

T δ  2  T δ 1+γ u (s)2 s ε (u ) dt ≤ 4 s ε ε 2 dt ≤ 4

2

1−γ u (s)2  2 ≤ (u )2 , cosu + u (s) − ε2 2 

and 

u ≥ hence

 0 − π2



(u (s))2 2(1 − γ ) , cosu + 2 ε 2 du

2(1−γ ) ε2



cosu + (u (s)) 2

2

≥ T − s.

(20)

(21)

(22)

from which (T − s) ≤ uπ(s) is bounded. Summing up, we see that as λ → −π , u(τ , λ ) arrives at − π2 as τ = s(λ ) and at 0 as τ = T (λ ). Both s(λ ) and T (λ ) go to infinity as λ → −π , while the difference (T − s) keeps bounded. Lemma 3. The function s(λ ) is a decreasing function of λ . Proof. Suppose that u1 (0) = λ1 , u2 (0) = λ2 , and −π < λ1 < λ2 ≤ − π2 , and that u1 and u2 are the corresponding solutions of (4)–(5) respectively, i.e., u1 = −

1 + γ sin τ δ sin u1 − u1 ε2 ε

(23)

subject to u1 (0) = λ1 and u1 (0) = 0, and u2 = −

1 + γ sin τ δ sin u2 − u2 , 2 ε ε

(24)

subject to u2 (0) = λ2 and u2 (0) = 0. Subtracting (23) from (24) and setting w = u2 − u1 yield w δ 1 + γ sin τ u 2 + u1 w = −2 sin − w cos (25) ε2 2 2 ε

8

C. Lu

with w(0) = λ2 − λ1 > 0, w (0) = 0. From (25) we see that w ≥ 0 as long as − implies that w ≥ 0 as long as u1 , u2 ∈

u1 < −π and w < π , which 2π < u2π + π −π , − 2 . Therefore, u2 becomes − 2 before u1 does, and hence s(λ1 ) > s(λ2 ) where ui (s(λi )) = − π2 . The lemma is proved. Our goal is to choose a sequence S = {λm }m=1,2,..., of λ such that T (λm ) ↑ ∞ and u (T (λm )) → α as λm ↓ −π where α = α (ε , γ , S) > 0. At this moment, it is seen only that s(λ ) ↑ ∞ as λ ↓ −π . Since T (λ ) − s(λ ) is bounded, we define sup

λ ∈(−π ,− π2 )

{T (λ ) − s(λ )} = b.

(26)

 2b  2b Let l = 2b π + 1 where π is the greatest integer less than or equal to π , so l π > 2b and l ≥ 1. We then prove the following result.

Lemma 4. Let λ ∈ −π , − π2 . Then there exists at least a real number μ < λ such that T (μ ) = T (λ ) + 2l π . Proof. For the number T (λ ) + 2l π , there exists at least a number μ ∈ (−π , 0) such that T (μ ) = T (λ ) + 2l π . We prove μ < λ . If this is not true, then it would be that λ ≤ μ . By Lemma 3, s(μ ) ≤ s(λ ), hence T (μ ) ≤ s(λ )+ b < T (λ )+ b ≤ T (λ )+ 2l π , a contradiction. Remark. From Lemma 4, we can make a sequence {λm }m=0,1,2, ... such that as m → ∞, λm ↓ −π , and T (λm+1 )−T (λm ) ≥ 4l π for m = 0, 1, 2, .... The technique employed in this chapter is to fully study the energy function E(τ ) =

[u (τ )]2 cos u(τ ) δ − − 2 ε2 ε

 τ 0

(u )2 dt.

(27)

Differentiating (27) and applying the original (4), one gets E  (τ ) = −

γ  u (τ ) sin u(τ ) sin τ , ε2

(28)

and hence, for the initial value problem (4)–(5) E(τ ) = E(0) −

γ ε2

 τ 0

u (t) sin u(t) sintdt.

(29)

Since E  (τ ) changes sign as τ increases, we cannot expect that it, as in [4], keeps positive while u stays in [λ , 0]. Definition. The scaled increment of the energy from 0 to τ is defined as G(τ , γ , δ ) = −

 τ 0

u (t) sin u(t) sintdt.

(30)

Chaos of a Pendulum

9

Recall that we have chosen λ0 such that T (λ0 ) = m0 π where m0 is the smallest positive odd integer in the domain of the function T . Then, we set λm such that for each λm there is an odd M with T (λm ) = M π and M ↑ ∞ as λm ↓ −π . Let τ = Mπ +t and v(t) = u(τ ) = u(M π + t). Then, v = u , v = u , and v(t) satisfies equation 1 − γ sint δ sin v = 0 v + v + ε ε2

(31)

v(0) = 0, v (0) = αm ,

(32)

with

where αm = u (T (λm )), and v(−M π ) = λm , v (−M π ) = 0. With the change of variables, for any τ > 0, G(τ , γ , δ ) =

 τ −Mπ −Mπ

v (r) sin v(r) sin rdr

(33)

Since τ = M π + t, we set G(t, γ , δ ) =

 t −M π

v (r) sin v(r) sin rdr,

(34)

which is the energy change from v(0) = λm to v(t). By integration by parts, for any integer m > −M, G(mπ ) = G(mπ , γ , δ ) =

 mπ −Mπ

cos v(r) cos rdr.

(35)

Lemma 5. Let k < n be two even integers and k > −M. Assume that ε and γ are given, then there exists a δ2 > 0 such that if δ < δ2 then G(nπ ) − G(kπ ) > 0 either v(t) ∈ (−π , − π2 ) or v(t) ∈ [ π2 , π ).

Proof. Suppose v(t) ∈ −π , − π2 . Then t < 0 and n < 0. It suffices to prove that G((k + 2)π , γ ) > G(kπ , γ ) for any even integer k > −M. Let r = kπ + x. Then, G((k + 2)π ) − G(kπ )

 π   3 2

=

=

0



π 2

0

+

π

π 2

 cos x

+

2π

π

+

 2π 3π 2

 cos v(kπ + x) cos xdx

 [cosv(kπ + x) − cosv((k + 1)π − x)]− dx. [cos v((k + 1)π + x) − cosv((k + 2)π − x)]

10

C. Lu

By the mean value theorem, [cos v(kπ + x) − cosv((k + 1)π − x)] −[cosv((k + 1)π + x) − cosv((k + 2)π − x)] = − sin v(ξ )v (ξ )(2x − π ) + sinv(η )v (η )(2x − π ) = [sin v(ξ )v (ξ ) − sin v(η )v (η )](π − 2x) where ξ ∈ (kπ + x, (k + 1)π − x) and η ∈ ((k + 1)π + x, (k + 2)π − x). Thus, G((k + 2)π ) − G(kπ ) =



π 2

0

[sin v(ξ )v (ξ ) − sin v(η )v (η )](π − 2x)dx

(36)

Since   1 − γ sint δ  sin v − v sin v [sin v(t)v (t)] = (v ) cosv + − ε2 ε   δ 1 − γ sint 2 = v v cos v − v sin v − sin v < 0 ε ε2 



 2

as v ∈ (−π , − 34π ] for δ < δ21 and   1 − γ sint δ  sin v − v [sin v(t)v (t)] = (v )2 cos v + − sin v < 0 ε2 ε

(37)

for v ∈ (− 34π , − π2 ] and δ < δ22 , it follows that sin v(ξ )v (ξ ) − sinv(η )v (η ) > 0 for ξ < η , and hence [sin v(ξ )v (ξ ) − sin v(η )v (η )](π − 2x) > 0.

(38)

 (k+2)π

Thus kπ cos v costdt > 0 as long as v ∈ (−π , − π2 ] and δ < δ2 = min{δ21 , δ22 }. Similarly, if v(t) ∈ [ π2 , π ), then t > 0 and k > 0. Thus, v < 0 and [sin v(t)v (t)] = (v )2 cos v + v sin v < 0. From (36), G((k + 2)π , γ ) − G(kπ , γ ) > 0, because [sin v(ξ )v (ξ ) − sin v(η )v (η )] < 0 and (π − 2x) < 0. This proves the lemma. Lemma 6. G((−M + 1)π , γ ) is exponentially small as M → ∞. Proof. Consider (4) with the initial condition u(0) = λm and u (0) = 0, where λm satisfies T (λm ) = M π . Recall that the time spent for u from − π2 to 0 is finite, i.e., if u(s(λ )) = − π2 and u(T ) = 0 then T (λ ) − s(λ ) is bounded by the constant b defined as in the proof of Lemma 3. Thus, Tλm = Mπ implies s(λm ) > M π − b. Introduce

Chaos of a Pendulum

11

another change of variable, w = u − (−π ). So the original initial value problem becomes (1 + γ sint) sin w δ = 0, (39) w + w − ε ε2 with w(0) = β , w (0) = 0 (40) where β = λm + π > 0. We shall study the asymptotic behavior of w as β → 0. Noting that w is positive, we see that w keeps positive for t > 0. Consider the solution w from β to π2 , correspondingly, u varies from λ to − π2 . Then, for w ∈ [β , π2 ], π2 w ≤ sin w ≤ w, and hence, (1 + γ )w δ 2(1 − γ )w ≤ w + w ≤ . πε 2 ε ε2

(41)

Consequently, 4(1 − γ ) 2 (w − β 2 ) ≤ (w )2 + 2 πε 2 From this, we see

 τ δ 0

ε

(w )2 dt ≤ 2

1+γ 2 (w − β 2 ). ε2

 2(1 + γ )  ≤ . ε w2 − β 2 w

(42)

(43)

Integrating the inequality with respect to t from 0 to τ , ln from which,

w+

  2(1 + γ ) w2 − β 2 τ, ≤ β ε

√  2(1+γ ) 2 2 w+ w −β ≤ βe ε τ.

(44)

(45)

On the other hand, differentiating equation produces 1 + γ sin τ  δ γ cos τ u cosu − sin u u = − u − ε ε2 ε2

(46)

or

1 + γ sin τ  δ γ cos τ w = − w + w cos w + sin w. (47) 2 ε ε ε2 From (47), it is seen that w ≥ 0 whenever w = 0 for w ∈ (0, π2 ]. This shows that w ≥ 0 as long as w ∈ (β , π2 ], or u ≥ 0 for u ∈ (−π + β , − π2 ]. Thus, (w )2 + 2

 τ δ 0

ε

(w )2 dt ≤ (w )2 +

2δ τ (w )2 ε

(48)

12

C. Lu

From inequality (42),   2 1−γ 1−γ 2  ≥  .  ≥     2 2 ε π 1+ δ τ ε π 1 + 2δ M π w −β w

ε

Then, w+

ln

(49)

ε

  2τ  w2 − β 2 1−γ   ≥   β ε π 1 + 2δ M π

(50)

ε

Setting w =

π 2

and τ = s(λm ) ≥ M π − b in (50), we get 2  ε (Mπ −b) 2 2 w+ w −β ≥ βe

from which, we have

π ≥ βe hence,

β ≤ πe

− 2ε (M π −b)

2 ε (Mπ −b)





1−γ π 1+ 2εδ Mπ

(



1−γ π 1+2 δε M π

(

1−γ π 1+ 2εδ M π

(

),

(51)

),

(52)

) ≤ π e− 2ε (Mπ −b)



1−γ π

(53)

This shows that β is exponentially small as M → ∞. In fact, on any finite interval [0, k] ⊂ [0, M π ], w is still exponentially small. Thus, for any fixed k > 0, 1 + cos u = 2 1 + cos(−π + w) = 1 − cosw < w2 is exponentially small as M → ∞. Therefore, G((−M + 1)π , γ ) =− =2

 π 0



π 2

cos u(t) costdt = −

sin

0



π 2

0

+

 π

cos u costdt

π 2

u(t) − u(π − t) u(t) + u(π − t) sin costdt 2 2

<0 π −t) is exponentially small, since sin u(t)−u( ≈ 2 proves the lemma.

u(t)−u(π −t) 2

is exponentially small. This

Remark. The upper bound of β = w(0) = u(0, λm ) + π obtained in the above proof is independent of δ . Thus, √

w ≤ βe

2(1+γ ) τ ε

≤ π e− ε (Mπ −b) 2



1−γ π +

√

2(1+γ ) τ ε

(54)

Chaos of a Pendulum

13

for w ≤ π /2. Hence, (w )2 ≤ 2

2 1+γ 2 1+γ (w − β 2 ) ≤ 2 2 β 2 e 2 ε ε

≤ 2π 2

1 + γ − 4ε (Mπ −b) e ε2



1−γ 2 π +

√

2(1+γ ) τ ε

√

2(1+γ ) τ ε

.

This shows that both w and w are exponentially small on [N π , M π ] as M → ∞ for a fixed positive integer N. Note that the above lemmas are true for all ε , γ ∈ [0, 1) and δ < δ0 = min{δ1 , δ2 }. As γ , δ → 0, the following lemma holds. Lemma 7. For given ε > 0, limγ ,δ →0, α (ε , γ , δ ) = α (ε , 0) = ε2 . Proof. Let γ → 0 in (13). Similarly, we can prove the similar results hold for the initial value problem (4)–(6) as follows: 1. As u (0) = σ → 0+ , T = T (α ) → ∞, where u(T ) = 0 and u (T ) is bounded. 2. Let u(s) = −π /2. Then, T − s is bounded, and Sup(T − s) = c as σ → 0 3. The energy function defined by (27) is E(τ ) =

α2 γ 1 + 2− 2 2 ε ε

 τ 0

u (t) sin u(t) sintdt.

(55)

4. The  increment of the energy function can also be defined as above, G(τ , γ ) = − 0τ u (t) sin u(t) sintdt. Let k < n be two even integers and k > −M. Assume that ε and γ are given, then there exists a δ0 > 0 such that if δ < δ0 then G(nπ , γ ) − G(kπ ) > 0 either v(t) ∈ (−π , − π2 ) or v(t) ∈ [ π2 , π ). 5. σ is exponentially small, and therefore, G((−M + 1)π , γ ) is exponentially small as M → ∞ for odd integer M. To see this, we begin with (41). In this case, we consider the initial condition, w(0) = 0, w (0) = σ . Integrate the inequality (1 + γ )w  δ 2(1 − γ )w  w ≤ w w + (w )2 ≤ w πε 2 ε ε2

(56)

with respect to t from 0 to τ to get 4(1 − γ ) 2 w ≤ (w )2 + πε 2

 τ δ 0

hence, (w )2 ≤ 2

ε

(w )2 dt − σ 2 ≤ 2

1+γ 2 (w + σ1 ) ε2

1+γ 2 w ε2

(57)

(58)

14

C. Lu

where σ1 =

α2ε2 . 2(1+γ )

Then, from which 

and w+

w w2 + σ1

≤

 2(1 + γ ) , ε

√  2(1+γ ) w2 + σ1 ≤ σ1 e ε τ .

(59)

(60)

To estimate σ1 , or equivalently, σ , we consider the equation x +

δ  1+γ x − 2 x=0 ε ε

(61)

Since (1 − γ sint) sin w ≤ (1 + γ )w for w ∈ (0, π /2], it follows that

δ δ (1 + γ sint) sin w (1 + γ )w w + w − ≥ w + w − 2 ε ε ε ε2

(62)

where w is the solution of (39) with w(0) = 0, w (0) = σ . Let x(t) be the solution of (61) with x(0) = 0, x (0) = σ . Then, (1 + γ sint) sin x 1+γ δ δ x + x − ≥ x + x − 2 x = 0. ε ε2 ε ε

(63)

This shows that x is a lower solution of (39), and hence w(t) ≥ x(t) where  √   √   εσ − 4(1+γ )−δ −δ t/ε 4(1+γ )−δ −δ t/ε +e e . x(t) =  4(1 + γ ) − δ

(64)

Setting t = M π − b, one establishes σ is exponentially small as M → ∞. In the next section, we study the case γ = δ = 0, in particular, the function G(t, ε , γ ) when γ = δ = 0. 3.1.1 The Case  = ı=0 Evaluating the function G for γ , δ = 0 is very difficult. However, it is possible to estimate it for sufficiently small γ and δ . To begin with, we set γ = δ = 0 in (4) to obtain 1 (65) u + 2 sin u = 0. ε Let u(τ ) = u(T + t) = v0 (t) with T = N π is the value such that u(N π ) = 0 at the first time, where N is an integer. Then v0 (t) is the solution of (65) with the initial condition v0 (0) = 0, v0 (0) = α . (66)

Chaos of a Pendulum

15

For simplicity, we denote v(t) = v0 (t) in this section. Notice that T is either T (λ ) or T (σ ). In the former case, v (−T ) = 0, v(−T ) → −π and T = M π → ∞ as α → ε2 ; in the latter case, v(−T ) = −π , v (−T ) → 0, and T → ∞ as α → ε2 . The advantage of the case γ = δ = 0 is that the (65) with the initial condition v(0) = 0 and v (0) = ε2 has an implicit solution for v < 0 and t < 0  v

√

0

√ dv 2t = ε 1 + cosv

(67)

An elementary integration gives 1 + sin 2v t = eε . v cos 2

(68)

Noting that v ∈ (−π , π ), we find from (68), 2t

cos v =

8e ε

2t

(1 + e ε )2

− 1.

(69)

Thus, we see that v (t) = g(t, ε ) is bounded for t ∈ R. In addition, from these two expressions, we see that for fixed ε > 0, | sin v|, |v | ≈ e−t/ε as t → ∞, and | sin v|, |v | ≈ et/ε as t → −∞. Therefore, we conclude that v ∈ L2(−∞,∞) and denote g(ε ) = Define a function, G−m =

1 2

 ∞

G−m =

(70)

(sin v sint)v dt.

(71)

cos v costdt

(72)

−∞

 0

By integration by parts,

4 ε

−mπ

(v (s))2 ds =

 0 −mπ

Note that this is not the increment of energy from t = −mπ to t = 0 for the case γ , δ = 0. G−m = 8

 0 −mπ

2t

eε

2t

(1 + e ε )2

costdt = 8

 mπ 0

e− ε

2t 2t

(1 + e− ε )2

costdt

(73)

By integration by parts, 1 1 G−m =− − + 2m 4ε 1 + e− ε π 2

 mπ 0

sint 2t

1 + e− ε

dt.

(74)

16

C. Lu 1

Expanding

2t 1+e− ε

= ∑(−1)k e−

2kt ε

and integrating

sint 2t

1+e− ε

, one gets

 ∞ π ε2  G−m 1 1 k − 2km ε 1 + e . + 2 + =− − (−1) ∑ 2m 4ε ε 2 + 4k2 1 + e− ε π 2 k=1

(75)

As m → ∞, we see G−m converges. Denote limm→∞ G−m = G−∞ . Then, 

 ∞ 1 1 k 2 + ε ∑ (−1) 2 G−∞ = 4ε . 2 ε + 4k2 k=1

(76)

Noting that the even expansion of the function e− 2 has a Fourier cosine series for x ∈ [0, π ] t

∞

ε

e− 2 x ∼ a0 + ∑ ak cos kx

(77)

k=1

where   ε 1 π −ε x 2  e 2 dx = 1 − e− 2 π , π 0 πε    2 π −ε x 4 ε k − ε2 π e 2 cos nxdx = · 2 e 1 − (−1) . ak = π 0 π 4k + ε 2

a0 =

(78) (79)

Since the even expansion of the function is continuous, by the Fourier theorem, at x=π  4ε ∞   ε ε 2  1 k − ε2 π e− 2 π = − e 1 − e− 2 π + (−1) . (80) ∑ 4k2 + ε 2 πε π k=1 from which  ∞ ε 1  (−1)k e− 2 π π e− 2 π ∑ 4k2 + ε 2 = ∑ 4k2 + ε 2 + 4ε − 2ε 2 1 − e− 2 π k=1 k=1 ε

∞

ε

(81)

Thus, 

G−∞

 ∞ 1 1 k 2 + ε ∑ (−1) 2 = 4ε 2 ε + 4k2 k=1   ∞ ε επ 1 1 . + + ε2 ∑ 2 = 4ε e− 2 π 2 2 4 k=1 ε + 4k

(82)

(83)

This shows that G−∞ > 0. For v > 0 and t > 0, formulas (67)–(69) still hold. For any positive integer n, consider the function,

Chaos of a Pendulum

17

Gn =

 nπ 0

(sin v sint)v dt =

 nπ

cos v costdt.

(84)

0

Noting that v(t) = −v(−t) for t ≥ 0 in the case γ = δ = 0, we see Gn = G−n . Therefore, G∞ = limn→∞ Gn = G−∞ > 0. Denote G−∞ = 2 f (ε )

(85)

There exists a positive integer n0 such that G−n > f (ε ) and Gn > f (ε ) for all integer n ≥ n0 . Especially, we choose two even integers N0 , K0 > n0 such that G−N0 , GK0 > f (ε ), v(−N0 π ) < − π2 , v(K0 π ) > π2 , f (ε ) , 4

(86)

1 f (ε ), 32

(87)

cos(v(K0 π )) > − sin v(K0 π , α (0)) < and

1 . (88) 32 This is possible because of v(t, α (0)) → π as t → ∞. Since v > 0 for all t ∈ (−∞, ∞), we see v(t) < − π2 for t < −N0 π and v(t) > π2 for t > K0 π . Also, v (K0 π , α (0)) <

G−N0 + GK0 =

 K0 π −N0 π

(sin v sint)v dt =

 K0 π −N0 π

 K0 π −N0 π

cos v cot dt > f (ε )

(v (t))2 dt ≥ g(ε )

(89)

(90)

3.1.2 Crossing Equilibria To prove the two theorems stated in the following, we need another lemma. Lemma 8. Let u(τ ) be a solution of (4)–(5), τ ≥ 0, with the first zero u(M π ) = 0 for an odd positive integer M. If u > 0 and u < π for all τ > 0, then π − u and u are exponentially small as τ → ∞. Proof. Since u > 0, it follows that u(t) → π as t → ∞. Let w = π −u and u(t0 ) > π /2 for some t0 > 0. Then the (4) becomes (1 + γ sint) sin w δ =0 w + w − ε ε2

(91)

18

C. Lu

with w(t0 ) < π /2 and w (t0 ) < 0. Suppose x(t) is a solution of x −

2(1 − γ )x =0 ε 2π

(92)

with x(t0 ) = w (t0 ) and x (t0 ) = w (t0 ). Then, (1 + γ sint) sin w 2(1 − γ )x δ (w − x) = − w + − >0 ε ε2 ε 2π

(93)

as long as w ≥ x for 0 < x, w < π2 . Hence, w > x for t > t0 . Note that 

x = ae

2(1−γ ) t ε2π

 −

+ be

2(1−γ ) t ε2π

(94)

for some constants a and b. By the assumption u → π as t → ∞, it must be that a = 0. This shows that b < 0 and  u ≤ b

2(1 − γ ) − e ε 2π  −

π − u ≤ −be



2(1−γ ) t ε2π

2(1−γ ) t ε2π

,

.

(95)

(96)

The proof of the lemma is complete. Theorem 1. For any given ε > 0, there exists numbers γ01 and δ01 depending on ε such that for any γ ∈ (0, γ01 ) and for any δ ∈ (0, δ01 ) there is at least a sequence {λm }m=1,2,3··· of λ ∈ (−π , 0) such that solutions u(τ ; λm ) of (4)–(5) increases monotonically on some interval [0,tm ] and u(tm ; λm ) = π . Proof. Since ε > 0 is fixed, we drop the letter ε from all expressions of the functions, i.e., denote α (ε , γ , δ ) = α (γ , δ ), G(t, ε , γ , δ ) = G(t), etc. Recall that α (γ , δ ) = limm→∞ u (T (λm )) which equals limλ →−π u (T (λ )) = limλn →−π u (nπ ), where n denotes some odd integers, and limδ ,γ →0 α (γ , δ ) = 2ε . Let v(t) be the solution of (31)–(32) so that v (−M π ) = 0, v(−M π ) + π is sufficiently small. Also, v(t) denotes the solution of (65)–(66) in the last section with α (0) = 2/ε . In this case, the equation takes the form

δ 1 − γ sint u + u + sin u = 0 ε ε2

(97)

and the energy function E(t) =

(u )2 δ + 2 ε

 t 0

(u )2 d τ −

cosu . ε2

(98)

Chaos of a Pendulum

19

The change of the energy function is 

γ t  u sin τ sin ud τ ε 2 t1 γ = 2 {G(t2 ) − G(t1 )} ε

E(t2 ) − E(t1 ) =

The idea in this case is to choose the two integers N0 and K0 sufficiently large so that G(K0 π ) − G(−N0 π ) > f (ε ) where f (ε ) is given in the last section. Since v0 (t) > 0 for t ∈ (−∞, ∞) and v0 (t) → −π as t → −∞ and v0 (t, α ) → π as t → ∞. For N0 and K0 , chosen as in the above section such that v(−N0 π , α (0)) + π2 = −r1 and v(K0 π , α (0)) − π2 = r2 where r1 and r2 are some small positive constants. Set 

σ = min

 1 1 1 1 f (ε ) , r1 , r2 , sin v(K0 π , α (0)), v (K0 π , α (0)) , 2(K0 + N0 )π 2 2 2 2

(99)

there exist γ0 , δ0 > 0 and α (γ , δ ) (very close to α (0)) such that for any γ ∈ (0, γ0 ) ⊂ (0, 1) and δ < δ0 and for t ∈ [−N0 π , K0 π ], the solution v(t, γ ) of (108)–(109) satisfies 1 (100) |v(t, α (γ , δ )) − v0 (t)| < σ 2 and 1 |v (t, α (γ , δ )) − v0 (t)| < σ . (101) 2 It implies that f (ε ) , (102) |v(t, α (γ , δ )) − v(t, α (0))| < 2(K0 + N0 )π 1 π v(−N0 π , α (γ , δ )) < v(−N0 π , α (0)) + σ < − , 2 2 and

σ π > . 2 2 Furthermore, we can require γ0 and δ0 sufficiently small such that v(K0 π , α (γ , δ )) > v(K0 π , α (0)) −

sin v(K0 π , α (γ )) <

1 f (ε ), 16

(103) (104)

(105)

1 16

(106)

(v0 )2 dt < min{ f (ε ), g(ε )}/ max{N, K}

(107)

v (K0 π , α (γ )) < and  K0 π −N0 π

(v )2 dt −

 K0 π −N0 π

for all γ < γ0 and δ < δ0 .

20

C. Lu

Then for any γ < γ0 and for any δ < δ0 there exists an m0 = m0 (γ , δ ) > 0 such that |u (T (λm )) − α (γ , δ )| is sufficiently small and u(T (λm )) = 0 for m > m0 (note that the choice of γ0 and δ0 is independent of m0 ). Set τ = M π + t and u(τ ) = u(M π + t) = v(t) where M is an odd integer with T (λm ) = M π and u (T (λm )) = 0. Then, (4) takes the form v +

δ  1 − γ sin t v+ sin v = 0, ε ε2

(108)

subject to the initial conditions v(0) = 0, v (0) = α (γ , δ ).

(109)

We now consider the initial value problem (108)–(109), which is equivalent to the initial value problem (4)–(5) with λ = λm for t ∈ [−N0 π , K0 π ]. The energy function is  (v )2 δ t  2 1 E(t, λm ) = (v ) ds − 2 cos v (110) + 2 ε 0 ε The derivative of the energy function E(t, λm ) takes the form E  (t, λm ) =

γ (sin v(t) sint)v (t). ε2

(111)

In addition, we can require that for m > M0 , sin v((K0 π , u (T (λm )) <

1 f (ε ) 8

(112)

and 1 v (K0 π , u (T (λm )) < . 8

(113)

From now on, we fix this γ < γ0 . Since limλm →−π v (T (λm )) = α (γ , δ ), for the same σ given above, there exists an M0 such that for m ≥ M0 , 1 |v(t, u (T (λm )) − v(t, α (γ ))| < σ 2

(114)

for t ∈ [−N0 π , K0 π ] and cos λm = cos v(−M π ) < −1 +

γ f (ε ) 4

(115)

Chaos of a Pendulum

21

(note that γ has been chosen), where Mπ = T (λm ) is the value of t where v = 0. Thus, |v(−N0 π , u (T (λm ))) − v(−N0 π , α (0)| ≤ |v(−N0 π , u (T (λm ))) − v(−N0 π , α (γ )| +|v(−N0 π , α (γ )) − v(−N0π , α (0))| < σ , +|v(−N0 π , α (γ )) − v(−N0 π , α (0))| < σ , so   1 π  v(−N0 π , u (T (λm )) < v −N0 π , α (0) + r1 < − . 2 2 Similarly, v(K0 π , u (T (λm ))) >

π 2

(116)

(117)

for m ≥ M0 . Moreover, |v(t, u T (λm )) − v(t, α (γ ))| <

f (ε ) , 2(K0 + N0 )π

(118)

f (ε ) (K0 + N0 )π

(119)

hence |v(t, u T (λm )) − v(t, α (0))| <

for all t ∈ [−N0 π , K0 π ] and for m ≥ M0 . Recall that the choice of λm , we write T (λm ) = M π where M is an odd integer. Of course, there are infinitely many m s which make inequalities (116), (117), and (118) true. In order to compete the proof of Theorem 1, it suffices to choose one of those m s, say m for simplicity. Hence, correspondingly, T (λm ) = M π , v(−M π , γ ) = λm , v (−M π , γ ) = 0, and v(0) = 0 in terms of the new variable t. We fix γ < γ0 , δ < δ0 and an m for which the three inequalities (116), (117), and (118) hold. The energy change from v(−M π ) = λm to v(K0 π ) is E(K0 π ) − E(−M π )

(120)  δ K0 π

[v (K0 π )]2 cos v(K0 π ) cos v(−M π ) − + + (v )2 ds 2 ε2 ε2 ε −Mπ  (−M+1)π  −N π  K0 π  0 γ cosv(t) costdt. = 2 + + ε −Mπ (−M+1)π −N0 π =

(121) (122)

It is observed that the first integral is exponentially small and that the second integral is positive. As to the third integral, we have  K π   K0 π  0      −N π cos v(t, u (T (λm ))) costdt − −N π cos v(t, α (0)) costdt  0

≤

 K0 π

−N0 π

0

| cos v(t, u (T (λm ))) − cosv(t, α (0))| costdt ≤ f (ε )/4.

22

C. Lu

Thus, we have

 K0 π −N0 π

cos v(t, u (T (λm ))) costdt > f (ε ),

(123)

and hence E(K0 π ) − E(−M π ) > ≥

γ δ f (ε ) − ε2 ε

 K0 π −Mπ

(v )2 ds

γ δ f (ε ) − f (ε ). 2 ε 16ε

Also, v (K0 π ) > 0. We now can claim that v(t, u (T (λm )) must exceed π at some t > K0 π . Since K0 is even, it follows that E  > 0 for t ∈ (K0 π , (K0 + 1)π ). So E(t, λm ) > E(K0 π ) for t ∈ (K0 π , (K0 + 1)π ). In the interval (K0 π , (K0 + 1)π ) (v )2 cos v(t) δ − + 2 ε2 ε > E(K0 π ) >

 t 0

(v )2 ds

− cos v(−Mπ ) δ + ε2 ε

(124)  −Mπ 0

(v )2 ds +

γ δ f (ε ) − f (ε ). (125) 2 ε 16ε

from which (v )2 cos v(t) − 2 ε2  t δ γ δ − cosv(−M π ) ≥− (v )2 ds + + 2 f (ε ) − f (ε ). 2 ε −Mπ ε ε 16ε  −N



Noting −M0π (v )2 dt is exponentially small from the remark above and Kt 0 π (v )2 d τ is also exponentially small by the lemma in this section, we can choose δ small enough so that  δ t γ (v )2 ds < 2 f (ε ) (126) ε −Mπ 8ε and

δ γ f (ε ) ≤ 2 f (ε ). 16ε 8ε

(127)

We then see that v > 0, because cos v(−M π ) < −1 + 4γ f (ε ) implies

γ 1 − cos v(−Mπ ) γ + 2 f (ε ) > 2 + 2 f (ε ) ε2 ε ε 2ε

(128)

Chaos of a Pendulum

and hence

23

[v (t)]2 − cosv(−M π ) + γ f (ε ) cos v(t) + > 2 ε2 ε2 3γ γ + 2 f (ε ) > 2 f (ε ). 2ε 4ε

(129) (130)

Is it possible that v (t) = 0 before v = π for t > (K0 + 1)π ? Since we know that the energy change, on the interval [kπ , (k + 2)π ], is positive for even positive integers k and that E  ≥ 0 on the interval [nπ , (n + 1)π ] for any positive even integer n, it is sufficient to consider the case t ∈ (kπ , (k + 1)π ) where k > K0 is an odd integer. Suppose that E(kπ ) > E(K0 π ) for an odd k > K0 . (It is true for k = K0 + 1). Then,

δ (v )2 cos v − 2 − E(kπ ) = − 2 ε ε

 t

γ (v ) d τ + 2 ε kπ  2

 t kπ

(sin v sint)v dt

(131)

where t ∈ (kπ , (k + 1)π ). The absolute value of the energy change from t = kπ to t satisfies     γ  t γ t 1 γ   (sin v sint)v dt  < 2 (132) | sin t|dt < 2 ε 2  kπ ε kπ 64 8ε because that v < 0 and v ≥ 0 as long as v < π implies that 0 < sin v(t) < sin v(K0 π ) and 0 < v (t) < v (K0 π ) for t ∈ (kπ , (k + 1)π ). Also,

δ ε

 t kπ

(v )2 d τ ≤

πδ 4ε

(133)

This shows that for t > K0 π and as long as v ≥ 0 (note that the case v = 0 is included), (v )2 cos v γ πδ − 2 > E(K0 π ) − 2 f (ε ) − 2 ε 8ε 4ε >

(134)

1 − γ4 f (ε ) γ γ πδ + 2 f (ε ) − 2 f (ε ) − ε2 ε 8ε 4ε

(135)

from which, if δ is small enough, (v )2 γ > 2 f (ε ). (136) 2 4ε  Therefore, we conclude that v > 2 εγ2 f (ε ) as long as v ≥ 0 and v < π for

∈ ((k + 1)π , (k + 2)π ). t ∈ (kπ , (k + 1)π ). If t > (k + 1)π for k odd, then E  > 0 for t Therefore, (136) is valid. Thus, we again obtain v > 2

γ ε2

f (ε ). Repeat this  argument and by mathematical induction, we conclude that v > 2ε γ2 f (ε ) as long as

24

C. Lu

v ≥ 0 and v < π . Thus, there must be a tm depending on λm such  that v(tm , λm ) ≥ π 

for all m ≥ M0 , because v(t) = v(K0 π ) + Kt 0 π v ds > v(K0 π ) + 2t εγ2 f (ε ). Finally, we relabel the subsequence {λm}m=M0 ,... as {λm }m=1,2,... . This proves Theorem 1. Similarly, one can prove the same result holds for the second initial value problem (4)–(6). Theorem 2. For any given ε > 0, there exists numbers γ02 and δ02 depending on ε such that for any γ ∈ (0, γ02 ) and for any δ ∈ (0, δ02 ) there is at least a sequence {σm } of σ such that σm → 0 and the solution u(τ ; σm ) of (4)–(6) increases monotonically on some interval [0,tm ] and u(tm ; σm ) = π .

3.1.3 First Local Maximum Theorem 3. For any ε > 0, there exist two real numbers γ03 > 0 and δ03 > 0 depending on ε such that for any γ ∈ (0, γ03 ) and for any δ ∈ (0, δ03 ) there is at least a sequence{λm}m=1,2,3··· of λ ∈ (−π , 0) such that the solution u(τ ; λm ) of (4)–(5) increases monotonically on some interval [0,tm ] and u (tm ; λm ) = 0 with u(tm ; λm ) < π . Theorem 4. For any ε > 0, there exist two real numbers γ04 > 0 and δ04 > 0 depending on ε such that for any γ ∈ (0, γ04 ) and for any δ ∈ (0, δ04 ) there is at least a sequence {σm }m=1,2,3··· of σ such that σm → 0 and solutions u(τ ; λm ) of (4)–(6) increases monotonically on some interval [0,tm ] and u (tm ; λm ) = 0 with u(tm ; λm ) < π . Proof. We only prove Theorem 4 in the chapter, because the proof of Theorem 3 is similar. 1. The monotone property of the solution, u ≥ 0 leads to u(t) = 0 with u > 0 and u < 0. Let u(T ) = 0 with u (T ) > 0. Then T = T (σ ) and T → ∞ as σ → 0. 2. Choose σn such that T (σn ) = nπ where n is a positive even integer. Set u(t) = u(nπ + τ ) = w(τ ). Then (4)–(6) becomes 1 + γ sin τ δ w + w + sin w = 0 ε ε2

(137)

with w(−nπ ) = 0 and w (−nπ ) = σn . Since u (T ) = w (0) = ρn is bounded, we will consider the initial value problem w(0) = 0, w (0) = ρ

(138)

where ρ > 0. 3. The scaled increment of the energy function E(τ ) − E(0) =

 τ γ 0

ε2

sin u sintdt

(139)

Chaos of a Pendulum

becomes H(τ ) = −

25

 τ 0

sin w sintw dt = −

Set t = nπ + τ . H(t) = −

 t−nπ −nπ

 τ −nπ

sin w sin tw dt.

sin w sin tw dt

(140)

(141)

An identical argument as above shows that H(mπ ) − H(pπ ) < 0

(142)

for any two even integers m > p > −n and for either w ∈ (−π , −π /2) or w ∈ (π /2, π ). 4. As σn → 0, equivalently, T (σn ) → ∞, σn is exponentially small with an  −N0 π  2 exponential upper bound independent of δ . This implies −n π (w ) dt has an upper bounded independent of δ for any fixed even integer N0 as n → ∞. 5. Suppose u is a solution of (4)–(6) with u(nπ ) = 0 for some positive even integer n. If u > 0 and u < π for all τ > 0, then π − u and u are exponentially small as  τ → ∞. Also, Kτ0 π (u )2 dt is bounded with an upper bound independent of δ . 6. In this case the energy change

γ E(t2 ) − E(t1 ) = − 2 ε

 t t1

u sin τ sin ud τ = −

γ [G(t2 ) − G(t1 )]. ε2

(143)

First, choose N0 and K0 sufficiently large so that G(K0 π ) − G(−N0 π ) ≥ 3 f (ε )/2 for setting γ = δ = 0. Then choose γ0 , δ0 > 0 small enough such that G(K0 π ) − G(−N0 π ) ≥ f (ε ) for γ < γ0 and δ < δ0 . Then we will use the lemma proved in this section and lemmas in the previous sections to show that the major contribution to the change of the velocity is less than − 2γε 2 f (ε ), namely, as long as u ≥ 0 and u < π,

2  2  γ (144) u (t) − u (−N0 π ) ≤ − 2 f (ε ) 2ε This shows that w must become zero before u = π . Let γ0 = min{γ0i } and δ0 = min{δ0i } for i = 1, 2, 3, 4. Then for δ < δ0 and for γ < γ0 , Theorems 1–4 hold. We are now ready to prove the main result.

3.2 Proof of Main Theorem To begin with the proof, first let m1 = 1. Apply Theorem 1, there is a λ1 ∈ (−π , 0) such that u(t, λ1 ) crosses π increasingly. Denote the largest interval containing λ1 by (ξ1 , η1 ) such that for any λ1 ∈ (ξ1 , η1 ), u(T0 , λ1 ) = 0 and u(T1 , λ1 ) = π . Since u (T1 ) = 0, we see T1 → ∞ as λ1 → ξ1 or η1 , which can be shown by using Theorem 3.

26

C. Lu

Also, it must be that u (T1 , λ1 ) → 0 as λ1 → ξ1 or η1 . Choose a λ1 ∈ (ξ1 , η1 ) such that T1 (λ1 ) = 2kπ for a sufficiently large integer k. By the periodic property of the sine function, we set u = 2π + v and t = T1 + τ . We then apply Theorem 4. There is an μi ∈ (−π , 0) such that its first local maximum v(t, μi ) < π with v (t, μi ) = 0 for i = 0, 1. Equivalently, u(t, λ2 ) increasingly crosses π and reaches its first local maximum before 2π . Denote the largest interval of these λ2 by (ξ1 , η1 ). We see that (ξ2 , η2 ) ⊂ (ξ1 , η1 ). If m1 = 2, we want u(t) to cross 3π and ready to get downward before 4π . This time, we use apply the periodic property of the sine function and by Theorem 2 to obtain a value of λ , λ2 such that u(t, λ2 ) crosses 2π increasingly. Let u(T2 , λ2 ) = 2π and notice that T2 (λ2 ) → ∞ as λ2 → (ξ2 , η2 ), where (ξ2 , η2 ) is the largest open interval containing such λ2 . Also note that u (T2 , λ2 ) → 0. Again, by the periodic property of the sine function and by Theorem 3. Thus, we get an open interval (ξ2 , η2 ) such that u(t, λ ) increases monotonically until it crosses 3π and ready to downward before 4π for λ ∈ (ξ2 , η2 ). Continuing this argument, we obtain an open interval (ξm 1 , ηm 1 ) such that for any λ ∈ (ξm 1 , ηm 1 ), the solution u(t, λ ) of (4)–(5) increasingly crosses (2m1 − 1)π and reaches its first local maximum before the line u = 2m1 π . Denote (ξm 1 , ηm 1 ) = (λ1− , λ1+ ). To get m2 clockwise rotations, we need show that after reaching its local maximum at T1 with u (T1 ) = 0. We can choose T1 = (2m − 1)π for a sufficiently large integer m and u(T1 ) is sufficiently close to the line u = 2m2 π . Then we apply the periodic property of the sine function and set u(t) = v(τ ) + 2m2 π , t = 2mπ + τ , and w(τ ) = −w(τ ). Then, sin u = − sin w, u = −w , u = −w , and sin t = sin τ . Note that w (τ ) and u(t) satisfy the same equation. Also, u(T1 ) − 2m2 π = v(0) < 0, and v (0) = 0. To make pendulum counterclockwise 2m2 full rotations is equivalent to let w monotonically cross the line w = (2m2 − 1)π until it reaches its first local maximum before the line w = 2m2 π . The proof is similar to the above argument to get an interval (λ2− , λ2+ ) ⊂ (λ1− , λ1+ ) such that u(t, λ ) would have the movement described in the main theorem corresponding to the first two positive integers m1 and m2 if λ ∈ (λ2− , λ2+ ). Continuing the similar argument by mathematical induction, − + we get the nested interval (λn− , λn+ ) ⊂ (λn−1 , λn−1 ) ⊂ · · · ⊂ (λ2− , λ2+ ) ⊂ (λ1− , λ1+ ). − − + + Since the distance λ j−1 − λ j > 0 and λ j−1 − λ j > 0, we can prove that there must be at least one λ ∈ (λ j− , λ j+ ) for all j = 1, 2, · · · , n, · · · such that u(t, λ ) corresponds to the movement of the pendulum described by the main theorem of the chapter. The proof of the main theorem is now complete.

4 Conclusion The above proof of the existence of the chaotic motions for the pendulum with an oscillating support shows that the starting point of the pendulum (u(0) = λm ) is so important and sensitive. A very little change on λm could result in a new sequence of {mi }. Therefore, to get a desired sequence of motions, it is a very delicate work to put the pendulum in the right spot, which reflects the sensibility of the systems

Chaos of a Pendulum

27

in its initial conditions. Of course, we may study a different initial conditions such as u(0) = 0 and u (0) = λ and let λ vary, which can be transformed to the initial conditions studied in the paper. The proof in the chapter still require either γ , δ , and ε sufficiently small because when estimating the change in energy, we need an implicit solution of the reduced equations. However, numerical works show that the chaos may occur when neither is small [3, 4]. Finally, as the chaos theory indicated, there must be countably many periodic solutions and an uncountable set of nonperiodic solutions, which can be seen from the motions of the pendulum corresponding to the infinite sequence of positive integers {mi }, i = 1, 2, · · · .

References 1. Guckenheimer J, Holmes P (1982) Nonlinear oscillations, dynamical systems, and bifurcations of vectors. Springer, New York 2. Greenhill AG (1959) The applications of Elliptic functions. Dove Publishing Inc., New York 3. Hastings SP (1993) Use of “simple shooting” to obtain chaos. Phys D 62:87–93 4. Hastings SP, McLeod JB (1993) Chaotic motion of a pendulum with oscillatory forcing. Am Math Mon 100:563–572 5. Hastings SP, McLeod JB (1991) On the periodic solutions of a forced second-order equation. J Nonlin Sci 1:225–245 6. Lu C (2006) Chaotic motions of a parametrically exited pendulum. Comm Nonlin Sci Num Sim 11:861–884 7. Lu C (2007) Choas of a parametrically undamped pendulum. Comm Nonlin Sci Num Sim 12:45–57 8. Melnikov VK (1963) On the stability of the center for time periodic solutions. Trans Moscow Math Soc (Trudy) 12:3–52 9. Wiggins S (1988) On the detection and dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations. SIAM J Appl Math 48:262–285

Energy and Nonlinear Dynamics of Hybrid Systems Katica R. (Stevanovi´c) Hedrih

Abstract The transmission of energy between subsystems coupled in hybrid system is very important for different applications. For first as an introduction, by using the author’s previously published references and that of her students, a short survey of an analytical study of the energy transfer between coupled subsystems is presented as a basis of this chapter. An analytical study of the mechanical energy transfer between two coupled subsystems, as well as, between two or more coupled rotation motions is presented. For starting, an analytical analysis of the mechanical energy transfer between a linear and a nonlinear oscillators of a hybrid system (see Refs. by Hedrih (Stevanovi´c) 2002 [10, 11, 15–18, 20, 24]) in the free, as well as forced, vibrations of a different types of interconnections between subsystems is presented. Coupling element between subsystems of the considered hybrid systems are standard light elements with elastic, viscoelastic, hereditary, or creeping properties as well as dynamical constrain element realized by rolling element with inertia properties. Using Krilov–Bogolyubov–Mitropolskiy’s asymptotic method, both the solutions in the first approximation and the system of nonlinear-coupled differential equations for the corresponding number of excited amplitudes and phases of multifrequency free as well as forced regimes are derived. By means of this asymptotic approximation of differential equations for the amplitudes and phases for forced vibrations of the coupled oscillators, the mutual influence of the nonlinear harmonics and energy transient were analyzed. The Lyapunov exponents corresponding to the each of two eigen like nonlinear modes are expressed by using energy of the corresponding eigen time components. A generalization of an analytical analysis of the transfer energy between linear and nonlinear oscillators for forced vibrations with different type constraints as a

K.R. (Stevanovi´c) Hedrih () Head of Department for Mechanics and Faculty of Mechanical Engineering, Mathematical Institute SANU Belgrade, University of Niˇs, Knez Mihailova 36/III, 11009 - Belgrade, Serbia e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 2, © Springer Science+Business Media, LLC 2012

29

30

K.R. (Stevanovi´c) Hedrih

couple between two subsystems, each of them with one degree of freedom, is done. A mathematical analogy between discrete and complex discrete-continual hybrid systems is pointed out. In the second part, an analytical analysis is extended to the transfer energy between plates for free and forced transversal vibrations of a visco and nonlinear elastically connected double plate system. The analysis showed that the visco- and nonlinear elastic connection between plates caused the appearance of two-frequency like regime of time function, which corresponds to one eigen amplitude function of one mode, and also that time functions of different vibration modes are coupled, as well as energy transfer between plates in one eigen mode appear. Next, as an author’s new research result, an analytical study of the energy transfer between two coupled-like string belts interconnected by light pure elastic layer in the axially moving sandwich double belt system in the free vibrations is presented.

1 Introduction 1.1 Importance The study of the transfer of energy between subsystems coupled into hybrid system is very important for different kinds of applications. For first as an introduction, by use author’s previously published references as well as by her students, a short review of an analytical study of the energy transfer between coupled subsystems is presented as a basis of this lecture. An analytical study of the mechanical energy transfer between two coupled subsystems, as well as between two or more coupled rotation motions, also, is presented. For starting, an analytical analysis of the mechanical energy transfer between a linear and a nonlinear oscillators of a hybrid system (see Refs. by Hedrih (Stevanovi´c) [10, 11, 15–18, 20, 24, 31, 34–38, 43]), in the free, as well as forced vibrations of a different type of interconnections (see Ref. by Goroˇsko and Hedrih (Stevanovi´c) [1–4]) between subsystems is presented. Coupling elements between subsystems of the considered hybrid systems are standard light elements with elastic, viscoelastic, hereditary, or creeping properties, as well as no light dynamical constraint element realized by rolling element with inertia properties. Using well known Krilov–Bogolyubov–Mitropolskiy’s asymptotic method, [73–80] both, the solutions in the first approximation and the system of nonlinear coupled differential equations for the corresponding number of excited amplitudes and phases of multifrequency free as well as forced regimes are derived. By means of this first asymptotic approximation of ordinary differential equations for the amplitudes and phases for forced vibrations of the coupled oscillators in resonant frequency intervals, the mutual influence of the nonlinear harmonics and energy

Energy and Nonlinear Dynamics of Hybrid Systems

31

transient were analyzed. The Lyapunov exponents corresponding to the each of two eigen like nonlinear modes are expressed by using energy of the corresponding eigen time components. A generalization of an analytical analysis of the transfer energy between linear and nonlinear oscillators for forced vibrations with different type constraints as a couple between two subsystems, each of them with one degree of freedom is done. A mathematical analogy between discrete and complex discrete-continual hybrid systems (see Ref. by Hedrih (Stevanovi´c) [42]) is pointed out. Mathematical analogy and phenomenological mapping (see Refs. by Hedrih (Stevanovi´c) [34,46]) between different mechanical systems on the basic of the discretizations by subdynamics or subcomponents of dynamics are used also for transfer energy analysis. For second, the study of the transfer energy between subsystems containing deformable body coupled in hybrid system is very important for different applications. An analytical analysis of the transfer energy between plates for free and forced transversal vibrations of a viscoelastically connected double plate system is pointed out. The analytical analysis showed that the viscoelastic connection between plates caused the appearance of two-frequency-like regime of time functions, which corresponds to one eigen amplitude function of one mode, and also that time functions of different vibration modes, in linear system, are uncoupled, but energy transfer between plates in one eigen mode appears. It was shown for each shape of vibrations. Series of the two Lyapunov exponents corresponding to the one eigen amplitude mode are expressed by using energy of the corresponding eigen amplitude time component. In the same second part, an analytical analysis is extended to the transfer energy between plates for free and forced transversal vibrations of a visco- and nonlinear elastically connected double plate system. The analysis showed that the viscoand nonlinear elastic connection between plates caused the appearance of twofrequency-like regime of time function, which corresponds to one eigen amplitude function of one mode, and also that time functions of different vibration modes are coupled, as well as energy transfer between plates in one eigen mode appears. More than two resonant jumps in the amplitude-frequency as well as in phasefrequency curves appeared and caused more than two resonant jumps of the energy and corresponding influence between nonlinear modes, as nonlinear phenomena interactions. Using the analytical asymptotic approximation of the amplitudes and phases of multifrequency particular solutions of such a dynamics, it is possible to analyze transfer energy between nonlinear modes in stationary and nonstationary regimes passing through resonant frequency intervals. Next, as an author’s new research result, an analytical study of the energy transfer between two coupled-like string belts interconnected by light pure elastic layer in the axially moving sandwich double belt system, in the free vibrations is presented. On the basis of the obtained analytical expressions for the kinetic and potential energy of the belts and potential energy of the light pure elastic distributed layer numerous conclusions are derived. For the pure linear elastic double belt system no transfer energy between different eigen modes of transversal vibrations of the axially moving double belt system appears, but in each of the set of the infinite

32

K.R. (Stevanovi´c) Hedrih

numbers eigen modes, there are transfer energy between belts. The corresponding free transversal vibrations are like two frequency, when changes of the potential energy of the booth belts are four frequency, and potential energy interaction is one frequency in the each eigen mode. Changes of the kinetic energy of the both belts of the sandwich double axially moving belt system are two frequency-like oscillatory regimes with two time multiplicities of the eigen frequencies of the corresponding eigen amplitude mode.

1.2 Literature Survey 1. The study of the transfer of energy between subsystems coupled in hybrid system (see Refs. Heedrih (Stevanovi´c) [7, 8, 19–23, 25–27, 32, 33, 40, 47, 48], Hedrih (Stevanovi´c) and Simonovi´c [58, 60, 61] and Hedrih (Stevanovi´c) and Hedrih A. [51, 52]) is very important for different applications. Two papers by the author (see Refs. Heedrih (Stevanovi´c) [8, 25, 26] presents analytical analysis of the transfer of energy between plates for free and forced transversal vibrations of an elastically connected double-plate system. Energy analysis of vibro-impact system dynamics with curvilinear trajectories and no ideal constraints was done by Jovi´c in 2009 and in 2011 in his two theses [115, 120], for Magistar of science as well for doctors of sciences degrees. Potential energy and stress state in material with crack was study by Jovanovi´c and presented in his Doctor’s Degree Thesis [109, 118] in 2009. Energy analysis of the nonlinear oscillatory motions of elastic and deformable bodies was done by Hedrih (Stevanovi´c) in her doctor’s degree thesis [100,101] in 1975. The energy analysis of longitudinal oscillations of rods with changeable cross-sections was original research results in 1995 presented by Filipovski in his magistar of sciences degree thesis [11] (for all see References from list in Appendix – References), 2. When, at an international conference ICNO in Kiev in 1969, my professor of mechanics and mathematics, D. P. Raˇskovi´c ([88,89]) (see Refs. Raˇskovi´c (1965, 1985) presented me to academician Yuri Alekseevich Mitropolskiy (1917–2008) (see Refs. Mitropolskiy ([73–80]) and when I started really to understand the differences between linear and nonlinear phenomena in dynamics of mechanical real systems, I knew I was on the right path of research which enchanted me ever more by understanding new phenomena and their variety in nonlinear dynamics of realistic engineering and other dynamical systems. (First, my knowledge about properties of nonlinearity and the nonlinear function I obtained in gymnasium from my excellent professor of mathematics Draginja Nikoli´c and during my research Matura work on the subject of Nonlinear elementary functions and their graphics as a final high-school examination.) For beginning of this chapter, a review survey of original results of the author and of researchers from Faculty of Mechanical Engineering University of Niˇs (see References [97–122] from list in Appendix – References), inspired and/or

Energy and Nonlinear Dynamics of Hybrid Systems

33

obtained by the asymptotic method of Krilov–Bogolyubov–Mitropolyskiy, and as a direct influence of professor Raˇskovi´c scientific instruction and also by published Mitropolskiy’s papers and monographs [73–80], as well as publications by Kiev Mathematical institute scientists in area of nonlinear and stochastic dynamics. These results have been obtained during realizations of the series of the research projects supported by Ministry of Sciences of Serbia, Faculty of mechanical engineering University of Niˇs and Mathematical Institute SANU Belgrade (see List of Projects (period 1967–2011) in Appendix – References -List of Projects [123–134]). These results have been published in scientific journals and were presented on the scientific conferences and in the bachelor degree works (see Stevanovi´c, (1967)), Magister of sciences theses (see [99,102,104,106,110,111,113,114]), and doctoral dissertations (see (Stevanovi´c) [101,103,105,107,116–119,121,122]) supervised by Mitropolskiy (in period from 1972 to 1975) or by Raˇskovi´c (in period from 1964 to 1974), and by Hedrih in period from 1976 to 2001 year as well. In area of stochastic stability, a scientific support by series of consultation to researchers was given by S.T. Ariaratnam (Canada) and A. Tylikowski (Polad) papers. The original results contain asymptotic analysis of the nonlinear oscillatory motions of elastic bodies: beams, plates, shells, and shafts (see References by (Stevanovi´c) [5, 6, 11–15, 24, 29–35, 38–40, 62–65, 93–95]). Also, late a series of new research results are obtained by Janevski in 2003 and by Simonovic [53–61] in 2008 an in 2011. The multifrequency oscillatory motion of elastic bodies was studied. Corresponding systems of partial differential equations of system dynamics, as well as system of first approximation of ordinary differential equations for corresponding numbers of amplitudes and phases of multifrequency regimes of elastic bodies nonlinear oscillations were composed. The characteristic properties of nonlinear systems passing through coupled multifrequency resonant state and mutual influences between excited modes were discovered. In the same cited papers, amplitude-frequency and phase frequency curves for stationary and nonstationary coupled multifrequency resonant kinetic states based on the numerical experiment on the system of ordinary differential equations in first approximation are presented. Resonant jumps are pointed out in the both series of graphical presentation: amplitude-frequency and phase-frequency curves for the case of the resonant interactions between modes in the same frequency resonant intervals. Using ideas of averaging and asymptotic methods Krilov–Bogoliyubov– Mitropolyskiy in the Doctoral dissertation and in References (see Refs. Hedrih (Stevanovi´c) [5–50]), the author gives the first asymptotic approximations of the solutions for one-, two-, three- and four-frequency vibrations of nonlinear elastic beams, shaft, and thin elastic plates, as well as of the thin elastic shells with positive constant Gauss’s curvatures and finite deformations, and system of the ordinary differential equations in first asymptotic approximation for corresponding numbers of amplitudes and phases for stationary and nonstationary vibration regimes.

34

K.R. (Stevanovi´c) Hedrih

Some results of an investigation of multifrequency vibrations in single-frequency regime in nonlinear systems with many degrees of freedom and with slow-changing parameters are presented by Stevanovi´c and Raˇskovi´c article (1974). Application of the Krilov–Bogolyubov–Mitropolskiy asymptotic method for study of elastic bodies nonlinear oscillations and energetic analysis of the elastic bodies oscillatory motions give new results in theses by Stevanovi´c in 1975. One-frequency transversal oscillations of thin rectangular plate with nonlinear constitutive material stressstrain relations and nonlinear transversal vibrations of a plate with special analysis of influence of weak nonlinear boundary conditions are contents of the articles by Hedrih (1979, 1981). First approximation of an asymptotic particular solution of the nonlinear equations of a thin elastic shell with positive Gauss’ curvature in two-frequency regime is pointed out in the article by Hedrih (1983). Two-frequency oscillations of the thin elastic shells with finite deformations and interactions between harmonics have been studied by Hedrih and Miti´c (1983), and multifrequency forced vibrations of thin elastic shells with a positive Gauss’s curvature and finite displacements by Hedrih (1984). Also, on the mutual influence between modes in nonlinear systems with small parameter applied to the multifrequencies plate oscillations are studied [54, 62, 63, 65]. Multifrequency-forced vibrations of thin elastic shells with a positive Gauss’ curvature and finite deformations and initial deformations influence of the shell middle surface to the phase-frequency characteristics of the nonlinear stationary forced shell’s vibrations and numerical analysis of the four-frequency vibrations of thin elastic shells with Gauss’ positive curvature and finite deformations are content of reference by Hedrih and Miti´c (1985). Also, initial displacement deformation influence of the thin elastic shell middle surface to the resonant jumps appearance was investigated by same authors Hedrih and Miti´c (1987). By means of the graphical presentations from the cited References, analysis was made and some conclusions about nonlinear phenomenon in multifrequency vibrations regimes were pointed out. Some of these conclusions are quoted here: Nonlinearities are the reason for the appearance of interaction between modes in multifrequency regimes; in the coupled resonant state, one or several resonant jumps appear on the amplitudefrequency and phase-frequency curves; these resonant jumps are from smaller to greater amplitudes and vice versa. Unique trigger of coupled singularities (see Refs. [28, 30, 50, 96]) with one unstable homoclinic saddle type point, and with two singular stable center type points appear in one frequency stationary-resonant kinetic state. It is visible on the phase-frequency as well as on the amplitude-frequency graphs for stationaryresonant state. In the case of the multifrequency-coupled resonant state and in the appearance of the more resonant-coupled modes in resonant range of corresponding frequencies, unique trigger of coupled singularities and multiplied triggers of coupled singularities (see Refs. by Hedrih, 2004, 2005) appear. Maximum number of triggers of coupled singularities is adequate to number of coupled modes and resonant frequencies of external excitations. Multiplied triggers contain multiple unstable saddle homoclinic points in the mapped phase plane as the number of resonant frequencies

Energy and Nonlinear Dynamics of Hybrid Systems

35

of external excitations. For example, if a four-frequency-coupled resonant process in u-v plane is in question, four homoclinic saddle-type points appear. The appearance of these unstable homoclinic saddle points requires further study, since it induces instability in a stationary nonlinear multifrequency kinetic process. By use a double circular plate system, presented in the Refs. [53–61], the multifrequency analysis of the nonlinear dynamics with different approaches and by use different kinetic parameters of multifrequency regimes is pointed out. Series of the amplitude-frequency and phase-frequency graphs as well as eigen-time functions–frequency graphs are obtained for stationary resonant states and analyzed according to present singularities and triggers of coupled singularities, as well as resonant jumps. An analogy between nonlinear phenomena in particular multifrequency stationary-resonant regimes of multi-circular plate system nonlinear dynamics, multibeam system nonlinear dynamics, and corresponding regimes in chain system nonlinear dynamics is identified (see References by Hedrih (Stevanovi´c) listed in the reference list from period 1972–2010). Using differential equations systems of the first approximation of multifrequency regime of stationary and no stationary-resonant kinetic states, we analyzed the energy of excited modes and transfer of energy from one to other modes. On the basis of this analysis, the question of excitation of lower frequency modes by higher frequency mode in the nonlinear multifrequency vibration regimes was opened. 3. In many engineering systems with nonlinearity, high-frequency excitations are sources of the appearance of multifrequency-resonant regimes with highfrequency modes as well as low-frequency modes. It is visible from many experimental research results and also theoretical results (see Refs. [81–87]). In the monographs written by Nayfeh [81–87], a coherent and unified treatment of analytical, computational, and experimental methods and concepts of modal nonlinear interactions is presented. This monograph is an obvious extension of Nayfeh’s and Balachandran’s well-known monograph titled by Applied Nonlinear Dynamics (1995). These methods are used to explore and unfold in a unified manner the fascinating complexities in nonlinear dynamical systems. Through the mechanisms discussed in this monograph, energy from high-frequency sources can be transferred to the low-frequency modes of supporting structures and foundations, and the result can be harmful large-amplitude oscillations that decrease their fatigue lives. The interaction between amplitudes and phases of the different modes in the nonlinear systems with many degrees of the freedom as well as in the deformable body infinite numbers frequency vibrations with free and forced regimes is observed theoretically by averaging asymptotic methods Krilov–Bogoliyubov–Mitropolyskiy (1955, 1964, 1968, 1976 and 2003). This knowledge has great practical importance. Application of the Krilov–Bogolyubov–Mitropolskiy asymptotic method as well as energy approach given in monographs by Mitopolskiy (see Refs. [73–80]) for study of the elastic bodies nonlinear oscillations and energy analysis of the elastic bodies oscillatory motions give new results listed in the previous part.

36

K.R. (Stevanovi´c) Hedrih

In the conclusion of this part, we can summarize the following: Oscillatory processes in dynamical systems depend on systems character; in such systems, energy is also transformed from one form to another and has different flows inside a dynamical system; transformation of kinetic energy into potential energy and vice versa occurs in conservative systems, but when linear systems are in question, the energy carried by a considered harmonic (mode) of adequate frequency remains constant during a dynamical process, as does the total systems mechanical energy; there is no mutual influence between harmonics, and the system may be presented by partial oscillators, the number of which is equal to the number of oscillations freedom degrees, or to the number of free vibrations own circular frequencies; during that the total mechanical energy of a single partial oscillator remains constant and the transformation of kinetic energy into potential occurs; in sash linear system, transfer energy between modes does not occurs (see Reference by Raˇskovi´c (1965)). When nonlinear conservative systems are in question, such conclusion as for linear systems would be incorrect. The theoretical and experimental studies reveal that the interactions between widely separated nonlinear modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. The theoretical results show also that damping may be destabilizing. The different types of nonlinear phenomena in single degree of freedom nonlinear system dynamics are investigated between other researchers. 4. An experimental and theoretical study of the response of a flexible cantilever beam to an external harmonic excitation near the beam’s third natural frequency is presented and in addition. Malatkar and Nayfeh (2003) noted that the energy transfer between the third and first modes is very much dependent upon the closeness of the modulation (or Hopf bifurcation) frequency to the firstmode natural frequency. In earlier studies by Nayfeh and coworkers [81–87], the modulation frequency was close to the first-mode natural frequency, and therefore large first-mode swaying was observed. Nayfeh developed a reducedorder analytical model by discretizing the integral partial-differential equation of motion. Identifying, evaluating, and controlling dynamical integrity measures in nonlinear mechanical oscillators are topics for researchers, presented in the Ref. [92]. Also some references by Hedrih [37] contain the energy transfer between coupled oscillators and a conclusion that energy transfer can be a measure of the dynamical integrity of hybrid systems as well as subsystems. Energy transfer in the complex system is subject of research published papers [66] and also [8, 13, 25, 32, 58]. In the paper by Lenci, S. and Rega, G., (2005) dimension reduction of homoclinic orbits of buckled beams via the nonlinear normal modes technique is presented. The problem of detecting the homoclinic orbits of an initially straight buckled beam is addressed. Two families of boundary conditions are identified and investigated in detail. A hierarchy of reduced order, single degree of freedom, models is determined. In the series of the papers [70, 71], the problem of detecting the homoclinic orbits applied to the different engineering system dynamics is investigated and

Energy and Nonlinear Dynamics of Hybrid Systems

37

obtained original research results. In the Refs. [68, 69], resonant nonlinear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic, and cubic nonlinearities are investigated for a class of shallow symmetric structural systems. Nonlinear orthogonality of the modes and activation of the associated interactions are clearly dual problems. 5. In the Refs. [51, 52], the expressions for the kinetic and potential energy as well as energy interaction between chains in the double DNA chain helix are obtained and analyzed for a linearized model. Corresponding expressions of the kinetic and potential energies of these uncoupled main chains are also defined for the eigen main chains of the double DNA chain helix. By obtained expressions, we concluded that there is no energy interaction between eigen main chains of the double DNA chain helix system. Time expressions of the main coordinates of the two eigen main chains are expressed by time, and eigen circular frequencies are obtained. Also, generalized coordinates of the double DNA chain helix are expressed by time correspond to the sets of the eigen circular frequencies. These data contribute to better understanding of biomechanical events of DNA transcription that occur parallel with biochemical processes. Considered as a linear mechanical system, DNA molecule as a double chain helix has its eigen circular frequencies and that is its characteristic. Mathematically, it is possible to decouple it into two chains with their set with corresponding eigen circular frequencies which are different. This may correspond to different chemical structure (the order of base pairs) of the complementary chains of DNA. We are free to propose that every specific set of base pair order has its eigen circular frequencies and its corresponding oscillatory energy, and it changes when DNA chains are coupled in the system of double chain helix. Oscillations of base pairs and corresponding oscillatory energy for specific set of base pairs may contribute to conformational chances of DNA double helix and its unzipping and folding.

1.3 Energy Exchange in Spring Pendulum System For introducing to the problem of the energy transfer or transient in the hybrid nonlinear systems, it is useful to take, for simple analysis, into consideration the change energy between parts of the energy carrying on the generalized coordinates φ and ρ in the very known system, known under name spring pendulum system, with two degree of freedom. For the analysis of the energy in the spring pendulum, we can write the kinetic and potential energies in the forms:  1  Ek = m ρ˙ 2 + (ρ + )2 φ˙ 2 2

38

K.R. (Stevanovi´c) Hedrih

and

1 (1) E p = cρ 2 + mg (ρ + ) (1 − cos φ ) 2 where: m is mass of the pendulum,  length of pendulum string-neglected mass spring in the static equilibrium state of the pendulum, and c spring axial rigidity and φ and ρ are respectfully, angle and extension part of length of the string-spring of the pendulum with comparison of the sprig length in static equilibrium state of the pendulum, taken as the generalized coordinates of the system. For the linearized case for kinetic energy, after neglecting small member – part of kinetic energy on the generalized coordinate φ – we can taking into account the following expression: ∗

Expression Ek2 = 12 m(ρ + )2 φ˙ 2 changes into approximation

1  2 Ek2 ≈ m φ˙ . (2) 2 Only for small oscillations – perturbations from equilibrium position – it is possible to use approximation of the expression for kinetic and potential energy in the form:  2  1  1 1 and E p ≈ cρ 2 + mgφ 2 Ek ≈ m ρ˙ 2 + φ˙ (3) 2 2 2 For that linearized case, the generalized coordinates are normal coordinates of the small oscillations of the spring pendulum around equilibrium position ρ = 0, φ = 0, and coordinates are decoupled. In this linearized case of the spring pendulum model, the energy carried on the these normal coordinates are uncoupled and transfer or transient of the total energy don’t appeared between proper parts of the separate normal coordinate and on the separate processes defined by normal coordinates are conservative systems each with one degree of the freedom. In this case, in each of the coordinate, there are conversion of the energies from kinetic to potential, but the sum of the both of one normal coordinates is constant. 1 1 (4) Ekρ ≈ mρ˙ 2 and E pρ ≈ cρ 2 2 2 1  2 1 Ekφ ≈ m φ˙ and E pφ ≈ mg φ 2 (5) 2 2 This is visible from system of the differential equations in the linearized form: c m g φ¨ + ω12 φ = 0 where ω12 = . 

ρ¨ + ω22 ρ = 0 where ω22 =

(6)

but for the nonlinear case the interaction between coordinates is present and then energy transient appears.  1  2 m ρ˙ + 2 φ˙ 2 + ρ 2φ˙ 2 + 2ρ φ˙ 2 and 2 1 E p = cρ 2 + mg  (1 − cos φ ) + mg ρ (1 − cos φ ) 2 Ek =

(7)

Energy and Nonlinear Dynamics of Hybrid Systems

39

We can separate the following parts: 1. Kinetic and potential energies carrying on the coordinate ρ are: 1 1 Ekρ = mρ˙ 2 and E pρ = cρ 2 + mgρ 2 2

(8)

By analyzing these previous expressions, we can see that with these expressions for decoupled oscillator with coordinate ρ , we have pure linear oscillator or harmonic oscillator with coordinate ρ and frequency ω22 = mc , and separated process is isochronous. 2. Kinetic and potential energies carrying on the coordinate φ are 1 Ekφ = m2 φ˙ 2 and E pφ = mg (1 − cos φ ) 2

(9)

By analyzing these previous expressions, we can see that with these expression for decoupled oscillator with coordinate φ , we have pure nonlinear oscillator with coordinate φ , and separated process is no isochronous. For a linearyzed case, this oscillator has eigen frequency ω12 = g . 3. Then, formally, we can conclude that in the spring pendulum, we have coupled two oscillators, one pure linear with one degree of freedom, and second nonlinear, also with one degree of freedom. In the hybrid system, these oscillators are coupled and mechanical energy of the coupling contain two parts: one kinetic energy and second potential energy. Then, in the coupling, hybrid connections with static and dynamic kinetic properties are introduced. Kinetic and potential energies of the coordinate φ and ρ interaction in the nonlinear hybrid model are: 1 Ek(φ ,ρ ) = m [ρ + 2] ρ φ˙ 2 and E p(φ ,ρ ) = −mgρ cos φ 2

(10)

For a nonlinear case, ordinary differential equations are in the following form:

ρ¨ + ω22 ρ = −g (1 − cos φ )

(11)

2 1 φ¨ + ω12 φ = ω12 (φ − sin φ ) − 2 ρ˙ φ˙ (ρ + ) − 2 ρ (ρ + 2) φ¨  

(12)

or in nonlinear approximation forms for small oscillations around zero coordinates ρ = 0, φ = 0 or around stable equilibrium position of the spring pendulum are 

φ2 φ4 φ6 φ8 − + − + . . ... (13) 2 24 6! 8!  3 2 1 φ φ5 φ7 − + − . . . . − 2 ρ˙ φ˙ (ρ + ) − 2 ρ (ρ + 2) φ¨ (14) φ¨ + ω12 φ ≈ −ω12 3 5! 7!  

ρ¨ + ω22 ρ ≈ −g

40

K.R. (Stevanovi´c) Hedrih

If we introduce phase coordinate, then we can write: v = ρ˙

ν˙ = −ω22 ρ − g (1 − cos φ ) u = φ˙ u˙ = −ω12 φ + ω12 (φ − sin φ ) −

2 ˙ 1 ρ˙ φ (ρ + ) − 2 ρ (ρ + 2) φ¨ 2 

(15)

or in the approximation v = ρ˙  v˙ ≈

−ω22 ρ − g

φ2 φ4 φ6 φ8 − + − + . . ... 2 24 6! 8!

u = φ˙  u˙ ≈

−ω12 φ

− ω12

φ3 φ5 φ7 2 1 − + − . . . . − 2 ρ˙ φ˙ (ρ + ) − 2 ρ (ρ + 2) u˙ (16) 3 5! 7!  

From system equations (11)–(12), as well from their approximations (13)–(14), we can see that their right-hand parts are nonlinear and are functions of generalized coordinates, as well as of the generalized coordinates first and second derivatives. Also we can see that generalized coordinates φ and ρ around their zero values, when ρ = 0, φ = 0 at the stable equilibrium position of the spring pendulum, and that also they are main coordinates of the linearized model. It is of reason that the asymptotic averaged method is applicable for obtaining first asymptotic approximation of the particular solutions, and it is possible to use for energy analysis of the transfer energy between energies carried by generalized coordinates φ and ρ in this nonlinear system with two degree of freedom, but formally, we can take into account that we have two oscillators, one nonlinear and one linear each with one degree of freedom as two subsystems coupled in the hybrid system with two degree of freedom, by hybrid connection realized by statically and dynamical connections. This interconnection have two parts of energy interaction between subsystems expressed by kinetic and potential energies in the forms expressed by (10). Taking into consideration some conclusion from considered system of the spring pendulum, we can conclude also that it is important to consider more simple case of the coupling between linear and nonlinear systems with one degree of freedom with different types of the coupling realized by simple static or dynamic elements, for to investigate hybrid phenomena in the coupled subsystems.

Energy and Nonlinear Dynamics of Hybrid Systems

41

2 Energy Analysis and Free Vibration Nonlinear System When nonlinear conservative systems are in question, such conclusion as for linear systems that no interaction between submotion components would be incorrect. The theoretical and experimental studies reveal that the interactions between widely separated modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. Kinetic energy and potential energy in first asymptotic approximation for nonlinear conservative system nonlinear modes using normal coordinates of unperturbed corresponding linear system are (see Ref. [20]):

 s=n s=n  Ek = ∑ Eks = ∑ ξ˙s2 +g ξ1 , ξ2 , . . . , ξs , ξr , .., ξn−1 , ξn , ξ˙1 , ξ˙2 , . . . , ξ˙s , ξ˙r , .., ξ˙n−1 , ξ˙n s=1

Ep =

s=1

s=n 

∑

 ωs2 ξs2 + f (ξ1 , ξ2 , . . . , ξs , ξr , .., ξn−1 , ξn )

(17)

s=1

where

ξs = as cos (θs + ψs ) s = 1, 2, . . . ..n

(18)

are first asymptotic approximations of normal coordinates, and as are amplitudes, and θs + ψs are phases as the functions of time and which are calculate from differential equations first approximations (see Ref. [78]).

2.1 Nonlinear Oscillator Kinetic and potential energies and Rayleigh dissipative function of nonlinear oscillator with one degree of freedom and generalized coordinate x1 are: 1 Ek(1) = m1 x˙21 , 2 1 1 E p(1) = c1 x21 ± c˜1 x41 2 4 1 2 Φ(1) = b1 x˙1 (19) 2 where m1 is masses, c1 is the spring rigidity coefficient of the linear elasticity low, and c˜1 the spring rigidity coefficient of the nonlinear elasticity low, upper sign (+) for hard and lower sign (−) for soft nonlinearity, b1 coefficient of the system linear   dumping force. For this nonlinear oscillator, it is right, dtd Ek(1) + E p(1) = −2Φ(1) , and for the case of the free vibrations. For this case, differential equation is in the following form: 2 3 x1 x¨1 + 2δ1x˙2 + ω12 x1 = ∓ω˜ N1

(20)

42

K.R. (Stevanovi´c) Hedrih

upper sign (−) for hard and lower sign(+) for soft nonlinearity. where c1 b1 c˜1 ω12 = , 2δ1 = , ω˜ 2 = . m1 m1 N1 m1

(21)

and characteristic equation of the basic liner equation, correspond to previous (20),

have the following characteristic numbers: λ1,2 = −δ1 ∓ i ω12 − δ12 = −δ1 ∓ ip1 for the small damping coefficient δ1 < ω1 , and solution for free vibrations is in the form: x1 (t) = R01 e−δ1 t cos(p1 t + α01 ). To obtain approximation by using averaged method, we propose solution in the following form: x1 (t) = R1 (t) e−δ1 t cos Φ1 (t)

(22)

where R1 (t) and Φ1 (t) are unknown functions. Also, we can write: Φ1 (t) = p1t + φ1 . After averaging with respect to the full phase Φ1 (t), we obtain the following system of the averaged first-order differential equations: R˙ 1 (t) = 0 3 2 2 φ˙1 (t) = ± ω˜ R (t) e−2δ1t 8p1 N1 1

(23)

upper sign (+) for hard and lower sign (−) for soft nonlinearity. After integration, we obtain for amplitude and phase the following first approximation: R1 (t) = R01 = const

φ1 (t) = ∓

3 ω˜ 2 R2 e−2δ1 t + α01 , 16δ1 p1 N1 01

for δ = 0

(24)

upper sign (+) for hard and lower sign (−) for soft nonlinearity, and for full phase Φ1 (t): 3 Φ1 (t) = p1t ∓ ω˜ 2 R2 e−2δ1t + α01 , for δ = 0 (25) 16δ1 p1 N1 01 (−) for strong and (+) for soft nonlinearity, and solution in the first averaged approximation form is: x1 (t) = R01 e−δ1 t cos Φ1 (t)  − δ t 1 x1 (t) = R01 e cos p1t ∓

3 2 2 −2δ1 t ˜ ω R e + α01 , 16δ1 p1 N1 01

for δ = 0

(26)

upper sign (−) for hard and lower sign (+) for soft nonlinearity, we can see that amplitude of the solution in the first averaged approximation form is in the form R01 e−δ1t and that phase Φ1 (t) is also function of the time, and also frequency 2 R2 e−2δ1 t , for δ = 0, upper sign (−) for hard and lower p˜ 1 (t) = p1 ∓ 8p3 1 ω˜ N1 1 01 sign (+) for soft nonlinearity., is changeable with time in the first asymptotic approximation obtained by averaged method. By using previous obtained first asymptotic averaged approximation of the solution, we obtain Lyapunov exponent in the form:

Energy and Nonlinear Dynamics of Hybrid Systems



1 2 1 2 λ1 = lim ln x1 (t) + 2 x˙1 (t) = −δ1 < 0 t→∞ 2t ω1 or in the form

 2 ˜λ1 = lim 1 ln x2 (t) + ωN1 x4 (t) + 1 x˙2 (t) 1 t→∞ 2t ω12 1 ω12 1 

˜λ1 = lim 1 ln Esist = −δ1 < 0 t→∞ 2t 2m1 ω12

43

(27)

(28)

In our research, we can investigate system with small nonlinearity and small vibrations around periodic vibrations.

2.2 Linear Oscillator Kinetic and potential energies and Rayleigh dissipative function (see Ref. by Hedrih (Stevanovi´c) (2002) [31]) of linear oscillator with one degree of freedom and generalized coordinate x2 are: 1 1 1 Ek(2) = m2 x˙22 , E p(2) = c2 x22 and Φ(2) = b2 x˙22 2 2 2

(29)

where m2 is mass, c2 is the spring rigidity coefficient of the linear elasticity low, b2 coefficient of the system linear dumping force.   For this system, it is possible to show that: dtd Ek(2) + E p(2) = −2Φ(2). For this case, the differential equation is in the following form: x¨2 + 2δ2 x˙2 + ω22 x2 = 0, where

ω22 = mc22 , 2δ2 = mb22 , and with characteristic numbers: λ1,2 = −δ2 ∓ i ω22 − δ22 for the small damping coefficient δ2 < ω2 . Solution for free vibrations is: x2 (t) = R0 e−δ2 t cos (p2t + α2 ) .

(30)

2.3 Hybrid Systems with Staitc Constraints Kinetic and potential energies and Rayleigh dissipative function (see Ref. by Hedrih (Stevanovi´c) (2002) [31]) of the hybrid system, containing two subsystems – one linear oscillator and one nonlinear oscillator, with two degree of freedom expressed by generalized coordinates x1 and x2 (see Fig. 1a∗ ) are: 1 1 m1 x˙21 + m2 x˙22 2 2 1 2 1 4 1 1 E p = c1 x1 ± c˜1 x1 + c (x1 − x2 )2 + c2 x22 2 4 2 2 Ek =

(31) (32)

44

K.R. (Stevanovi´c) Hedrih

Fig. 1 Two hybrid systems containing coupled subsystems by (a∗ ) static constraint, coupled by linear spring with rigidity c and (b∗ ) dynamical constraint, coupled by rolling element of the mass m – dynamic coupling: one nonlinear (left) and second linear (right)

Φ=

1 2 1 2 b1 x˙ + b2 x˙ 2 1 2 2

(33)

where m1 and m2 are masses, c1 , c, and c2 are the spring rigidity coefficients of the linear elasticity law, and c˜1 the spring rigidity coefficient of the spring nonlinear elasticity law, where in (32) upper sign (+) for hard and lower sign (−) for soft nonlinearity. b1 and b2 coefficient of the system linear dumping forces. For this system, it is possible to show that: dtd (Ek + E p) = −2Φ. Energy interaction in this hybrid system, containing two coupled subsystems by statical constraint is potential energy of the spring for coupling nonlinear and linear subsystem and is expressed in the form: 1 E p(1,2) = c (x2 − x1)2 2

(34)

Coupled system of differential equations of the hybrid system containing two subsystems, one nonlinear and one linear, are in the forms:   2 3 x1 x¨1 + 2δ1 x˙2 + ω12 + a21 x1 − a21x2 = ∓ω˜ N1   2 x¨2 + 2δ2 x˙2 + ω2 + a22 x2 − a22x1 = 0 (35) where are: upper sign (−) for hard and lower sign (+) for soft nonlinearity; and. . .

ωi2 =

ci bi c c˜1 2 , 2δi = , a2i = , ω˜ N1 = , i = 1, 2. mi mi mi m1

(36)

Taking into account that consideration of the homogeneous system does not lose generality of the phenomena, next our considerations are applied to this homogeneous hybrid system. For the basic linear equations of the coupled system of the differential equations of the hybrid system containing two subsystems, one linearized and one linear, are in the form:   x¨1 + 2δ1x˙1 + ω12 + a21 x1 − a21 x2 = 0   x¨2 + 2δ2x˙2 + ω22 + a22 x2 − a22 x1 = 0 (37)

Energy and Nonlinear Dynamics of Hybrid Systems

45

and for case that linearized and linear systems are equal (ω12 = ω22 and δ1 = δ2 and a21 = a22 , we can define characteristic equation with roots – characteristic numbers: λ1,2 = −δ ∓ ip1 and λ3,4 = −δ1 ∓ i p˜1 for the small damping coefficient δ1 < ω1 , it is possible to write: λ1,2 = −δ1 ∓i ω12 − δ12 = −δ1 ∓i p˜1

λ3,4 = −δ1 ∓i ω12 + 2a21 − δ12 = −δ1 ∓i p˜ 1

where: ω12 − δ12 for the small damping coefficient δ1 < ω1 . p˜2 = p˜1 = ω12 + 2a21 − δ12 for the small damping coefficient δ1 < ω1 .

p1 =

Corresponding solution of the linear-coupled subsystem into system, we can write in the following two-frequency form: x1 (t) = e−δ t [R01 cos(p1t + α01 ) + R02 cos ( p˜2 t + α02 )] x2 (t) = e−δ t [R01 cos(p1t + α01 ) − R02 cos ( p˜2 t + α02 )]

(38)

where amplitudes R0i and phases α0i are constants depending of initial conditions. By using averaged method, the first approximation of the solution of the hybrid system, containing coupled nonlinear and linear system, we propose in the forms: x1 (t) = e−δ t [R1 (t) cos Φ1 (t) + R21 (t) cos Φ2 (t)] x2 (t) = e−δ t [R1 (t) cos Φ1 (t) − R21 (t) cos Φ2 (t)]

(39)

where amplitudes Ri (t) and phases Φi (t), i = 1, 2 are unknown functions. Also, we can write: Φi (t) = pit + φi . Then after application averaging method and averaging obtained ordinary differential equations with respect to the full phase Φi (t), we obtain the following system of the first asymptotic approximation of the system differential equations for amplitudes Ri (t) and phases Φi (t): R˙ 1 (t) = 0   2 3 2 φ˙1 (t) = ± ω˜ N1 R1 (t) + 2R22 (t) e−2δ1t 16p1 R˙ 2 (t) = 0  2  3 2 R2 (t) + 2R21 (t) e−2δ1t φ˙2 (t) = ± ω˜ N1 16 p˜ 2 where upper sign (+) for hard and lower sign (−) for soft nonlinearity.

(40)

46

K.R. (Stevanovi´c) Hedrih

After integration of the previous system of ordinary differential equations (40) in first asymptotic approximation in the case that damping is different them zero, δ1 = 0 we obtain the following expressions for two amplitudes Ri (t) and two corresponding phases Φi (t), in first asymptotic approximation: R1 (t) = R01 = const

φ1 (t) = ∓

 2  3 2 R01 + 2R202 e−2δ1 t + α01 , ω˜ N1 32δ p1

for δ1 = 0

R2 (t) = R02 = const

φ1 (t) = ∓

 2  3 2 2R01 + R202 e−2δ1 t + α02 , ω˜ N1 32δ p˜ 2

for δ1 = 0

(41)

where upper sign (−) for hard and lower sign (+) for soft nonlinearity. The first asymptotic approximation of the solutions in two frequency regime in averaged form of the hybrid system dynamics is in the following form are:   2  −2δ t 3 −δ t 2 2 ω˜ + α01 + x1 (t) = e R01 cos p1t ∓ R + 2R02 e 32δ p1 N1 01   2  3 2 ω˜ N1 +e−δ t R02 cos p˜2t ∓ 2R01 + R202 e−2δ t + α02 32δ p˜2 for δ1 = 0



 2  −2δ t 3 2 2 ω˜ + α01 − x2 (t) = e R01 cos p1t ∓ R + 2R02 e 32δ p1 N1 01   2  3 2 ω˜ N1 −e−δ t R02 cos p˜2t ∓ 2R01 + R202 e−2δ t + α02 32δ p˜2 −δ t

for δ1 = 0

(42)

where upper sign (−) for hard and lower sign (+) for soft nonlinearity. We can see that amplitudes of the solution in the first approximation are in the form R0i e−δ t and that phases are also functions of the time, and also frequencies p1 (t) =p1 ∓

  3 2 R201 +2R202 e−2δ t ω˜ N1 16 p˜1

and

p˜2 (t) = p˜2 ∓

  3 2 2R201 +R202 e−2δ t ω˜ N1 16 p˜2 (43)

where upper sign (−) for hard and lower sign (+) for soft nonlinearity. Are changeable with time in the first approximation obtained by asymptotic averaged method. By using previous first asymptotic approximation of the solution in the two frequency regime, we can obtain Lyapunov exponents in the forms:

 1 2 1 2 λ1 = lim ln x1 (t) + 2 x˙1 (t) = −δ < 0 t→∞ 2t ω1

 1 2 1 2 λ2 = lim ln x2 (t) + 2 x˙2 (t) = −δ < 0 (44) t→∞ 2t ω2

Energy and Nonlinear Dynamics of Hybrid Systems

47

Also, taking into account that system is nonlinear, we can obtain Lyapunov exponents in the following forms:

 2 ˜λ1 = lim 1 ln x2 (t) + ω˜ N1 x4 (t) + 1 x˙2 (t) 1 t→∞ 2t ω12 1 ω12 1

 ˜λ1 = lim 1 ln Esubsist(1) = −δ < 0 (45) t→∞ 2t 2m1 ω12 For the nonhomogeneous case, we can define characteristic equation, with four roots: λ1,2 = −δˆ1 ∓ i pˆ1 and λ3,4 = −δˆ2 ∓ i pˆ2 , and solution of the linear-coupled system, we can write in the following form: (1)

ˆ

(1)

ˆ

(2)

ˆ

(2)

ˆ

x1 (t) = K21 e−δ1 t R01 cos ( pˆ1 t + α01 ) + K21 e−δ2 t R02 cos ( pˆ2t + α02 ) x2 (t) = K22 e−δ1 t R01 cos ( pˆ1 t + α01 ) + K22 e−δ2 t R02 cos ( pˆ2t + α02 )

(46)

(s)

where K2i are cofactors of the system, and amplitudes and phases, R0i and α0i , are constants. By using asymptotic averaged method, a first asymptotic approximation of the solution of the hybrid system, containing coupled nonlinear and linear system as subsystems, we propose solutions in the following forms: (1)

ˆ

(1)

ˆ

(2)

x1 (t) = K21 e−δ1 t R1 (t) cos Φ1 (t) + K21 e−δ2 t R02 cosΦ2 (t) (2)

ˆ

x2 (t) = K22 e−δ1 t R01 cos Φ1 (t) + K22 e−δ2 t R02 cos Φ2 (t) ˆ

(47)

where Ri (t) and Φi (t) are unknown functions. Also we can write: Φi (t) = pˆit + φi . And all next is similar as in previous considered part.

2.4 Hybrid Systems with Dynamic Constraints In Fig. 1b∗ , we can see a hybrid system containing two subsystems, one linear and one nonlinear coupled by dynamical constraint. Dynamical constraint consists of the one disk with mass m and mass inertia axial moment JC with possibility of rolling between two masses m1 and m2 of the subsystems. In our research, we can investigate small nonlinearity in the subsystem, and also in the hybrid system and also small nonlinear vibrations around periodic regimes. Kinetic energy of the coupling nonlinear and linear subsystems is in the following form:  1 Ek(1,2) = aˆ11 x˙21 + aˆ22x˙22 + 2x˙1x˙2 aˆ12 (48) 2

48

K.R. (Stevanovi´c) Hedrih

Fig. 2 Uncoupled subsystems: one nonlinear (a∗ ) and second linear (b∗ )

where

m m m JC JC JC , aˆ22 = + 2 , aˆ12 = − 2 . (49) + 4 4R2 4 4R 4 4R Then we have a hybrid system with coupled dynamic, but also linear, constraint between two subsystems as a resultant dynamic of two subsystem dynamics in mutual interactions. Kinetic and potential energies and Rayleigh energy dissipation function of the hybrid system, containing two subsystems – one linear oscillator and one nonlinear oscillator, with two degree of freedom expressed by generalized coordinates x1 and x2 (see Fig. 2a∗ ) are:    2 2  1 1 1 + x ˙ − x ˙ x ˙ x ˙ 1 2 2 1 + JC (50) m Ek = m1 x˙21 + m2 x˙22 + 2 2 2 2 2R aˆ11 =

1 1 1 E p = c1 x21 ± c˜1 x41 + c2 x22 2 4 2 1 1 Φ = b1 x˙21 + b2 x˙22 2 2

(51)

where upper sign (+) for hard and lower sign (−) for soft nonlinearity. Also, where m1 and m2 are masses, c1 , c, and c2 are the spring rigidity coefficients of the linear elasticity low, and c˜1 the spring rigidity coefficient of the nonlinear elasticity low, b1 and b2 coefficient of the system linear dumping forces. For this system, it is possible to show that is: dtd (Ek + E p ) = −2Φ. Energy interaction in this system is by kinetic energy of the rolling element for coupling nonlinear and linear subsystem and is expressed in the form: Ek =

 1 a˜11 x˙21 + a˜ 22x˙22 + 2a˜12x˙1 x˙2 2

(52)

where a˜11 = m1 + a˜12 =

JC JC m m + = a11 + aˆ11, a˜22 = m2 + + 2 = a22 + aˆ22, 4 4R2 4 4R

JC m − 2 = aˆ12 4 4R

(53)

Energy and Nonlinear Dynamics of Hybrid Systems

49

JC Coefficient a˜12 = m4 − 4R 2 is coefficient of the subsystems coupling, and the constraint is dynamical. Then, this coefficient is coefficient of inertia. When this coefficient is equal to zero, then the system coordinate x1 and x2 are decoupled and there are not energy of the coupling, but there are energy of the influence of the dynamic constraint by additional members. Kinetic energy of the first subsystem as a one part of the hybrid system is: Ek = 1 2 2 a˜11 x˙1 . Kinetic energy of the second subsystem as a one part of the hybrid system is: Ek = 12 a˜22 x˙22 . Kinetic energy of the coupling of the subsystems as a two parts of the hybrid system is: Ek = a˜12 x˙1 x˙2 . Additional part of the kinetic energy of the first subsystem – reduction of the dynamic constraint to the first subsystem Ek(1)d = 12 aˆ11 x˙21 . Additional part of the kinetic energy of the second subsystem – reduction of the dynamic constraint to the second subsystem Ek(2)d = 12 aˆ22 x˙22 . When the coefficient   JC of subsystems coupling equals zero, a˜12 = m4 − 4R 2 = 0, then subsystems do not have kinetic energy interaction, but have additional part of kinetic energy of the first subsystem – reduction of the dynamic constraint to the first subsystem and additional part of the kinetic energy of the second subsystem – reduction of the dynamic constraint to the second subsystem. System of differential equations is based on the kinetic and potential energy and Rayleigh energy dissipation function and is obtained in the following form: 2 3 x¨1 + κ1x¨2 + ω˜ 12 x1 + 2δ˜1 x˙1 = ∓ω˜ N1 x1

x¨2 + κ2x¨1 + ω˜ 22 x2 + 2δ˜2 x˙2 = 0

(54)

where upper sign (−) for hard and lower sign (+) for soft nonlinearity and following 2 = c˜1 = ω 2 m1 , 2δ˜ = bi , i = ˜ N1 notations: κ1 = aa˜˜12 , κ2 = aa˜˜12 , ω˜ 12 = a˜c111 , ω˜ 22 = a˜c221 , ω˜ N1 i a˜ 11 a˜11 a˜ii 11 22 1, 2 are introduced. For the basic linear equations of the linear dynamically coupled system of the differential equations of the hybrid system containing two subsystems, one linearized and one linear are in the form x¨1 + κ1 x¨2 + ω˜ 12 x1 + 2δ˜1 x˙1 = 0 x¨2 + κ2 x¨1 + ω˜ 22 x2 + 2δ˜2 x˙2 = 0

(55)

We can compose corresponding characteristic equation with four roots: λ1,2 = −δˆ1 ∓ i pˆ1 and λ3,4 = −δˆ2 ∓ i pˆ2 . It is not difficult to obtain eigen amplitude numbers and solutions of the basic linear-coupled system, we can write in the following form: (1)

ˆ

(1)

ˆ

(2)

ˆ

(2)

ˆ

x1 (t) = K21 e−δ1 t R01 cos ( pˆ1 t + α01 ) + K21 e−δ2 t R02 cos ( pˆ2t + α02 ) x2 (t) = K22 e−δ1 t R01 cos ( pˆ1 t + α01 ) + K22 e−δ2 t R02 cos ( pˆ2t + α02 ) (s)

(56)

where K2i are cofactors of the system, and amplitudes R0i and phases α0i , are constants, depending of initial conditions.

50

K.R. (Stevanovi´c) Hedrih

By using asymptotic averaged method, a first asymptotic approximation of the solution of the hybrid system dynamics, containing dynamical coupled nonlinear and linear system, we propose solutions in the following forms: (1)

ˆ

(1)

ˆ

(2)

ˆ

(2)

ˆ

x1 (t) = K21 e−δ1 t R1 (t) cos Φ1 (t) + K21 e−δ2 t R2 (t) cosΦ2 (t) x2 (t) = K22 e−δ1 t R1 (t) cos Φ1 (t) + K22 e−δ2 t R2 (t) cosΦ2 (t)

(57)

where amplitudes Rs (t) and phases Φs (t) are unknown functions. Also, we can write: Φi (t) = pˆ it + φi . After applying asymptotic averaging with respect to the full phases Φs (t), we obtain the system of the first asymptotic averaged approximation of the differential equations for amplitudes Ri (t) and phases Φi (t). After integrating the system of averaged differential equations, we obtain first approximation of the amplitudes Ri (t) and phases Φi (t) of the solution in the following form: R1 (t) = R01 = const 3 2   ω˜ N1 (1) (2) (1) (2) 16p1 K21 K22 − K22 K21   ˆ ˆ e−2δ2 t (1)  (2) 2 e−2δ1 t (1) 3 2 2 K21 [R01 ] + × K21 K21 [R02 ] + α01 2δˆ1 δˆ2

φ1 (t) = ∓

for δ1 = 0 R2 (t) = R02 = const 3 2   ω˜ N1 (2) (1) (2) (1) 16 pˆ 2 K21 K22 − K22 K21   ˆ ˆ e−2δ2 t (1)  (2) 2 e−2δ1 t (1) 3 2 2 × K21 K21 [R02 ] + α02 K21 [R01 ] + δˆ1 2δˆ2

φ2 (t) = −

for δ1 = 0

(58)

where upper sign (−) for hard and lower sign (+) for soft nonlinearity. Solution in the first averaged asymptotic approximation is not difficult to compose by use expression (57) and (58). By using previous first asymptotic approximation of the solution in the two frequency regime, we can obtain Lyapunov exponents in the forms:   λ1 = lim 2t1 ln x21 (t) + ω˜12 x˙21 (t) = −δˆ1 < 0 t→∞

1

  λ2 = lim 2t1 ln x22 (t) + ω˜12 x˙22 (t) = −δˆ2 < 0 t→∞

2

(59)

Energy and Nonlinear Dynamics of Hybrid Systems

51

Also, taking into account that system is nonlinear, we can introduce first Lyapunov exponent in the forms:

 ω˜ 2 4 1 2 1 λ˜ 1 = lim ln x21 (t) + N1 x (t) + x ˙ (t) t→∞ 2t ω˜ 12 1 ω˜ 12 1

 Esubsist(1) 1 λ˜ 1 = lim ln = −δˆ1 < 0 t→∞ 2t 2m1 ω˜ 12

(60)

3 Energy Analysis of Forced Nonlinear Systems 3.1 A Spring Pendulum For introducing to the problem of the energy transfer or transient in the hybrid nonlinear system forced dynamics, it is useful to take, for simple analysis, into consideration the change energy between parts of the energy carrying on the generalized coordinates φ and ρ in the spring pendulum system with two degree of freedom excited by external excitations. For the analysis of the energy in the spring pendulum in the forced regime excited by external one frequency excitation – generalized forces Mφ (t) = M0 cos(Ωφ t + ϑφ ) and Fρ (t) = F0 cos(Ωρ t + ϑρ ) – we can write the kinetic and potential energies in the forms (1). By taking into account all comments and asymptotic approximation as in the introductory part of this paper, as well as corresponding expressions (2)–(5), system of the differential equations of the linearized system is in the following form: where

  ρ¨ + ω22 ρ = h0ρ cos Ωρ t + ϑρ

(61)

c F0 , h0ρ = m m   φ¨ + ω12 φ = h0φ cos Ωφ t + ϑφ

(62)

ω22 =

M0 where ω12 = g , h0φ = m 2. Solutions of the linearized equations (61) an (62) are:

ρ (t) = R2 cos (ω2t + α02 ) +

h0 ρ 2 ω2 − Ω2ρ

  cos Ωρ t + ϑρ

(63)

φ (t) = R1 cos (ω1t + α01 ) +

h0 φ 2 ω1 − Ω2φ

  cos Ωφ t + ϑφ

(64)

For that linearized case, both chosen coordinates are main coordinates of the linearized model, and from solutions (63)–(64), we can see that free and also, forced

52

K.R. (Stevanovi´c) Hedrih

vibrations are uncoupled, and not interaction between free, and also forced modes of the vibrations. Then, we have two uncoupled oscillators with different eigen circular frequencies ω12 = g and ω22 = mc and different forced external excitation frequencies Ωφ and Ωρ and with possibilities of appearance two main uncoupled resonant regimes, when Ω2φ ,resonant = ω12 = g and Ω2ρ ,resonant = ω22 = mc . In this case, for linearized models and in the resonant cases, expressions for solutions are in the following forms:

ρ˙ 0 sin ω2 t + ω2     ω2 t sin ω2t + ϑρ − sin ω2 t sin ϑρ

(65)

φ˙0 sin ω1t + ω1     ω1 t sin ω1t + ϑφ − sin ω1 t sin ϑφ

(66)

ρ (t)|Ωρ ,resonant = ω2 = ρ0 cos ω2t + +

h 0ρ 2ω2

φ (t)|Ωφ ,resonant = ω1 = φ0 cos ω1t + +

h 0φ 2ω1

But, for the nonlinear case the interaction between coordinates is present and then energy transient appears. Expressions for kinetic and potential energies are in the same forms as presented and analyzed in first part for free vibrations and named by (6)–(10). Then, the expressions for coordinates are different and must be taken in the forms (65)–(66). By analyzing corresponding expressions, we can see that with these expressions for decoupled oscillator with coordinate ρ , we have pure linear oscillator or harmonic oscillator with coordinate ρ and frequency ω22 = mc , and separated process is isochronous. By analyzing these corresponding expressions, we can see that with these expressions for decoupled oscillators with coordinate φ , we have pure nonlinear oscillator with coordinate φ , and separated process is no isochronous. For a linearyzed case, this oscillator has eigen frequency ω12 = g . For forced nonlinear case, differential equations of the system nonlinear oscillations are in the following form:   ρ¨ + ω22 ρ = −g (1 − cos φ ) + h0ρ cos Ωρ t + ϑρ 2 φ¨ + ω12 φ = ω12 (φ − sin φ ) − 2 ρ˙ φ˙ (ρ + ) −    1 − 2 ρ (ρ + 2) φ¨ + h0φ cos Ωφ t + ϑφ 

(67)

(68)

or in nonlinear approximation forms for small oscillations around zero coordinates ρ = 0, φ = 0 or of the around stable equilibrium position of the spring pendulum

Energy and Nonlinear Dynamics of Hybrid Systems

φ2 φ4 φ6 φ8 − + − + . . ... + ≈ −g 2 24 6! 8!   +h0ρ cos Ωρ t + ϑρ

53



ρ¨

+ ω22 ρ



φ¨ + ω12 φ ≈ −ω12 −

(69)

φ3 φ5 φ7 − + − . . .. − 3 5! 7!

  1 2 ˙ ρ˙ φ (ρ + ) − 2 ρ (ρ + 2) φ¨ + h0φ cos Ωφ t + ϑφ 2 

(70)

If we introduce phase coordinate, then we can write: v = ρ˙

  v˙ = −ω22 ρ − g (1 − cos φ ) + h0ρ cos Ωρ t + ϑρ

u = φ˙ 2 ˙ ρ˙ φ (ρ + ) − 2   1 − 2 ρ (ρ + 2) u˙ + h0φ cos Ωφ t + ϑφ 

u˙ = −ω12 φ + ω12 (φ − sin φ ) −

(71)

or in the approximation v = ρ˙



v˙ ≈ −ω22 ρ − g u = φ˙ u˙ ≈



−ω12 φ −

  φ2 φ4 φ6 φ8 − + − + . . . .. + h0ρ cos Ωρ t + ϑρ 2 24 6! 8!

− ω12

φ3 φ5 φ7 − + − . . .. − 3 5! 7!

  1 2 ˙ ρ˙ φ (ρ + ) − 2 ρ (ρ + 2) u˙ + h0φ cos Ωφ t + ϑφ 2  

(72)

From system of the differential equations (67)–(68), as well as from their approximations (68)–(70), we can see that their right-hand parts are nonlinear and are functions of generalized coordinates, as well as of the generalized coordinates first and second derivatives with respect to time and function of time. Also, we can see that generalized coordinates φ and ρ around their zero values, when ρ = 0, φ = 0 at the stable equilibrium position of the spring pendulum are also main coordinates of the linearized model. It is reason that the asymptotic averaged method is applicable for obtaining first asymptotic approximation of the solutions.

54

K.R. (Stevanovi´c) Hedrih

Then it is possible that first asymptotic approximations of the solutions of the system of nonlinear differential equations (67)–(68) take into account in the following asymptotic approximations for the small spring pendulum forced elongations in the form:   ρ = aρ (t) cos ω1 t + ϕρ (t)   φ = aφ (t) cos ω2 t + ϕφ (t) (73) where amplitudes aρ (t) and aϕ (t) and phases ϕρ (t) and ϕϕ (t) are defined by system of first order nonlinear differential equations in first asymptotic approximation in the following form: a˙ρ (t) = 

  h 0ρ  sin ϕρ (t) − ϑρ ω2 + Ωρ

ϕ˙ ρ (t) = ω2 − Ωρ −

  h  0ρ  cos ϕρ (t) − ϑρ aρ (t) ω2 + Ωρ

    a2 (t) h h 0φ  sin ϕφ (t) − ϑφ +  0φ  ρ 2 sin ϕφ (t) − ϑφ a˙φ (t) ≈ −   2 ω1 + Ωφ 3 ω 1 + Ωφ     a2ρ (t) h ω1  cos ϕφ (t) − ϑφ +  0φ ϕ˙ φ (t) ≈ ω1 − Ωφ + − 1− 12 22 2aφ (t) ω1 + Ωφ +

  a2 (t) h  ρ 2 cos ϕφ (t) − ϑφ  0φ  3aφ (t) ω1 + Ωφ

(74)

where Ωφ ≈ ω1 and Ωρ ≈ ω2 are external excitation frequencies in the resonant rages corresponding eigen frequencies of corresponding linearized system. Previous system of four nonlinear and first-order differential equation in the first asymptotic approximation are obtained by asymptotic Krilov–Bogoliyubov– Mitropolyskiy method and for small amplitudes of external excitations and in the resonant rages of the both frequencies. Taking into consideration some conclusion from considered system of the spring pendulum, we can conclude, also, that it is important to consider more simple case of the coupling between linear and nonlinear systems with one degree of freedom with different types of the coupling realized by simple static or dynamic elements, for to investigate hybrid phenomena in the nonlinear system forced dynamics. Also, it is possible to use for energy analysis of the transfer energy between energies carried by generalized coordinates φ and ρ in this nonlinear system forced dynamics with two degree of freedom, but formally, we can take into account that, we have two oscillators, one nonlinear and one linear each with one degree of freedom as two subsystems coupled in the hybrid system with two degree of freedom, by hybrid connection realized by statical and dynamical connections. This interconnection have two parts of energy interaction between subsystems expressed by kinetic and potential energy in the form (10).

Energy and Nonlinear Dynamics of Hybrid Systems

55

Taking into consideration some conclusion for considered system of the spring pendulum forced oscillations, we can conclude also that it is important to consider more simple case of the coupling between linear and nonlinear systems with one degree of freedom with different types of the coupling realized by simple static or dynamic elements, for to investigate hybrid phenomena in the system forced dynamics.

3.2 A Nonlinear Oscillator For to obtain asymptotic approximation of the nonlinear differential equation (73) by using asymptotic methods Krilov–Bogoliyubov–Mitropolyskiy, we propose solution in the first approximation in the following form: x1 (t) = R1 (t) e−δ1 t cos Φ1 (t)

(75)

where amplitude R1 (t) and phase Φ1 (t) are unknown functions and defined by system of the first-order differential equations in the following form: R˙ 1 (t) = −

h01 eδ1 t sin φ1 (t) p1 + Ω 1 ( τ )

(76)

3ω˜ 2 h01 eδ1t sin φ1 (t) φ˙1 (t) = p1 − Ω1 ± N1 R21 (t)e−2δ1t + 8p1 R1 (t) [p1 + Ω1 (τ )]

(77)

where upper sign (+) for hard and lower sign (−) for soft nonlinearity. Also, where: Φ1 (t) = p1t + φ1 , and for the case that frequency of external excitation is in the frequency interval of resonant range of the eigen frequency of the corresponding linearyzed system, Ω1 ≈ p1 . By using previous first asymptotic approximation of the solution (74)-(76)-(77) in the single frequency regime, we can obtain Lyapunov exponent in the form:

 1 2 1 2 λ1 = lim ln x1 (t) + 2 x˙1 (t) = −δ1 < 0 (78) t→∞ 2t ω1 In our research, we can investigate small nonlinearity and small vibrations around periodic vibrations in regimes of stationary resonant ranges, and far of resonant frequency range. For forced vibrations and for work of the external excitation force and damping force, we can write that: AFTPw(1)

=

TP

   FW (1) x˙ p(1) dx p(1) (t) = − b1 (x˙P1 )2 dt TP

0

F(t) AT (1) P

=

TP 0

(79)

0

F(1) (t)dx p (t) = −

TP 0

F(1) (t)x˙P1 dt

(80)

56

K.R. (Stevanovi´c) Hedrih F(t)

In linear systems AFTw(1) and AT (1) , these works for one period of the external P P excitation are equal, and in result of the appearance of the pure periodic forced vibrations with frequency of the external one frequency excitation. But in the nonlinear system when external excitation frequency is outside of the resonant frequency range intervals, and in the system appear pure periodic forced vibrations with external excitation frequency, then we can conclude that these works are equal. But, in nonlinear systems, it is evident that under the influence of the pure one-frequency external excitation, it is a possible appearance of different forced vibration regimes, as double periodic as well as chaotic like and stochastic like regimes, and this need to find relations between these works, of the external excitations and damping force. Also, it needs to investigate energy used to chaotic like and stochastic like forced regime appearance.

3.3 A Linear Oscillator Expressions of kinetic and potential energies, and Raleigh energy dissipation function of linear oscillator, see Fig. 2b∗ , with one degree of freedom and generalized coordinate x2 are same as expression (29). For this case, ordinary differential equation is in the following form: x¨2 + 2δ2 x˙2 + ω22 x2 = h02 cos(Ω2 t + ϑ02 ), where ω22 = mc22 , 2δ2 = mb22 , h02 = mF22 and with characteristic eigen numbers: λ1,2 = −δ2 ∓ i ω22 − δ22 for the small damping coefficient δ2 < ω2 . F(t)

In linear systems AFTPw(1) and ATP (1) , these work for one period of the one period of the external excitation are equal, and in result of the appearance of the pure periodic forced vibrations with frequency of the external one frequency vibrations: AFTw(2) P

=

TP

   FW (2) x˙ p(2) dx p(2) (t) = − b2 (x˙P2 )2 dt TP

0

F(t) AT (2) P

=

TP 0

(81)

0

F(2) (t)dx p(2) (t) = −

TP

F2 (t)x˙P2 dt

(82)

0

3.4 Hybrid System with Static Constraints Expressions for kinetic and potential energies, and Rayleigh function of energy dissipation of the hybrid system (see Fig. 3), containing two subsystems – one linear oscillator and one nonlinear oscillator, in results, with two degree of freedom are

Energy and Nonlinear Dynamics of Hybrid Systems

57

Fig. 3 Hybrid system containing coupled subsystems by (a∗ ) static constraint, coupled by linear spring rigidity c and (b∗ ) dynamical constraint, coupled by rolling element of the mass m – dynamic coupling: one nonlinear (left) and second linear (right) excited by external excitations

expressed by generalized coordinates x1 and x2 and in the forms (31), (32) and (33). For this system, it is possible to show that: dtd (Ek + E p ) = −2Φ is for free vibrations → → − → → − and dtd (Ek(1) + E p(1)) = −2Φ(1) + ( F 1 , − v 1) + ( F 2, − v 2 ) is for forced vibrations. Energy interaction in this hybrid system, containing two coupled subsystems by statically coupling (spring) element, is potential energy of the spring for coupling nonlinear and linear subsystems, and is expressed in the form (34). System of the coupled ordinary differential equations of the hybrid system dynamics, containing two subsystems, one nonlinear and one linear are in the forms:   2 3 x¨1 + 2δ1x˙2 + ω12 + a21 x1 − a21 x2 = ∓ω˜ N1 x1 + h01 cos (Ω1t + ϑ01 )

(83)

  x¨2 + 2δ2x˙2 + ω22 + a22 x2 − a22 x1 = h02 cos (Ω2t + ϑ02 )

(84)

where upper sign (−) for hard and lower sign (+) for soft nonlinearity, and are:

ωi2 =

ci bi c c˜1 F0i 2 , 2δi = , a2i = , ω˜ N1 = , h0i = i = 1, 2. mi mi mi m1 mi

(85)

Taking into account that consideration of the homogeneous system does not loose generality of the phenomena, our next consideration is to use this homogeneous hybrid system as a basic system to the nonhomogeneous with small nonlinearity members. By use the corresponding basic linear differential equations of the coupled system of the differential equations of the hybrid system containing two subsystems, one linearized and one linear, we can obtain a characteristic equation with four characteristic numbers, same as in part 2.3. By using asymptotic method

58

K.R. (Stevanovi´c) Hedrih

Krilov–Bogoliyubov–Mitropolyskiy, we propose solution in the first approximation in the following form: x1 (t) = e−δ t [R1 (t) cos Φ1 (t) + R2 (t) cos Φ2 (t)] x2 (t) = e−δ t [R1 (t) cos Φ1 (t) − R2 (t) cos Φ2 (t)]

(86)

where amplitudes and phases, Ri (t) and Φi (t), i = 1, 2 are unknown time functions and defined by system of the first order differential equations in the following form: h01 eδ1 t sin φ1 (t) (p1 + Ω1 (τ ))   1 2 δ1 t (R1 (t))2 e−3δ1 t + 3 (R2 (t))2 e−(2δ2 +δ2 )t − φ˙1 = p1 − Ω1 ± ω˜ N1 e 16p1

R˙ 1 (t) = −

−

h01 eδ1 t cos φ1 (t) (p1 + Ω1 (τ ))1 R1 (t)

h02 eδ2 t sin φ2 (t) (p2 + Ω2 (τ ))   ω˜ 2 φ˙2 = p2 − Ω2 ± N1 eδ2t 3 (R1 (t))2 e−(2δ1 +δ2 )t + (R2 (t))2 e−3δ2 t + 16p2

R˙ 2 (t) = −

+

h02 eδ2 t cos φ2 (t) (p2 + Ω2 (τ )) R2 (t)

(87)

where: upper sign (+) for hard and lower sign (−) for soft nonlinearity, and Φi (t) = pit + ϑ0i + φi , Ωi ≈ pˆi , i = 1, 2. By using previous first asymptotic approximation of the two amplitudes and two phases of first asymptotic approximation of solution, we can obtain Lyapunov exponents in the forms: 

1 1 ln x21 (t) + 2 x˙21 (t) = −δ1 < 0 t→∞ 2t ω1

 1 1 λ2 = lim ln x22 (t) + 2 x˙22 (t) = −δ2 < 0 t→∞ 2t ω2

λ1 = lim

(88)

Also, taking into account that system is nonlinear: 

ω˜ 2 4 1 2 1 λ˜ 1 = lim ln x21 (t) + N1 x (t) + x ˙ (t) = −δ 1 < 0 t→∞ 2t ω12 1 ω12 1 

Esubsist(1) 1 = −δ1 < 0 λ˜ 1 = lim ln t→∞ 2t 2m1 ω12

(89)

Energy and Nonlinear Dynamics of Hybrid Systems

59

3.5 Hybrid System with Dynamical Coupling (Rolling) Element In Fig. 3b∗ , a hybrid system excited by two frequency external excitation, and containing two coupled oscillators, one linear and one nonlinear coupled by dynamically coupling in the form of rolling element is shown. Expressions for kinetic and potential energies, and Rayleigh function of energy dissipation of the hybrid system (see Fig. 3b∗ ), containing two subsystems – one linear oscillator and one nonlinear oscillator, in results, with two degree of freedom are expressed by generalized coordinates x1 and x2 and with dynamic coupling by rolling element, are same as expressions (48) (49), (51), (51), (52), and (53). This system is excited by external two frequency excitation Fi (t) = F0i cos(Ωi t + ϑi ), i = 1, 2 applied to the subsystems into hybrid system. For this system, it is possible to show that is: dtd (Ek + E p ) = −2Φ for free vibrations, and for forced vibrations that is:

→  d  − →  − → →  Ek(1) + E p(1) = −2Φ(1) + F 1 , − v 1 + F 2, − v2 . (90) dt The kinetic energy of the dynamical, coupling, by rolling element is expressed by JC (48)–(49). Coefficient a˜12 = m4 − 4R 2 is coefficient of the subsystems dynamical coupling, and this coefficient is coefficient of mass inertia with source in coupling rolling element. When this coefficient is equal to zero, then the system coordinate x1 and x2 are decoupled and there are not energy of the coupling, but there are energy of the influence of the dynamic coupling by mass of the additional member in the system. Then subsystems haven’t kinetic energy interaction, but hybrid system have additional part of the kinetic energy of the first subsystem – reduction of the dynamic coupling element to the first subsystem and additional part of the kinetic energy of the first subsystem – reduction of the dynamic coupling element to the second subsystem. Kinetic energy of the first subsystem as a one part of the hybrid system is: Ek = 1 2 a ˜ 2 11 x˙1 , kinetic energy of the second subsystem as a one part of the hybrid system kinetic energy is: Ek = 12 a˜22 x˙22 . Kinetic energy of the coupling of the subsystems as a two parts of the hybrid system is: Ek = a˜12 x˙1 x˙2 . Additional part of the kinetic energy of the first subsystem – reduction of the dynamic coupling element to the first subsystem is Ek(1)d = 12 aˆ11 x˙21 and additional part of the kinetic energy of the second subsystem – reduction of the dynamic coupling element to the second subsystem is Ek(2)d = 12 aˆ22 x˙22 . After introducing the following notations:

κ1 =

a˜12 a˜12 2 c1 c1 , κ2 = , ω˜ = , ω˜ 2 = , a˜11 a˜22 1 a˜11 2 a˜22

c˜1 bi F0i 2 2 m1 ω˜ N1 = = ω˜ N1 , 2δ˜i = , h0i = i = 1, 2 a˜11 a˜11 a˜ii a˜ii

(91)

60

K.R. (Stevanovi´c) Hedrih

the system of ordinary differential equations of the forced dynamics of the hybrid system obtain the following form: 2 3 x¨1 + κ1 x¨2 + ω˜ 12 x1 + 2δ˜1x˙1 = ∓ω˜ N1 x1 + h01 cos (Ω1 t + ϑ01 )

x¨2 + κ2 x¨1 + ω˜ 22 x2 + 2δ˜2x˙2 = h02 cos (Ω2 t + ϑ02 )

(92)

where upper sign (−) for hard and lower sign (+) for soft nonlinearity. For the basic linear equations of the dynamically coupled system of the differential equations (92) of the hybrid system containing two subsystems, one linear and one linearized, have characteristic equation with four roots: λ1,2 = −δˆ1 ∓ i pˆ1 and λ3,4 = −δˆ2 ∓ i pˆ 2 with discussion of their values. By using asymptotic method Krilov–Bogoliyubov– Mitropolyskiy, we propose solution in the first approximation in the form (52), where amplitudes Ri (t) and phase Φi (t), i = 1, 2 are unknown functions and determined by system of the first-order differential equations in the following first asymptotic approximation form: R˙ 1 (t) = e−δ1t

 (2) (2) h01 K21 + κ2 K22 Δ˜ 12 [ pˆ1 + Ω1 (τ )]

sin φ1 (t)

φ˙1 (t) = pˆ1 − Ω1 (τ ) +

 (2) (2) 2

 K21 + κ2 K22  3ω˜ N1 ˆ (1) 3 e−2δ1 t K21 [R1 (t)]2 ∓ 16Δ˜ 12 pˆ1    (2) 2 2 −2δˆ2 t (1) +2e K21 K21 [R2 (t)] − −e−δ1t

 (2) (2) h01 K21 + κ2 K22

sin φ1 (t) Δ˜ 12 [ pˆ1 + Ω1 (τ )] R1 (t)

 (1) (1) h02 K22 + κ1K21 sin φ2 (t) R˙ 2 (t) = −eδ2 t Δ˜ 21 [ pˆ2 + Ω2 (τ )]

φ˙2 (t) = pˆ2 − Ω2 (τ ) +

 (1) (1) 2

 3ω˜ N1 K22 + κ2 K21  ˆ (1) 2 (2) ∓ 2e−2δ1 t K21 K21 [R1 (t)]2 16Δ˜ 21 pˆ2    ˆ (2) 3 +e−2δ2 t K21 [R2 (t)]2 + +eδ2 t

 (1) (1) h02 K22 + κ1 K21 Δ˜ 21 [ pˆ2 + Ω2 (τ )] R2 (t)

cos φ2 (t)

(93)

Energy and Nonlinear Dynamics of Hybrid Systems

61

where: upper sign (+) for hard and lower sign (−) for soft nonlinearity, and also, Φi (t) = pˆit + φi , Ωi ≈ pˆi , i = 1, 2, τ = ε t slow-changing time and determinants: 





 

(1) (1) (2) (2) (1) (1) (2) (2) K21 + κ2 K22 − K22 + κ2 K21 K21 + κ1 K22 Δ˜ 12 = K21 + κ1K22 Δ˜ 12 == 0 







 (2) (2) (1) (1) (2) (2) (1) (1) Δ˜ 21 = K21 + κ1K22 K21 + κ2 K22 − K22 + κ2 K21 K22 + κ1 K21 Δ˜ 21 = 0 (94) By using previous system of first asymptotic approximation of amplitudes and phases of solution first asymptotic approximation, we can obtain Lyapunov exponents in the forms:

 1 2 1 2 λ1 = lim ln x1 (t) + 2 x˙1 (t) = −δˆ1 < 0 t→∞ 2t ω˜ 1

 1 1 λ2 = lim ln x22 (t) + 2 x˙22 (t) = −δˆ2 < 0 (95) t→∞ 2t ω˜ 2 Also, taking into account that system is nonlinear, we can introduce first Lyapunov exponents in the forms:

 2 ˜λ1 = lim 1 ln x2 (t) + ω˜ N1 x4 (t) + 1 x˙2 (t) = −δˆ1 < 0 1 t→∞ 2t ω˜ 12 1 ω˜ 12 1

 Esubsist(1) 1 λ˜ 1 = lim ln (96) = −δˆ1 < 0 t→∞ 2t 2m1 ω˜ 12

4 Concluding Remarks on Energy Analysis 4.1 Energy Analysis of Nonlinear System Numerical analysis of the series of the amplitude-frequency [a2 (t) = e−δ2 t R2 (t), ˆ Ω1 (τ ) = ν1 (τ )] and [a2 (t) = e−δ2 t R2 (t), Ω2 (τ ) = ν2 (τ )] and phase-frequency curves [ϕ1 (t) = φ1 (t), Ω1 (τ ) = ν1 (τ )] and [ϕ2 (t) = φ2 (t), Ω2 (τ ) = ν2 (τ )] for stationary and nonstationary resonant regimes: (a∗ ) amplitude-frequency curves for linear one-frequency stationary and nonstationary regime process for different velocities of forced excitation frequency change passing through the resonant range; (b∗ ) amplitude-frequency curves for nonlinear-like one-frequency stationary and nonstationary regime process – oscillatory process for different velocities of ˆ

62

K.R. (Stevanovi´c) Hedrih

forced excitation frequency change passing through the resonant range in both directions – increasing and decreasing frequency; (c∗ ) amplitude-frequency and (d∗ ) phase-frequency curves of a stationary resonant state of like two-frequency nonlinear oscillations of a nonlinear system with two degree of freedom obtained by integration of the system differential equations for the amplitudes and phases in the first asymptotic approximation show that interactions between modes appears. Numerical analysis illustrates the characteristic phenomena of a like two-frequency regime of coupled resonant states under stationary conditions for the system with small nonlinearity. We can notice the appearance of the singularity trigger with stable knots and homoclinic unstable saddle-type points along amplitude-frequency and phase-frequency characteristics for the resonant frequency interval of the resonant frequency interactions. The appearance of multiple resonant jumps, typical for multifrequency-coupled resonant states, is also noticeable. The appearance of homoclinic points of unstable saddle type points to the appearance of stochasticlike and chaotic-like processes in subsystems of hybrid system have source in coupled resonant states multifrequency oscillation regimes. This requires further study separate for each particular case. Under the conditions of nonlinear system multifrequency forced oscillations, and by using the asymptotic method of Krilov–Bogolyubov–Mitropolyskiy, the appearance of “own circular frequencies stroll” may be noticed. Resonant frequency ranges dependent on the character of nonlinearity, and on the initial conditions and momentary adequate nonlinear harmonics and amplitudes and phases are formed in that way. That is the reason why, besides the notion “resonant state”, we introduce notions “passage through the coupled resonant states” and “coupled resonant states”. With the appearance of the own circular frequency stroll, a mutual influence shown either in adequate harmonics amplitudes, frequencies, and phases increase or decrease appears. As the energy (kinetic, potential, and the energy of dissipation caused by dissipative forces, as well as the energy of forced multifrequency forces work) “carried” by a nonlinear harmonics of a corresponding oscillations “stroll” frequency depends both on the amplitudes square and on the square of its time derivatives, or frequency, the harmonic amplitudes, phases, or frequencies change during the oscillatory process and regime itself as well as the interaction between them causes the energy change. The appearance of energy transfer from one harmonic onto other or others of higher or lower frequencies can also be noticed here. On amplitude-frequency and phase-frequency multifrequency forced oscillations diagrams, we can notice the appearance of one or more resonant jumps which point to the appearance of a resonant energy jumps, both kinetic and potential, carried by a nonlinear harmonic. We can see that while one harmonic jumps to higher amplitudes, the other one to lower ones. The energy jumps indicated on energy-frequency graph of one nonlinear mode is similar as corresponding jumps in amplitude frequency graphs as well as in phase-frequency graphs. On certain harmonics frequencies, a resonant jump of energy carried by the observed nonlinear harmonics, onto a lower or a higher value, happens. At the same time, similar resonant jumps of energy in the opposite or the same direction happen on other harmonics.

Energy and Nonlinear Dynamics of Hybrid Systems

63

If the stationary multifrequency forced oscillations amplitudes and phases, an appearance of amplitude trigger and of coupled amplitudes triggers of coupled stationary singularities in an amplitudes combination stable–unstable–stable. That way, we have a trigger and/or a coupled triggers of harmonic-energies in a corresponding set of fixed frequencies from the harmonics’ coupled resonant frequencies range. The analysis of total mechanical energy of a nonlinear system is also significant, as well as the analysis of the kinetic and potential energy of each nonlinear harmonics, in the singular and bifurcation states, and especially of those corresponding two unstable and hyperbolically, as well as homoclinical orbits. The question about the analysis of energy and its transfer between harmonics under conditions under which chaotic-like and stochastic-like vibrations appear in deterministic nonlinear dynamical system remains open. In nonlinear systems, we can observe the idea of equivalent systems exchange by the use of elementary linear simple oscillators which would be uncoupled and would make an equivalent replacement for a linearized system. And after that, we may, using the asymptotic methods of nonlinear mechanics, for instance, the method of Krilov–Bogolyubov–Mitropolyski, compose a system of necessary approximation of the first-order ordinary differential equations for nonlinear oscillations harmonics amplitudes and phases that are close to an unperturbed oscillations. From such a system of adequate approximation differential equations for amplitudes and phases that are mutually coupled by non-linear members, we may, using either quantitative or qualitative analysis, derive certain conclusions about the flows and the transfer of energy by following the phase and harmonics trajectories through the phase space of dynamical systems state. A generalization of an analytical analysis of the transfer energy between linear and nonlinear oscillators for free vibrations with different type of coupling as a couple between two subsystems each of them with one degree of freedom is also important, but it is new task.

5 Energy Analysis of Hybrid Complex Structures 5.1 A Double Plate System 5.1.1 Partial Differential Equations By using the model of double plate system with viscoelastic layer (similar as in Refs. Hedrih (2005, 2006)), we can consider the energy transfer between plates. For that reason, we use corresponding derived partial differential equations and

64

K.R. (Stevanovi´c) Hedrih

w1 ( x, y, t)

b h1

y

a

c,b

h1

h1 h1

h1 x h1

h2

w2

(x, y , t )

h1

Fig. 4 Double plate system with viscoelastic layer: structure and noted corresponding kinetic parameters and coordinate systems

corresponding analytical results and expressions for solutions of the transversal displacements of the both plates vibrations. This double plate system is presented in Fig. 4. The governing system of the coupled partial nonlinear differential equations for free double plates oscillations is in the following form (see Ref. [12]):

 ∂ 2 w1 (x, y,t) ∂ w2 (x, y,t) ∂ w1 (x, y,t) 4 − + c(1) ΔΔw1 (x, y,t) − 2δ(1) − ∂ t2 ∂t ∂t −a2(1) [w2 (x, y,t) − w1 (x, y,t)] = = ±εβ(1) [w2 (x, y,t) − w1 (x, y,t)]3 + q˜(1) (x, y,t)

 ∂ 2 w2 (x, y,t) ∂ w2 (x, y,t) ∂ w1 (x, y,t) 4 + c ΔΔw (x, y,t) + 2 δ − + 2 (2) (2) ∂ t2 ∂t ∂t +a2(2) [w2 (x, y,t) − w1 (x, y,t)] = = ±εβ(2 [w2 (x, y,t) − w1 (x, y,t)]3 + q˜(2) (x, y,t)

(97)

where in first partial differential equation in the system (1) upper sign (+) for hard and lower sign (−) for soft nonlinearity, and in the second partial differential equation in the system (1) upper sign (−) for hard and lower sign (+) for soft nonlinearity. Also in the previous system of partial differential equations: wi (x, y,t), i = 1, 2 are plate small transverse deflections (with means, as has been discussed in books [88] and small compared to the plates thickness, hi , i = 1, 2,) and that plates vibrations occur only in the orthogonal direction with respect to the parallel middle surfaces of the plates passing through their parallel contours with same boundary plates conditions; a2(i) = ρichi , 2δ(i) = ρibhi i = 1, 2 and c4(i) = ρDi hi i , i = 1, 2 with Di =

Ei h 3 , 12(1− μ 2 )

i = 1, 2 corresponding bending cylindrical rigidities of the plates,

Energy and Nonlinear Dynamics of Hybrid Systems

65

and ΔΔ = ∂∂x4 + 2 ∂ x∂2 ∂ y2 + ∂∂y4 is differential operator; Ei modulus of elasticity, μi Poisson’s ratio and Gi shear modulus, ρi plate mass distribution. The plates are interconnected by a viscoelastic layer with constant surface stiffness c and with constant surface damping force coefficient b distributed along all plates’ surfaces. For the solutions of the governing system of the corresponding coupled partial differential equations (97) for forced double plate system oscillations, we take into account the eigen amplitude functions W(i)nm (x, y), i = 1, 2, n, m = 1, 2, 3, 4, . . ..∞ and the time expansion with the coefficients in the form of the unknown time functions T (t), i = 1, 2, n, m = 1, 2, 3, 4, . . . .∞ describing their time evolution: 4

4

wi (x, y,t) =

4

∞

∞

∑ ∑ W(i)nm (x, y) T(i)nm (t), i = 1, 2

(98)

n=1 m=1

where the eigen amplitude functions W(i)nm (x, y), i = 1, 2, n, m = 1, 2, 3, 4, . . . .∞ are the same, for both plates in the system, as in the case with decoupled plates problem (see Ref. [12]). Then after introducing expression (98) into governing system of the coupled partial differential equations for forced double plates oscillations in the form (97): and after multiplying first and second equation with W(i)sr (x, y)dxdy and after integrating along the middle plate surface and taking into account orthogonality conditions and corresponding equal boundary conditions of the plates, we obtain the mn-family of the systems containing coupled two ordinary differential equations for determination series of the unknown time functions T(i)nm (t), i = 1, 2, n, m = 1, 2, 3, 4, . . ..∞. We take into consideration, the case, when external distributed two-frequencies force is applied and distributed along upper surfaces of upper plate with both frequencies near eigen circular frequencies of coupled plate system presented by linearized model, Ωinm ≈ pˆinm , i = 1, 2. In this case, the lower plate is free of load. We can conclude that external excitation frequencies are in the resonant frequency interval close to the resonant frequency of corresponding linear double plate system. We suppose that the functions of external excitation at nm-mode of oscillations are the two-frequency process in the form: q˜(i)nm (t) = h01nm cos[Ω1nmt + φ1nm ] + h02nm cos[Ω2nmt + φ2nm ]

(99)

For this case of the external two-frequency excitation time functions T(i)nm (t), describing their time evolution of the transversal displacements of the plate middle surface points are in the following forms (see Refs. by and [12, 54, 57, 58]): (1)

(2)

T(i)nm (t) = Kinm e−δ1nmt R1nm (t) cos Φ1nm (t) + Kinm e−δ2nmt R2nm (t) cos Φ2nm (t) (100) ˆ

ˆ

s where Kijnm cofactors of determinant corresponding to basic linear homegenous coupled system, δˆinm real parts of the corresponding pair of the roots of the characteristic equation and amplitudes Rinm (t) and full phases Φinm (t) = Ωinmt + φinm (t) unknown time functions which, we are going to obtain using the Krilov– Bogolyubov–Mitropolyskiy asymptotic method (see Refs. [73–80]). It is taken into account that defined task satisfy all necessary conditions for applying asymptotic

66

K.R. (Stevanovi´c) Hedrih

method Krilov–Bogolyubov–Mitropolskiy concerning small parameter and that external excitation frequencies Ω1nm ≈ pˆ1nm and Ω2nm ≈ pˆ2nm are in the resonant frequency intervals of the corresponding eigen frequencies of unperturbed linear system solution. By applying the asymptotic method Krilov–Bogolyubov–Mitropolskiy (1965), we obtain the system of the first-order four ordinary differential equations according ˆ unknown amplitudes Rinm (t) = eδinm t ainm (t) and full phases Φinm (t) = Ωinmt + φinm (t) in the first asymptotic approximation in the following forms (see Refs. [6, 53, 67, 95]):  

(2) (2) (1) (1) δ(1) K22nm + δ(2) K21nm K22nm − K21nm

 a˙1nm (t) = − a1nm (t) + (2) (1) (2) (1) K22nm K21nm − K21nmK22nm (2)

+

h K

22nm 01nm  cos φ1nm (2) (1) (2) (1) (Ω1nm + pˆ 1nm) K22nm K21nm − K21nmK22nm

  φ˙1nm (t) = pˆ− nm1 Ω1nm − 

(2) (2) ε Ψ (Wnm ) β(1) K22 + β(2)K21  ±

 (2) (1) (2) (1)  K22 K21 − K21 K22 Ω+ 1nm pˆ nm1

(1)

(1)

K22nm −K21nm

3 3 8

a21nm (t) +

 

2 1 (1) (1) (2) (2) K22nm −K21nm K22nm −K21nm a22nm (t) − 2 (2)

−

K22 h01nm

 sin φ1nm (2) (1) (2) (1) (Ω1nm + pˆ nm1) K22nm K21nm − K21nmK22nm a1nm (t)

 

(1) (1) (2) (2) δ(1) K22nm + δ(2) K21nm K22nm − K21nm

 a˙2nm (t) = − a2nm (t) + (1) (2) (2) (1) K22nm K21nm − K22nmK21nm (1)

+

h K

22nm 02nm  cos φ2nm (1) (2) (2) (1) (Ω2nm + pˆ2nm) K22nm K21nm − K22nm K21nm

φ˙2nm (t) = ( pˆ 2nm − Ω2nm ) −

 (1) (1) ε Ψ (Wnm ) β(1) K22nm + β(2)K21nm ·

± (1) (2) (2) (1) (Ω2nm + pˆ 2nm) K22nm K21nm − K22nmK21nm

(101)

Energy and Nonlinear Dynamics of Hybrid Systems

67





2

 3 3 1 (1) (1) (2) (2) (2) (2) 2 2 −K K22nm −K21nm a1nm (t)+ K22nm −K21nm K a (t) − 2 22nm 21nm 8 2nm (1)

−

K22nm h02nm 

sin φ2nm (1) (2) (2) (1) (Ω2nm + pˆ 2nm) K22nm K21nm − K22nm K21nm a2nm (t)

where upper sign (+) for hard and lower sign (−) for soft nonlinearity, and also, a b

Ψ(Wnm ) =

00 a b 00

4 W(1)nm (x,y)dxdy

is coefficient of influence of ideal elastic layer non-

2 W(1)nm (x,y)dxdy

linearity. 5.1.2 Kinetic Energy of Plates The expressions of kinetic energies of the plates are in the following forms: (i)

Ek = (i) Ek

1 2





ρi

V

1 = ρ i hi 2

 A



∂ wi (x, y,t) ∂t ∞

2 dzdA 2

∞

∑ ∑ W(i)nm (x, y) T˙ (i)nm (t)

dA, i = 1, 2

(102a)

n=1 m=1

or in the form: ∞ ∞  2 1 (i) Ek = ρi hi ∑ ∑ M(1)nm (x, y) T˙ (i)nm (t) , i = 1, 2 2 n=1 m=1

(102b)

where M(i)nmsr =



 W(i)nm (x, y)W(i)sr (x, y) dA =

A

0 M(1)nm

sr = nm sr = nm

(103)

The kinetic energy of the one plate we can express in the form of the sum by (i) components Ek,nm belong to corresponding mn-family mode n, m = 1, 2, 3, 4, . . ..∞ in the following form: ∞ ∞ (i) (i) Ek = ∑ ∑ Ek,nm , i = 1, 2 (104) n=1 m=1

(i)

where the kinetic energy components Ek,nm , i = 1, 2 belong to corresponding mn-family mode n, m = 1, 2, 3, 4, . . ..∞ was expressed by derivatives of the eigen component time functions belong to same corresponding mn-family mode  2 1 (i) (i) Ek,nm = ρi hi M(1)nm (x, y) T˙ (i)nm (t) = M(i)nmsr E˜ k,nm , i = 1, 2 2

(105)

68

K.R. (Stevanovi´c) Hedrih

(i) Also, we can introduce reduced component of the kinetic energy E˜ k,nm .i = 1, 2 belong to corresponding mn-family mode n, m = 1, 2, 3, 4, . . ..∞ in the following form:

(i) E˜ k,nm =

(i)

Ek,nm M(1)nm (x, y)i

=

2 ρ i hi  ˙ T(i)nm (t) , i = 1, 2 2

(106)

5.1.3 Potential Energy of Plates The potential energy of the plate is equal to energy of the deformation of elastic plate in the vibration state and expression, we can write in the following form: E p = Ad =

1 2



[εx σx + εy σy + εz σz + γxy τxy + γxz τxz |γyz τyz ] dV

(107)

V

where εx , εy , εz , γxy , γxz , γyz are tensor strain components, σx , σy , σz , τxy , τxz , τyz are tensor stress components of the plate strain and stress vibration state. Tensor stress components τzx and τzy are small, as tensor strain components γzx and γzy are also small and can be neglected in the comparison with other members in expression for work of elastic deformation of the thin plate. Then, we can express the work of elastic plate deformation on the simpler form. Also, we take into account that plates are thin and that stress state is planar and that we can calculate with middle plate surface, and make averaging with respect to the middle plate surface (see Ref. [88]) and for the work of elastic plate deformation, we can write the following approximate expression: D Ep ≈ 2



 A

∂ 2 w (x, y,t) ∂ x2

∂ 2 w (x, y,t) +2μ ∂ x2



2

∂ 2 w (x, y,t) + ∂ y2

2

2  2  ∂ 2 w (x, y,t) ∂ w (x, y,t) dA (108) + 2 (1 − μ ) ∂ y2 ∂ x∂ y

After introducing solutions (98) in previous expression (108) for expression of the potential energy of the plates, we obtain the following: (i)

Ep ≈

Di ∞ ∞ ∞ ∞ ∑ ∑ ∑ ∑ Cnm,sr(i) T(i)nm (t) T(i)sr (t), i = 1, 2 2 n=1 m=1 s=1 r=1

where, 4 Cnm,sr(i) = knm

⎧ ⎪ ⎨ = M ⎪ ⎩ (i)nm

sr = nm 2 W(i)sr (x, y) dA sr = nm

  0 A

(109)

(110)

Energy and Nonlinear Dynamics of Hybrid Systems

69

The potential energies of the separate plates are in the following forms:  2 Di ∞ ∞ 4 knm M(i)mm T(i)nm (t) ∑ ∑ 2 n=1 m=1

(111)

2  ρi h i ∞ ∞ 2 ω(i)nm M(i)nm T(i)nm (t) ∑ ∑ 2 n=1 m=1

(112)

(i)

Ep ≈ or in the forms: (i)

Ep ≈ 2 where ω(i)nm =

Di 4 ρi hi knm .

The potential energy of the one plate, we can express in (i)

the form of the sum by components E p,nm belong to corresponding mn-family mode n, m = 1, 2, 3, 4, . . ..∞ in the following form: ∞

(i)

Ep =

∞

∑ ∑ E p,nm, i = 1, 2 (i)

(113)

n=1 m=1 (i)

where the energy components Ek,nm , i = 1, 2 belong to corresponding mn-family mode n, m = 1, 2, 3, 4, . . . .∞ was expressed by derivatives of the component time functions belong to same corresponding mn-family mode (i)

E p,nm ≈

2  ρ i hi 2 (i) ω Mnm T(i)nm (t) = M(i)nmsr E˜ p,nm 2 (i)nm

(114)

(i)

Also, we can introduce reduced component potential energy E˜ p,nm .i = 1, 2 belong to corresponding mn-family mode n, m = 1, 2, 3, 4, . . . .∞ (i) 2  1 E p,nm (i) 2 M(1)nm T(i)nm (t) = . E˜ p,nm = ρi hi ω(i)nm 2 M(i)nmsr

(115)

5.1.4 Potential Energy of Visco-Nonlinear Elastic Layer For analysis of the double plate system with visco-nonlinear elastic layer, we can write expression for the potential energy of the constraints between coupled plates in the form of the energy of deformation of the distributed elastic layer neglected mass and properties of inertia and neglected kinetic energy. Then expression for the potential energy of the coupling of the plates is in the form: E p(a,b)eyer =

 A

 c c˜ (w2 − w1 )2 ± (w2 − w1 )4 dA 2 4

(116)

70

K.R. (Stevanovi´c) Hedrih

After introducing solutions (98) in previous expression (116), and taking into account ortogonality conditions, for expression of the potential energy of the plate coupling, we obtain the following two next parts of expression for two pointed parts of the potential energy of the nonlinear coupling of the plates in the forms:  2 1 ∞ ∞ E p(1,2)ayer,linear = c ∑ ∑ M(1)nm T(2)nm (t) − T(1)nm (t) 2 n=1 m=1

(117)

4  1 ∞ ∞ E p,nm(1,2)ayer,non−linear = ± c˜ ∑ ∑ M˜ (i)nmnm T(2)nm (t) − T(1)nm (t) 4 n=1 m=1

(118)

where upper sign (+) for hard and lower sign (−) for soft nonlinearity of the coupling layer. The potential energy of the nonlinear elastic properties of the layer between plates, we can express in the form of the sum by components E p,nm(1,2)layer,linear and E p,nm(1,2)layer,non−linear belong to corresponding mn-family mode n, m = 1, 2, 3, 4, . . . .∞ in the form: E p(1,2)ayer =

∞

∞

∞

∞

∑ ∑ E p,nm(1,2)ayer,linear + ∑ ∑ E p,nm(1,2)ayer,non−linear

n=1 m=1

(119)

n=1 m=1

where the energy components E p,nm(1,2)ayer,linear and E p,nm(1,2)ayer,non−linear belong to corresponding mn-family mode was expressed by the eigen component time functions belong to same corresponding mn-family mode. Also, we can introduce reduced components of the potential energy of the light distributed elastic layer E˜ p,nm(1,2)ayer,linear and E˜ p,nm(1,2)ayer,non−linear , belong to corresponding mn-family mode like as: 2 E p,nm(1,2)ayer,linear 1  E˜ p,nm(1,2)ayer,linear = c T(2)nm (t) − T(1)nm (t) = 2 M(i)nm,nm

(120)

4 E p,nm(1,2)ayer,non−linear 1  E˜ p,nm(1,2)ayer,non−linear = ± c˜ T(2)nm (t) − T(1)nm (t) = 4 M˜ (i)nmnm (121) where upper sign (+ ) for hard and lower sign (-) for soft nonlinearity of the coupling layer.

5.1.5 Rayleigh Energy Dissipation For analysis of the double plate system with visco-nonlinear elastic layer, we can write expression for the Rayleigh function of the energy dissipation of the constraint between coupled plates in the form of the power of the damping force depending of velocity of the deformation of the distributed visco-nonlinear elastic layer neglected

Energy and Nonlinear Dynamics of Hybrid Systems

71

mass and properties of inertia and neglected kinetic energy. Then expression for the Rayleigh function of the energy dissipation in the viscoelastic layer of the plate system is in the form: Φ(1,2)eyer =



1 b (w˙ 2 − w˙ 1 )2 dA 2

A

1 Φ(1,2)eyer = b 2 1 Φ(1,2)eyer = b 2

  A





∂ w2 (x, y,t) ∂ w2 (x, y,t) − ∂t ∂t ∞

∑

(122) 2 dA, i = 1, 2

2   ˙ ˙ ∑ W(i)nm (x, y) T(1)nm (t) − T(2)nm (t) dA ∞

n=1 m=1

A

or in the form:  2 1 ∞ ∞ Φ(1,2)eyer = b ∑ ∑ M(1)nm (x, y) T˙ (2)nm (t) − T˙ (1)nm (t) , 2 n=1 m=1 i = 1, 2 Φ(1,2)ayer =

(123)

1 ∞ ∞ ∑ ∑ M(1)nmΦ˜ nm(1,2)ayer 2 n=1 m=1

where b(i)nmsr = b



 W(i)nm (x, y) W(i)sr (x, y) dA =

A

0 sr = nm bM(1)nm sr = nm

  Φ ˜ nm(1,2)ayer = 1 b T˙ (2)nm (t) − T˙ (1)nm (t) 2 = nm(1,2)ayer Φ 2 M(1)nm

(124)

5.1.6 Lyapunov Exponents and Concluding Remarks For each of the eigen plate time functions T(1)nm (t) and T(2)nm (t) and time processes in nm-mode, we can define Lyapunov exponents in the form:    2 2 1 ˙ 1 λnm(i) = lim ln T(i)nm (t) + 2 T(i)nm (t) t→∞ 2t ω˜ (i)nm ˜

 1 2Enm(i) ln 2 = − δ˜nm(1) + δ˜nm(2) , t→∞ 2t ω˜ (i)nm

λnm(i) = lim

i = 1, 2, n, m = 1, 2, 3, 4, . . . .∞

(125)

72

K.R. (Stevanovi´c) Hedrih

Also, by using analogy, we can define Lyapunov exponents of the plate energy interaction in the following form: ˜

 2E 1 nm(1,2) = − δ˜nm(1) + δ˜nm(2) < 0, ln t→∞ 2t ω(1)nm ω(2)nm

λnm(1,2) = lim

(126)

n, m = 1, 2, 3, 4, . . . .∞

5.1.7 Concluding Remarks For the case of the free vibrations of conservative system, these Lyapunov exponents are equal to zero. But by using this energy approach, we can introduce Lyapunov exponents of this type and way for coupled hybrid systems with different type of the material properties, as it is visco-nonlinear elastic or creep, and to use for investigation of the stability process, or deformable forms of the deformable body motion in the hybrid systems. Then, we can see that these Lyapunov exponents are measures of the processes integrity or system motion integrity. For the second case of a model of the double plate system with discontinuity in elastic layer considered as a model of the interface crack between two plates connected by thin elastic layer of the Winkler type and by using obtained results presented in Ref. by Hedrih (2006b), it is easy to conduct energy analysis of the transfer energy also using consideration from this paper and corresponding solutions from cited paper. For that case, it is necessary to take into account that all nm-families of the modes are in mutual interaction, because discontinuity in the elastic layer is special type of the strong nonlinearity. In that case, defined Lyapunov exponents obtain important role in analysis of the transfer energy between plates, including interaction between different nonlinear modes.

5.2 Energy Exchange in an Axially Moving Double-Belt System In this part, as an author’s new research result, an analytical study of the energy transfer between two coupled like-string belts interconnected by light pure elastic layer in the axially moving sandwich double belt system, in the free vibrations is presented (see Fig. 5, and Refs. [9, 40]). On the basis of the obtained analytical expressions for the kinetic and potential energy of the belts and potential energy of the light pure elastic distributed layer, numerous conclusions are derived. For the pure linear elastic double belt system, no transfer energy between different eigen modes of transversal vibrations of the axially moving double belt system, but in each from the set of the infinite numbers eigen modes, there are transfer energy between belts, and free transversal vibrations are like two-frequency, when change of the potential energy of the booth belts are four frequency, and potential energy interaction is one frequency in the each eigen mode. Changes of the kinetic energy of the both belts of the sandwich double

Energy and Nonlinear Dynamics of Hybrid Systems

73

Fig. 5 Transversal vibrations of the axially moving sandwich belt system. (a∗ ) Kinetic parameters of the transversal vibrations of the axially moving sandwich belt system. (b∗ ) Elementary segment of the axially moving sandwich belt system with length dx and notations of the kinetics parameters

axially moving belt system is two frequency like oscillatory regimes with two-time multiplicities of the eigen frequencies of the corresponding eigen amplitude mode. For concluding remarks, we can summarize research results obtained by energy analysis of the axially moving double belt system directing attentions of the reader to the author References (for detail see Refs. [9, 40]). Analogy values of the kinetic ˜ (i) (η ) energy E˜ k(i) (η ) and potential energy E˜ p(i) (η ) as well as Raleigh function Φ of the energy dissipation of the one belt of the considered axially moving sandwich double belt system are four frequency function with respect to η - coordinate in each of s-eigen modes of the double belt system transversal vibrations with infinite number of possible modes. These frequencies are double values of the both eigen frequencies of the corresponding s-mode 2 p˜s and 2 p˜s , s = 1, 2, 3, 4, . . . . . .., and values of the sum and of the difference of the two corresponding eigen s-mode of the double belt system transversal vibrations p˜s + p˜ s and p˜ s − p˜ s , s = 1, 2, 3, 4, . . .. . ... Analogy values of the system total kinetic energy E˜ k (η ) and system total ˜ η ) of the energy dissipation potential energy E˜ p (η ) as well as Raleigh function Φ( of the considered axially moving sandwich double belt system are two frequency functions with respect to η -coordinate in each s -eigen mode of the double belt system transversal vibrations infinite number of possible modes. These frequencies are double values of the both eigen frequencies of the corresponding s-mode 2 p˜ s and 2 p˜s , s = 1, 2, 3, 4, . . . . . ...of the basic belts dynamics. Analogy value of the potential energy E˜ p(1,2)(η ) of the elastic layer between belts of the axially moving double belt system – reduced analogue value of the potential energy of the interaction between belts (two subsystems coupled by elastic layer) in the hybrid system is one frequency function of η - coordinate in each eigen s -mode of the axially moving double belt system. These frequencies are double values of

74

K.R. (Stevanovi´c) Hedrih

the higher of two s-eigen frequencies of the corresponding s-mode and expressed by 2 p˜ s , s = 1, 2, 3, 4, . . .. . ... Research results presented in this paper are advances to the previous published results in the author-cited papers containing analytical and numerical research results concerning free and forced transversal vibrations of the axially moving sandwich double belt system. To the present question concerning the main aim of this research and about the usefulness of the obtained results can be answered that the primary and main aim of this research is in theoretical and methodological usefulness for university teaching process in the subject of Elastodynamics, as analytical results for introducing students with mechanisms of the transfer energy in the hybrid systems between subsystems, as well as about energy transformation inside of the sets of eigen modes. Considered axially moving, sandwich double belt system is a hybrid simple pure elastic and pure rheolinear systems with elegant possibilities to make an analysis of 0 the analogy in the plane ξ = x, η = c2v−v 2 x + t. 0

0

Acknowledgments This paper is dedicated to the memory of my Professors Dr. Ing. Math Danilo P. Raˇskovi´c (1910–1985, Serbia) and Academician Yuriy Alekseevich Mitropolskiy (1917–2008, Ukraine). Parts of this research were supported by the Ministry of Sciences of Republic Serbia through the Mathematical Institute SANU Belgrade Grants ON144002 “Theoretical and Applied Mechanics of Rigid and Solid Body. Mechanics of Materials” and OI 174001” Dynamics of hybrid systems with complex structures. Mechanics of materials”, and Faculty of Mechanical Engineering University of Niˇs.

References 1. Goroˇsko OA, Hedrih (Stevanovi´c) K (2001) Analitiˇcka dinamika (mehanika) diskretnih naslednih sistema, (Analytical Dynamics (Mechanics) of Discrete Hereditary Systems), University of Niˇs, Monograph, p. 426, YU ISBN 86-7181-054-2. 2. Goroˇsko OA, Hedrih (Stevanovi´c) K (2007) Construction of the Lagrange’s mechanics of the hereditary systems. Facta Universitatis Series Mechanics, Automatic Control and Robotics 6(1):1–23 3. Goroˇsko OA, Hedrih (Stevanovi´c) K (2007) Construction of the Lagrange’s mechanics of the hereditary systems, APM Saint Petersburg 2007 pp., Minisymposium Opening Lecture, The International Summer School APM –Advanced Problem in Mechanics, Saint Petersburg pp. 133–156. 4. Goroˇsko OA, Hedrih (Stevanovi´c) K (2008) Advances in development of the analytical dynamics of the hereditary discrete systems. J Phys: Conf Ser, 96: 012143. http://www.iop. org/EJ/main/-list=current/ 5. Hedrih (Stevanovi´c) K (1985) Multifrequency forced vibrations of thin elastic shells with positive Gauss´curvature and finite deformations. Theor Appl Mech (11):59-72. http://www.ssm. org.rs/WebTAM/arhives.html, http://www.ssm.org.rs/WebTAM/ private/arhiva/tam11.jpg 6. Hedrih (Stevanovi´c) K R. and Simonovi´c J., Multi-frequency analysis of the double circular plate system non-linear dynamics, Nonlinear Dynamics, Springer, NODY1915R2, (in press) DOI: 10.1007/s11071-011-0147-7 7. Hedrih (Stevanovi´c) K (1975) Selected Chapters from the Theory of Nonlinear Vibrations [Preprint in Serbian], Niˇs, Yugoslavia 8. Hedrih (Stevanovi´c) K (2007), Energy analysis in the nonlinear hybrid system containing linear and nonlinear subsystem coupled by hereditary element. Nonlinear Dynam Springer 51(1):127–140. doi: 10.1007/s11071-007-9197-2

Energy and Nonlinear Dynamics of Hybrid Systems

75

9. Hedrih (Stevanovi´c) K (2007) Transversal vibrations of the axially moving sandwich belts. Arch Appl Mech Springer 78(9):725–735. doi: 10.1007/s00419-006-0105-x 10. Hedrih (Stevanovi´c) K (2006) The frequency equation theorems of small oscillations of a hybrid system containing coupled discrete and continuous subsystems. Facta Univ. Ser Mech, Automatic Control Robot 5(1):25–41. http://facta.junis.ni.ac.yu/facta/ 11. Hedrih (Stevanovi´c) K., (2005), Partial Fractional Order Differential Equations of Transversal Vibrations of Creep-connected Double Plate Systems, in Monograph - Fractional Differentiation and its Applications, Edited by Alain Le Mahaute, J. A. Tenreiro Machado, Jean Claude Trigeassou and Jocelyn Sabatier, p. 780, U-Book, Printed in Germany, pp. 289–302. ISBN 3-86608-025-3 BARCOD 9-783866-0802063 12. Hedrih (Stevanovi´c) K., (2006), Transversal Vibrations of Double-Plate Systems, Acta Mechanica Sinica, Springer, (2006) 22, pp. 487-501 (hard cower and on line). ISSN 05677718 13. Hedrih (Stevanovi´c) K (2007) Double plate system with discontinuity in the elastic bonding layer. Acta Mech Sinica 23(2):221–229. (hard cower and on line) 14. Hedrih (Stevanovi´c) K (2005) Integrity of dynamical systems. J Nonlinear Anal 63:854–871 15. Hedrih (Stevanovi´c) K (2006) Modes of the homogeneous chain dynamics. Signal Process Elsevier 86:2678–2702 16. Hedrih (Stevanovi´c) K (2006) Transversal vibration of a parametrically excited beam: influence of rotatory inertia and transverse shear on stochastic stability of deformable forms c Freund Publishing House Ltd. Int J Nonlinear Sci Numerical Simulation and processes.  7(1):117–124 17. Hedrih (Stevanovi´c) K (2003) Frequency equations of small oscillations mixed systems of the coupled discrete and continuous subsystems, Mehanika tverdogo tela (Rigid Body Mechanics), Donetsk, UDC 531.1:534.012:534.013, ISSN 0321-1975, vip. 33, pp. 174–189. Ukraine 18. Hedrih (Stevanovi´c) K (2005) Eigen amplitude vectors and functions extended orthogonality of small oscillations mixed systems of the coupled discrete and continuous subsystems. Facta Univ Ser Mech Autom Control Robot 4(17):225–243. http://facta.junis.ni.ac.yu/facta/macar/ macar200501/macar200501-04.html 19. Hedrih (Stevanovi´c) K (1996) Interpretation of the transfer energy from high-frequenccy to low-frequenccy modes by averaging asymptotic method Krilov-Bogolyubo-Mitropolsky. Book of abstracts, the second international conference “Asymptotics in Mechanics” St. Petersburg Marine Technical University, Russia, 13–16 October 1996, p 29 20. Hedrih (Stevanovi´c) K (1997) Interpretation of the transfer of energy from high-frequency to low-frequency modes by averaging asymptotic method Krilov-Bogolybov-Mitropolsky. In: Nayfeh AH, Rozdestvensky KV (ed) AiM 96, Asymptotics in mechanics, Proceedings of the Second International Conference, p. 302, Saint-Petersburg State Marine Technological University, Saint Petersburg, 1997, pp. 97–104. ISBN 5-88303-069-6, Izdatelskiy centr SPbGMTU, 1997 21. Hedrih (Stevanovi´c) K (1995) Energijska analiza kinetike konstrukcija za razlicite modele materijala (Energy analysis of the sytucture kinetics for different models of the materials), Kratki prikazi radova, “Mehanika, materijali i konstrukcije”, SANU, Beograd, 1995, pp. 106– 107. (in Serbian) 22. Hedrih (Stevanovi´c) K (1997) Interpretation of the transfer of energy from high-frequency to low-frequency modes by averaging asymptotic method Krilov-Bogolybov-Mitropolsky. In: Nayfeh AH, Rozdestvensky KV (ed) AiM 96, Asymptotics in mechanics, Proceedings of the Second International Conference, p. 302, Saint-Petersburg State Marine Technological University, Saint Petersburg, 1997, pp. 97–104 23. Hedrih (Stevanovi´c) K (1997) Energetic analysis of oscillatory processes and the modes in nonlinear systems, solid mechanics, serbian-Greek symposium 1997, scientific meetings of the Serbian Academy of Sciences and Arts, Department of Technical Sciences, SANU Belgrade, vol LXXVII, Book 3, pp 137–146

76

K.R. (Stevanovi´c) Hedrih

24. Hedrih (Stevanovi´c) K (2005) Transversal vibrations of creep connected multi plate homogeneous systems. In: van Campen DH, Lazurko MD, van den Over WPJM (ed) CD Proceedings, Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven University of Technology, ID 11-428, pp. 1445–1454 25. Hedrih (Stevanovi´c) K (2008) Energy transfer in double plate system dynamics. Acta Mechanica Sinica 24(3):331–344. doi: 10.1007/s10409-007-0124-z, Springer Berlin/Heidelberg, ISSN (567-7718 (Print) 1614-3116 (Online) 26. Katica (Stevanovi´c) H (2008) Energy interaction between linear and nonlinear oscillators (Energy transient through the subsystems in the hybrid system), ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6, pp. 796–814 27. Hedrih (Stevanovi´c) K (1997) Energetic analysis of oscillatory processes and the modes in nonlinear systems. Solid mechanics, Serbian-Greek symposium 1997, scientific meetings of the Serbian Academy of Sciences and Arts, Department of Technical Sciences, SANU Belgrade, vol LXXVII, Book 3, pp 137–146 28. Hedrih (Stevanovi´c) K (2004) A trigger of coupled singularities. MECCANICA 39(3), 295–314. International Journal of the Italian Association of Theoretical and Applied echanics, Meccanica, Publisher: Springer 29. Hedrih (Stevanovi´c) K (2004) Contribution to the coupled rotor nonlinear dynamics, advances in nonlinear sciences, Monograph, Belgrade, Academy of Nonlinear Sciences, pp 229–259. (engleski). ISBN 86-905633-0-X UDC 530-18299(082) 51–73:53(082) UKUP 30. Hedrih (Stevanovi´c) K (2005b) Multifrequency forced vibrations of thin elastic Shell. In: van Campen DH, Lazurko MD, van den Over WPJM (eds) CD proceedings, fifth EUROMECH nonlinear dynamics conference, Eindhoven University of Technology. ID of contribution 21-439, pp 2417–2426. ISBN 90 386 2667 3, www.enoc2005.tue.nl 31. Hedrih (Stevanovi´c) K (2002) The dissipation function of a nonconservative system of mass particles. Tensor NS 63(2):176–186. Tensor Society, Japan. ISSN 0040-3504 32. Hedtrih (Stevanovi´c) K., (2008), Dynamics of coupled systems, Journal Nonlinear Analysis: Hybrid Systems, Volume 2, Issue 2, June 2008, Pages 310–334. doi:10.1016/j.nahs.2006.06. 003 in line at http://www.sciencedirect.com/science/journal/1751570X 33. Hedrih (Stevanovi´c) K (2006) Dynamics of sandwich structures, invited plenary lecture, recent advances in civil engineering. In: Proceeding of papers – Faculty of Civil Engineering, Subotica, 2006, E.III, vol 15, Abstract, pp 20, and CD – Full papers 30–39. YU ISSN 03526852. COBISS SR.ID. 14404098 34. Hedrih (Stevanovi´c) K (2005) Mathematical Analogy and Phenomenological Mapping: Vibrations of Multi Plate and Multi Beam Homogeneous Systems. In: Bereteu L, Cioara T, Toth-Tascau M, Vigaru C (eds) Scientific Buletletin of the “Politehnica” University of Timisoara, Romania, Transaction on Mechanics, Editura Politehnica, Tom 50(64), Special Issue, 11–18, ISSN 1224-6077, 2005 35. Hedrih (Stevanovi´c) K (2005c) Partial fractional order differential equations of transversal vibrations of creep-connected double plate systems. In: Le Mahaute A, Tenreiro Machado JA, Trigeassou JC, Sabatier J (ed) Monograph - Fractional differentiation and its applications, U-Book, pp. 289–302 36. Hedrih (Stevanovi´c) K (2003) Frequency equations of small oscillations mixed systems of the coupled discrete and continuous subsystems, Mehanika tverdogo tela (Rigid Body Mechanics), Donetsk, UDC 531.1:534.012:534.013, ISSN 0321–1975, vip. 33, pp. 174–189. Ukraine 37. Hedrih (Stevanovi´c) K (2005) Eigen amplitude vectors and functions extended orthogonality of small oscillations mixed systems of the coupled discrete and continuous subsystems. Facta Univer Ser Mech Automatic Control Robot 4(17):225–243. YU ISSN 05342009. UDC 534.1:534.012:534.013. http://facta.junis.ni.ac.yu/facta/macar/macar200501/ macar200501-04.html 38. Hedrih (Stevanovi´c) K (2004) Forced oscillations of the coupled discrete and continuous subsystems, the first international conference on computational mechanics (CM’04), In c 2004, BELGRADE, SERBIA AND memory of Prof.dr Mladen Berkovic, 46 MMM, 

Energy and Nonlinear Dynamics of Hybrid Systems

77

MONTENEGRO, 15–17 NOVEMBER 2004, BOOK OF PAPERS, Printing Faculty of Mathematics, University of Belgrade 1–16 ISBN 86-7589-042-7 39. Hedrih (Stevanovi´c) K (2004) A model of railway track of sandwich type and it’s dynamics excited by moving load. In: Proceedings of scientific-expert conference on Railways - XII RAILKON‘04, Faculty of Mechanical Engineering University of Niˇs, pp 149–154. 21 i 22 oktobra 2004 40. Hedrih (Stevanovi´c) K (2006) Transversal vibrations of the axially moving double belt system with creep layer, Preprints, 2nd IFAC workshop on fractional differentiation and its applications, 19–21 July 2006, Porto, Portugal, pp 230–235. +CD. IFAC WS 2006 0007 PT, ISBN 972-8688-42-3, 978-972-8688-42-4.ISBN. http:/www.iser.ipp.pl 41. Hedrih (Stevanovi´c) K (2005) Transveral vibrations of creep connected multi plate homoheneous systems. In: van Campen DH, Lazurko MD, van den Over WPJM (eds) CD Proceedings, Fifth EUROMECH nonlinear dynamics conference, eindhoven university of technology. ID of contribution 11–428, pp 1445–1454. ISBN 90 386 2667 3, www.enoc2005. tue.nl 42. Hedrih (Stevanovi´c) K (2004) Discrete continuum method, computational mechanics. In: Zao c 2004 ZH, Zuang MW, Zhong WX (eds) WCCM VI in conjunction with APCOM’04.  Tsinghua University Press & Springer, pp 1–11. CD. IACAM International Association for Computational Mechanics – www. iacm.info, ISBN 7-89494-512-9 43. Hedrih (Stevanovi´c) K (2004) Creep vibrations of a fractional derivative order constititive relation deformable bodies, PACAM VIII. La Habana, 2004. Appl Mech Americas 10:548– 551. ISBN 959-7056-20-8 44. Hedrih (Stevanovi´c) K (2004) Phase portraits and homoclinic orbits visualization of nonlinear dynamics of multiple step reductor/multiplier, proceedings, vol 2, The eleventh world congress in Mechanism and machine Sciences, IFToMM, China Machine press, Tianjin, 1– 4 April 2004, pp 1508–1512. ISBN 7-111-14073-7/TH-1438. http://www.iftomm2003.com Publisher: China Machine press, Tianjin, China 45. Hedrih (Stevanovi´c) K (2004) Contribution to the coupled rotor nonlinear dynamics, Advances in nonlinear Sciences, Monograph, Belgrade, Academy of Nonlinear Sciences, pp. 229–259. (engleski). ISBN 86-905633-0-X UDC 530-18299(082) 51–73:53(082) UKUP. 46. Hedrih (Stevanovi´c) K (2004) On rheonomic systems with equivalent holonomic conservative systems applied to the nonlinear dynamics of the watt’s regulator, proceedings, vol 2, The eleventh world congress in Mechanism and machine Sciences, IFToMM, China Machine press, Tianjin, 1–4 April 2004, pp 1475–1479. ISBN 7-111-14073-7/TH-1438. http://www. iftomm2003.com 47. Hedrih (Stevanovi´c) K (1995) Energijska analiza kinetike konstrukcija za razlicite modele materijala, Kratki prikazi radova, “Mehanika, materijali i konstrukcije”, SANU, Beograd, 1995, pp. 106–107 48. Hedrih (Stevanovi´c) K (1996) Interpretation of the transfer energy from high-frequenccy to low-frequenccy modes by averaging asymptotic method Krilov-Bogolyubo-Mitropolsky. Book of abstracts, the second international conference “Asymptotics in Mechanics” St. Petersburg Marine Tech.Univ., Russia, 13–16 October 1996, pp 29 49. Hedrih (Stevanovi´c) K (1997) Energetic analysis of oscillatory processes and the modes in nonlinear systems. Solid mechanics, serbian-Greek symposium 1997, scientific meetings of the Serbian Ac of Sci and Arts, Dep of Techn Sciences, Belgrade, vol LXXVII., Book. 3, pp 137–146 50. Hedrih (Stevanovi´c) K, Kneˇzevi´c R, Cvetkovi´c R (2001) Dynamics of planetary reductoe with turbulent damping. Int J Nonlinear Sci Numerical Simulations 2(3):265–277. ISSN 1565-1339. (in English), Freund Publishing House L.T.D. http://www.cs.uky.edu/\simjzhang/ NSNS/2001-02-03.html 51. Hedrih (Stevanovi´c) K, Hedrih AN (2009) Transfer of energy of oscillations trough the double DNA chain helix. In: Ambr´osio J, et al (eds) 7th EUROMECH solid mechanics conference. Lisbon, Portugal, 7–11 September 2009, CD –MS-24, Paper 315, pp 1–15 52. Hedrih (Stevanovi´c) K, Hedrih AN (2009) Transfer of energy of oscillations trough the double DNA chain helix. In: Ambr´osio J, et al. (eds) 7th EUROMECH solid mechanics conference,

78

K.R. (Stevanovi´c) Hedrih

Lisbon, Portugal, September 7–11, 2009, ESMC 2009 Book of Abstracts ‘Mini’Szmposia, iydanje Instituto Superior Tecnico, Lisbon, APMTAC, pp. 591’592.ISBN 9789899 626423 53. Hedrih (Stevanovi´c) K, Simonovic’ JD (2010) Non-linear dynamics of the sandwich double circular plate system. Int J Nonlinear Mech. doi:10.1016/j.ijnonlinmec.2009.12.007, Int J Nonlinear Mech 2009. 12. 007 54. Hedrih (Stevanivic) K, Simonovic J (2007) Transversal vibrations of a non-conservative double circular plate system. Facta Univ Series: Mech Automatic Control Robot 6(1):51–64 55. Hedrih (Stevanovi´c) K, Simonovi´c J (2008) Transversal vibrations of a double circular plate system with visco-elastic layer excited by a random temperature field. Int J Nonlinear Sci Numerical Simulation 9(1):47–50. http://www.ijnsns.com/2008/TOC9.1.doc 56. Hedrih (Stevanovi´c) K, Simonovi´c J (2007) Dynamical absorption and resonances in the sandwich double plate system vibration with elastic layer. Sci Tech Rev LVII(2):1–10 57. Hedrih (Stevanovi´c) K, Simonovi´c J (2009) Energy transfer through double curcular plate nonconservative system dynamics. In: Ambr´osio J, et al (eds) 7th EUROMECH solid mechanics conference. Lisbon, Portugal, 7–11 September 2009, CD –MS-24, Paper 294, pp 1–16 58. Hedrih (Stevanovi´c) K, Simonovi´c J (2009) Energy transfer through double curcular plate nonconservative system dynamics. In: Ambr´osio J, et al (eds), 7th EUROMECH solid mechanics conference. Lisbon, Portugal, 7–11 September 2009, ESMC 2009 Book of Abstracts ‘Mini’Szmposia, iydanje Instituto Superior Tecnico, Lisbon, APMTAC, pp 589–590, ISBN 9789899 626423 59. Hedrih (Stevanovi´c) K, Simonovi´c J (2006) Characteristic eigen numbers and frequencies of the transversal vibrations of sandwich system, first South-East European conference on computational mechanics, SEECCM-06, pp 90–94 60. Hedrih (Stevanovi´c) K, Simonovi´c J (2009) Energy transfer through double curcular plate nonconservative system dynamics. In: Ambr´osio J, et al (eds), 7th EUROMECH Solid Mechanics Conference. Lisbon, Portugal, 7–11 September 2009, CD –MS-24, Paper 294, pp 1–16 61. Katica (Stevanovi´c) H, Simonovi´c J (2009) Energy transfer through double curcular plate nonconservative system dynamics. In: Ambr´osio J, et al. (eds) 7th EUROMECH solid mechanics conference, Lisbon, Portugal, September 7–11, 2009, ESMC 2009 Book of Abstracts‘Mini’ Symposia, Instituto Superior Tecnico, Lisbon, APMTAC, pp 589–590, ISBN 9789899 626423 62. Hedrih (Stevanovi´c) K, Kozi´c P, Pavlovi´c R (1984) O uzajamnom uticaju harmonika u nelinearnim sistemima s malim parametrom, Recueil des travaux de l’Institut Mathem. Nouvell serie, Tome 4(12), 1984, pp. 91–102. Beograd 63. Hedrih (Stevanovi´c) K, Kozi´c P, Pavlovi´c R (1985) Stashionarniy i nestashionarniy R-chastotniy analiz kolebaniy sistem s konechnim chislom stepeni svobodu kolebaniy i vzaimnoe vliyanie garmonikov. Theoretical and Applied Mechanics 11, 73–84, 1985, Beograd 64. Hedrih (Stevanovi´c) K, Pavlovic R, Kozic P, Mitic Sl (1986) Stashionarniy i nestashionarniy chetiri-chastotniy analiz vinuzhdenih kolebaniy tonkoy uprugoy oboloscchki s nachalynimi nepravilynostyami. Theoretical and Applied Mechanics 12, 33–40, 1986, Beograd 65. Hedrih (Stevanovi´c) K, Miti´c Sl, Pavlovi´c R, Kozi´c P (1986) Mnogochastotnie vinuzhdenie kolebaniya tonkoy uprugoy oboloschki s nachalynimi nepravilynostyami, (Analiticheskiy analiz). Theoretical and Applied Mechanics 12, 41–58, 1986, Beograd 66. Kozmin A., Mikhlin Yu. and C. Pierre, Transient in a two DOF nonlinear Systems, Nonlinear Dynamics, 2006; Nonlinear Dynamics, Volume 51, Numbers 1–2 / January, 2008, DOI: 10.1007/s11071-007-9198-1 67. Lacarbonara W, Rega G, Nayfeh AH (2003) Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems. Int J Non-Linear Mech 38(6):851–872

Energy and Nonlinear Dynamics of Hybrid Systems

79

68. Lacarbonara W, Rega G (2003) Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems. Int J Non-Linear Mech 38(6):873–887 69. Lazarov BS, Thomsen JJ (2009) Using high-frequency vibrations and non-linear inclusions to create metamaterials with adjustable effective properties. Int J Non-Linear Mech 44(1):90–97 70. Lenci S, Rega G (2007) Dimension reduction of homoclinic orbits of buckled beams via the non-linear normal modes technique. Int J Non-Linear Mech 42(3):515–528 71. Lenci S, Rega G (2008) Control of the homoclinic bifurcation in buckled beams: Infinite dimensional vs reduced order modeling. Int J Non-Linear Mech 43(6):474–489, Jul 2008 72. Malatkar P, Nayfeh AH (2003) On the transfer of energy between widely spaced modes in structures. Nonlinear Dyn 31(2):225–242(18) 73. Mitropolyskiy YuA (1955), Nestashionarnie proshesi v nelinyeynih sistemah, AN USSR, Kiev. (in Russian) 74. Mitropolyskiy YuA, Mosseenkov BI (1968), Lekciyi po primenyeniyu metodov k recheniyu uravnyeniy v chastnih proizvodnih, Int. Math. AN USSR, Kiev. (in Russian) 75. Mitropolyskiy YuA (1964) Problemi asimptoti-cheskoy teorii nestashionarnih kolebaniy, Nauka Moskva. (in Russian) 76. Mitropolyskiy YuA, Mosseenkov BI (1976) Assimptoticheskie recheniya uravnyeniya v chastnih proizvodnih, Vichaya chkola Kiev. (in Russian) 77. Mitropolskiy YuA, Nguyen Van Dao (2003) Lectures on asymptotic methods of nonlinear dynamics. Vietnam National University Publishing House, Hanoi, p 494 78. Mitropolskiy YuA (1995) Some problems in the development in nonlinear mechanics theory and applications. Facta Univ Ser Mech Automatic Control Robot 1(5):539–560 79. Mitropolskiy YuA (1996) On application of asymptotic methods of nonlinear mechanics for solving some problems of oscillation theory. Facta Univ, Ser Mech Automatic Control Robot 2(6):1–9 80. Mitropolskiy YuA (1983) Adiabatic processes in nonlinear oscillation systems (Adiabaticheskie processi v nelinyeynih kolebatelynih sistemah), Simpozijum’83 – Nelinearni problemi dinamike, Arandjelovac, 23–25 Novembar 1983. IT-Druˇstvp za mehaniku Srbije, pp. I-1–I12. (in Russian) 81. Nayfeh AH, Balachandran B (1990) Experimental investigation of resonantly forced oscillations of a two-degree-of-freedom structure. Int J Non-Linear Mech 25(2–3):199–209 82. Nayfeh AH, Balachandran B (1989) Modal interactions in dynamical and structural systems. Appl Mech Rev, 42(11), Part 2:175–202. Review article 83. Nayfeh AH (1993) Transfer of energy from high-frequency to low frequency modes. In: Paper and poster abstracts, IUTAM symposium nonlinearity and chaos in engineering dynamics. Centre NDA UCL, London 84. Nayfeh AH, Balachandran B (1995) Applied nonlinear dynamics. Wiley, New York 85. Nayfeh AH (1996) Transfer of energy from high-frequency to low-frequency modes, book of abstracts. In: The second international conference “Asymptotics in Mechanics.” St. Petersburg Marine Technical University, Russia, p 44, 13–16 October 1996 86. Nayfeh AH (2000) Nonlinear interactions: analytical, computational and experimental methods. Wiley, New York 87. Nayfeh AH, Pai PF (2004) Linear and nonlinear structural mechanics. Wiley, New York 88. Raˇskovi´c D (1965) Teorija oscilacija, (theory of oscillations). In: Nauna knjiga (ed) p. 503 (in Serbian) 89. Raˇskovi´c D (1985) Teorija elastiˇcnosti (Theory of Elasticity), Nauˇcna knjiga, 1985, 414 90. Rega G (2004) Nonlinear vibrations of suspended cables–Part I: modeling and analysis. ASME Appl Mech Rev 57(6):443–478 91. Rega G (2004) Nonlinear vibrations of suspended cables–Part II: deterministic phenomena. ASME Appl Mech Rev 57(6):479–514 92. Rega G, Lenci S (2005) Identifying evaluating and controlling dynamical integrity measures in nonlinear mechanical oscillators. Nonlinear Anal 63(5–7):902–914. Springer 93. Stevanovi´c (Hedrih) K (1975) Application of the asymptotic method for the investigation of the nonlinear oscillations of elastic bodies – energy analysis of the oscillatory motions of

80

K.R. (Stevanovi´c) Hedrih

elastic bodies [in Serbian], Doctoral Dissertation, Faculty of Mechanical engineering, Niˇs, p 331. (in Serbian) – Supervision: Prof. dr. Ing. Math Danilo P. Raˇskovi´c (Yugoslavia) and academician Yuriy Alekseevich Mitropolskiy (Ukraine) 94. Stevanovi´c (Hedrih) K, Raˇskovi´c D (1974) Investigation of multi-frequencies vibrations in single-frequency regime in nonlinear systems with many degrees of the freedom and with slowchanging parameters. J Nonlinear Vibrat Probl 15:201–202 95. Stevanovi´c (Hedrih) K (1972) Two-frequency nonstationary forced vibrations of the beams. Math Phys 12:32 96. Zakrzhevsky M (2008) New concepts of Nonlinear Dynamics: complete bifurcation groups protuberances unstable periodic infinitums and rare attractors. J vibroengineering 10(4):421– 441. Latvia

Appendix – References: 97. Hedrih (Stevanovi´c) K (1975) Selected chapters from theory of nonlinear vibrations (in Serbian), Faculty of Mechanical Engineering, Niˇs, First Edition 1975, p 180 98. Hedrih (Stevanovi´c) K (1972) Study of methods of nonlinear vibrations theory (in Serbian), Poligraphy, Faculty of Mechanical Engineering, Niˇs, Preprint, p 500 99. Hedrih (Stevanovi´c) K (1972) Teorija nelinearnih oscilacija i primene na nelinearne sisteme automatskog upravljanja (Theory of non-linear oscillations and applications to nonlinear system automatic control), [in Serbian], Faculty of Technical Sciences in Niˇs,Supervisor D. Raˇskovi´c 100. Hedrih (Stevanovi´c) K (1972) Reˇsrnje jednaˇcone transverzalnih oscilacija jednoraspone grede asimptotskim metodama nelinearne mehanike (Solution of the equation of transversal oscillation one span beam by asymptotic method of nonlinear mechanics), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Technical Sciences in Niˇs, Supervision: Prof. dr. Ing. Math Danilo P. Raˇskovi´c (Yugoslavia) and academician Yuriy Alekseevich Mitropolskiy (Ukraine) 101. Hedrih (Stevanovi´c) K (1975) Application of the Asymptotic Method for the Investigation of the Nonlinear Oscillations of Elastic Bodies – Energy Analysis of the Oscillatory Motions of Elastic Bodies, [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervision: Prof. dr. Ing. Math Danilo P. Raˇskovi´c (Yugoslavia) and academician Yuriy Alekseevich Mitropolskiy (Ukraine) 102. Kozi´c P (1982) Izuˇcavanje nelinearnih torzijskih oscilacija vratila asimptotskom metodom (Investigation of non-linear torsion oscillations of shafts by asymptotic method), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Belgrade, Working Supervisor K. Hedrih (Stevanovi´c) 103. Kozi´c P (1990) Stabilnost diskretnih mehaniˇckih sistema pri dejstvu sluˇcajne pobude (Stability of discrete mechanical systems subjected by random excitations), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 104. Pavlovi´c R (1982) Prilog nelinearnim oscilacijama plitkih cilindriˇcnih ljuski (Contribution to the non-linear oscillations of the shallow shells), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih(Stevanovi´c) 105. Pavlovi´c RR (1990) Dinamiˇcka stabilnost kontinualnih sistema od kompozitnih materaijala pod dejstvu sluˇcajnih pobuda (Dynamic stability of continuous systems made from composite materials subjected to random excitation), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 106. Miti´c Sl (1985) Viˇsefrekventna analiza oscilovanja tankih elastiˇsnih ljuski sa konstantim krivinama i poˇcetnim nepravilnostima (Multi-frequency analysis of the vibrations of thin shells with constant curvature and initial inperfetions), [in Serbian], Magistar of Sciences Degree thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih(Stevanovi´c)

Energy and Nonlinear Dynamics of Hybrid Systems

81

107. Miti´c Sl (1989) Stabilnost deterministiˇckih i stohastiˇckih procesa u vibroudarnim sistemima Stability of desterministic and stochastic processes in vibroimpact systems), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 108. Jovanovi´c D (1989) Analiza naponskog i defprmacionog stanja ravno napregnutih ploˇca sa primenom na eliptiˇcno prstenastu ploˇcu (Stress and strain analysis of plane loaded plate with applications to elliptical annular plate), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 109. Miti´c Sn (1989) Analiza naponskog i defprmacionog stanja ravno napregnutih ploˇca (Stress and strain analysis of plane loaded plates), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 110. Pavlov B (1985) Novootkriveni fenomeni u nelunearnim dinamiˇckim sistemima sa analogijom na sisteme opisane Mayhieu-ovom diferencijalnom jednaˇcinom (New phenomena in non-linear dynamical systems with analogy tio the systems described by Matheu differential equations), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c). 111. Filipovski A (1995) Energijska analiza longitudinalnih oscilacija sˇtapova promenljivog preseka (Energy analysis longitudinal oscillations of rods with changeable cross sections), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 112. Peri´c Lj (2004) Prostorna analiza stanja napona i stanja deformacije napregntog piezokeramiˇckog materijala (Space analysis of stress and straon state of stressed piezoceramiv materials), Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 113. ∗ Janevski G (2003) Nelinearne oscilacije ploˇca od kompoyitnih materijala (Nonlinear vibrations of the composite plates), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor R. Pavlovi´c 114. Simonovi´c J (2008) Fenomeni dinamike mehaniˇckih sistema sloˇzenih struktura (Phenomen of Dynamics of Complex Structure Mecnanical Systems), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 115. Jovi´c S (2009) Energijska analiza dinamike vibroudarnih sistema (Energy analysis of vibroimpact system dynamics), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Technical Sciences in Kosovska Mitrovica, University of Priˇstina, Supervisor V. Raiˇcevi´c, Project ON144002 Leader K. Hedrih (Stevanovi´c) 116. Peri´c Lj (2005) Spregnuti tenzori stanja piezoeletriˇcnih materijala (Coupled tensors of the piesoelectric material states), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 117. Kneˇzevi´c R (2000) Nelinearni fenomeni u dinamici planetnih prenosnika (Nonlinear phenomena in dynamics of planetary gear transmission), [in Serbian], Doctor’s Degree Thesis, Niˇs, Yugoslavia, Faculty of Mechanical Engineering in Niˇs, (200). Supervisor K. Hedrih (Stevanovi´c) 118. Jovanovi´c D (2009) Potencijalna energija i stanje napona u materijalu sa prslinom (Potential energy and stress state in material with crack), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 119. ∗ Janevski G (2010) Dinamiˇcka stabilnost mehaniˇckih sistema pri dejstvu sluˇcajnih optere´cenja (Dynamic stability of mechanical system;oaded by random excitation), Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor Kozi´c P. 120. Jovi´c S (2011) Energijska analiza dinamike vibroudarnih sistema sa krivolinijskim putanjama i neidealnim vezama (Energy analysis of vibroimpact system dynamics with curvilinear paths and no ideal constraints), [in Serbian], Doctor’s Degree Thesis, Faculty of Technical Sciences in Kosovska Mitrovica, University of Priˇstina, Supervisor V. Raiˇcevi´c, Projects ON144002 and OI174001 Leader K. Hedrih (Stevanovi´c)

82

K.R. (Stevanovi´c) Hedrih

121. Simonovi´c J (2011) Dinamika i stabilnost hibridnih dinamiˇckih sistema (Dynamics and Stability of Dynamics Hybrid Systems), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 122. Veljovi´c Lj (2011) Nelinearne oscilacije giro-rotora (Non-linear oscillations of Gyro-rotors), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niˇs, Supervisor K. Hedrih (Stevanovi´c) 123. Oscillations of the System with many degree of Freedom and Elastic Bodies with Nonlinear Properties, Basic Scientific Found of Region Niˇs (1979-1981). Mechanical Engineering Faculty University of Niˇs 124. Oscillations of the Special Elements and Systems, B Basic Scientific Found of Region Niˇs (1981-1986). Some research results included in two Magistar of sciences theses of P. Kozi´c and R. Pavlovi´c. Mechanical Engineering Faculty University of Niˇs 125. Stochastic Processes in Dynamical Systems-Applications on the Mechanical Engineering Systems, Basic Scientific Found of Region Niˇs (1986-1989). Some research results included in Magistar of sciences theses of Sl. Miti´c and in two doctoral dissertations of P. Kozi´c and R. Pavlovi´c. Mechanical Engineering Faculty University of Niˇs 126. Nonlinear Deterministic and Stochastic Processes with Applications in Mechanical Engineering Systems, Ministry of Science and Technology Republic of Serbia, (1990-1995). Some research results included in two Magistar of sciences theses of Blagoj Pavlov and Alekdsandar Filipovski and in a doctoral dissertation of Sl. Miti´c. Mechanical Engineering Faculty University of Niˇs 127. Sub-Projects 5.1. Thema: Stress and Strain State of the Deformable Bodies and 5.2. Theme: Vector Interpretation of Body Kinetic Parameters, as a part of Project: Actual Problems on ˇ si´c Ministry of Sciences, Technology Mechanic (1990-1995), Project Leader prof. dr Mane Saˇ and Development of Republic Serbia. Some research results included in three Magistar of sciences theses of Ljubiˇsa Peri´c, Dragan Jovanovi´c and Sneˇzana Miti´c. Mathematical Institute SANU 128. Sub-Project: 04M03A Current Problems on Mechanics and Applications (1996-2000), as a part of Project: Methods and Models in Theoretical, Industrial and Applied Mathematics, Project Leader prof. dr Gradimir Milovanovi´c, Ministry of Sciences, Technology and Development of Republic Serbia. Mathematica Institute SANU 129. Project 1616 Real Problems on Mechanics (2002-2004), Basic Science-Mathematics and Mechanics, Ministry of Sciences, Technology and Development of Republic Serbia. Some research results included in two doctoral dissertations of Ljubiˇsa Peri´c and Dragan Jovanovi´c. Mathematical Institute SANU and Mechanical Engineering Faculty University of Niˇs. 130. Project ON1828 Dynamics and Control of active Structures (2001-2005), Basic ScienceMathematics and Mechanics, Ministry of Sciences, Technology and Development of Republic Serbia. Some research results included in two doctoral dissertations of Ljubiˇsa Peri´c and Dragan Jovanovi´c. Mechanical Engineering Faculty University of Niˇs. 131. Project ON144002 -Theoretical and Applied Mechanics of the Rigid and Solid Bodies. Mechanics of Materials (2006-2010). Support: Ministry of Sciences and Environmental Protection of Republic of Serbia. Some research results included in two Magistar of sciences theses of Srdjan Jovi´c, and Julijana Simonovi´c and in four doctoral dissertations of Dragan Jovanovi´c, Srdjan Jovi´c, Ljiljana Veljovi´c and Julijana Simonovi´c. Institution Coordinator: Mathematical Institute Serbian Academy of Sciences and Arts and Mechanical Engineering Faculty University of Niˇs. 132. ON174001 - Dynamics of hybrid systems with complex structures. Mechanics of materials. (2011-2014), Ministry of Sciences and Technology of Republic of Serbia. Some research results included in thre doctoral dissertations of Srdjan Jovi´c, Ljiljana Veljovi´c and Julijana Simonovi´c. Institution Coordinator: Mathematical Institute Serbian Academy of Sciences and Arts and Mechanical Engineering Faculty University of Niˇs.

Energy and Nonlinear Dynamics of Hybrid Systems

83

133. Project 1409 Stability and nonlinear oscillations of viscoelastic and composite continuous systems, (2002–2005). Support: Ministry of Sciences and Environmental Protection of Republic of Serbia. Some research results included in a Magistar of sciences these of Goran Janevski. Mechanical Engineering Faculty University of Niˇs. 134. OI 144023 Deterministic and stochastic stability of mechanical systems (2006–2010.), Ministry of Sciences and Technology of Republic of Serbia. Some research results included in a doctoral dissertations of Goran Janevski (supervisor Predrag Kozi´c). Mechanical Engineering Faculty University of Niˇs.

Characteristics Diagnosis of Nonlinear Dynamical Systems Liming Dai and Lu Han

Abstract This chapter is on the updated methodologies of diagnosing characteristics of nonlinear dynamic systems. The widely used and most applicable methods and approaches in characterizing the nonlinear behaviors of the systems are reviewed briefly. Characteristics of Lyapunov exponents and recently developed periodicity ratio approach are described in detail and compared. The applicability and efficiency of the approaches are presented and compared.

1 Introduction In the studies of nonlinear dynamic systems, which are usually governed by nonlinear second-order differential equations or a system of differential equations, the criteria for distinguishing the characteristics of the systems are crucial. The techniques providing high efficiency and accuracy in diagnosing and quantifying different characteristics such as chaos, periodicity, quasiperiodicity, and other nonlinear characteristics are always demanded in studying nonlinear dynamic systems. There are several methods available in the literature for determining the onset of chaotic oscillations and some predictive and diagnostic criteria for chaos are also reported [1–3]. Power spectral density is one of such methods that can be used to distinguish chaos from regular behavior of deterministic systems. This method may also be used to distinguish chaos from generic stationary stochastic behavior [4]. Another method, the Fractal Dimensions approach is one of the most popular methods in diagnosing characteristics of nonlinear dynamic systems. The approach identifies the chaotic attractors’ dimension [5–9]. In fact, all the characteristics of a dynamic system can be identified and classified by observing the phase diagrams, bifurcation diagrams [10], and Poincar´e maps [11, 12] of the

L. Dai () • Lu Han University of Regina, Regina, SK, S4S 0A2, Canada e-mail: [email protected]; [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 3, © Springer Science+Business Media, LLC 2012

85

86

L. Dai and Lu Han

system. However, the process of classification is tedious and not practically sound if a periodic–quasiperiodic–chaotic diagram is considered. Kolmogorov entropy is also used in the field as an index that indicates 0 for nonchaotic motion and positive for chaotic motion [13–15]. In some areas, topology can be used to study the structure of bounding torus surrounding the chaotic attractor [16]. The Hurst exponent [17] is found useful in characterizing the nonlinear systems. It can be employed to examine the existence of chaos by inspecting the autocorrelation of the time series. Among these, some are empirical methods that rely upon physical experiments, some depend on the data obtained from approximate mathematical models of the corresponding dynamical systems [18–20]. Besides these, several new attempts are found in diagnosing the nonlinear characteristics. Neural networks and fuzzy logic have been used in attempting to provide high speed, flexibility, and logical decision for real-time diagnosability of a system’s behavior [21]. Among all the diagnosing approaches, Lyapunov exponent approach is probably the most popular approach [22, 23] due to its efficiency and simplicity. Lyapunov exponents measures the sensitivity of a system to initial conditions and therefore classifies the system’s responses as either convergent or divergent. However, Lyapunov exponents cannot be used to distinguish quasiperiodicity and nonperiodicity of a system. A new diagnosing tool named the Periodicity-Ratio method is developed Dai and Singh [24]. This approach can be used to identify almost all the nonlinear characteristics and to be employed to plot the periodic-quasiperiodic-chaotic diagram efficiently for nonlinear dynamical systems.

2 Nonlinear Behavior Diagnosis This section will discuss the characteristics and applications of the most popular diagnosing methods.

2.1 Bifurcation Approach Bifurcation is widely used in the field of nonlinear dynamics for characterizing the nonlinear behaviors of the dynamic systems. It indicates the changes in qualitative or topological structure solutions of differential equations. More technically, consider the continuous dynamical systems described by ordinary differential equations such as x˙ = f (x, λ ), f : Rn × R → Rn ; where x represents a vectorial variable and λ is a parameter set. A local bifurcation specifically occurs at (x0 , λ0 ) if the Jacobian matrix df (x0 ,λ0 ) has an eigenvalue with zero real part. For discrete dynamical systems, consider the system xn+1 = f (xn , λ ), f : Rn × R → Rn . Then a local bifurcation occurs at (x0 , λ0 ) if the matrix df (x0 ,λ0 ) has an eigenvalue with a modulus equals to one. A bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden

Characteristics Diagnosis of Nonlinear Dynamical Systems

87

Fig. 1 Bifurcation diagram of logistic mapping

‘qualitative’ or topological change in its behavior [11]. Many bifurcations can finally lead to chaos such as the consequence of the explosive bifurcation is an on-off intermittent transition to chaos and some dangerous bifurcation can jump to a bounded or unbounded remote chaotic attractor. A very famous finding by Li and Yorke [25] disclosed a bifurcation with period three always implies the occurrence of chaos. Figure 1 shows an example of bifurcation in logistic map: xn+1 = rxn (1 − xn ) [26]. The bifurcation parameter r is the horizontal axis of the plot and the vertical axis shows the possible long-term population values of the logistic function. It can be seen from the figure that bifurcation may lead to chaos as the value r increases. With a bifurcation diagram, one may conveniently identify periodic responses of a system and evaluate the evolution of the system from simple harmonic to multiple periodic cases. However, quasiperiodicity and chaos can hardly be identified with merely implementation of bifurcation diagrams.

2.2 Fractal Dimension A fractal is “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,” [27] a property called self-similarity. A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion [28]. There are several examples of fractals, which are defined as portraying exact self-similarity, quasi self-similarity, or statistical self-similarity.

88

L. Dai and Lu Han

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). An attractor can be considered as a set toward which a dynamical system evolves over time. That is, points that get close enough to an attractor remain close even if slightly disturbed. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor, which is an attracting set that has zero measure in the embedding phase space and has fractal dimension. Chaotic attracters can be associated with fractals. The phase space of chaos can be fractals. The attractors from chaotic systems are known as strange attractors which have great detail and complexity, in comparing with the simpler fixedpoint attractors and limit cycles. The curves appear randomly within a strange attractor. Both continuous dynamical systems and discrete systems may have strange attractors. Strange attractors are considered as the third type of formula fractals, in addition to Julia and Mandelbrot sets. The responses of a nonlinear system on the strange attractors are usually considered as chaotic responses. The most appealing way of assigning a dimension to a set that can yield a fractal dimension to certain kinds of sets is the capacity dimension D0 [11]. A set of points that lies in a d-dimensional Cartesian space is covered with cubes of edge length r. Let Nr be the minimum number of cubes need to cover the set. The process is repeated successively by smaller values of r. The capacity dimension D0 can be ln N(r) given by D0 = lim ln(1/r) provided the limit exists. r→0

In practice of nonlinear characteristics diagnosis, the capacity dimension of a limit cycle considered as one, that of a two-period quasiperiodic orbit is two, and that of an m-period quasiperiodic orbit is m. Strange attractor, D0 always takes fraction or noninteger values. For example, the Lorenz attractor is a good example of a strange attractor whose capacity dimension is 2.06 ± 0.01 and therefore taken as chaotic system. Due to the definition of the capacity dimension, in practice however, it is not always easy to determine a fractal dimension for a system that has strange attractor.

2.3 Lyapunov Exponent The spectrum of Lyapunov exponents has proven to be one of the practically sound techniques for diagnosing chaotic systems. Lyapunov exponent is probably the most widely used index in characterizing the behaviors of nonlinear dynamic systems. The approach with Lyapunov exponent is based on the important characteristic that chaos of a nonlinear dynamic system is sensitivity to initial conditions. Lyapunov exponents count the average exponential rates of divergence or convergence of close orbits of a vibrating object in the phase space of a dynamic system [29]. Wolf [22] gave a powerful and efficient method for determining Lyapunov exponents from time series. Rong [30] investigated the principal resonance of a stochastic Mathieu oscillator to random parametric excitation and gave the conclusion that the

Characteristics Diagnosis of Nonlinear Dynamical Systems

89

instability of the stochastic Mathieu system depends on the sign of the maximum Lyapunov exponent. Lyapunov exponents were also used to analyze the numerical characteristic [31]. Lyapunov exponent is usually determined by experiments or computer simulations. The definition of Lyapunov exponent is associated with a measure of the average rates of expansion and contraction of trajectories surrounding a given trajectory. They are asymptotic quantities, defined locally in state space, and describe the exponential rate at which a perturbation to a trajectory of a system grows or decays with time at a certain location in the state space. They are useful in characterizing the asymptotic state of an evolution [11]. Nayfeh has clearly described the definition of Lyapunov exponent. Let X(t) such that X(t = 0) = X0 represent a trajectory of the system governed by the following n-dimensional autonomous system: x˙ = F(x; M)

(1)

where the vector x is made up of n state variables, the vector function F describes the nonlinear evolution of the system, and M represents a vector of control parameters. Denoting the perturbation provided to X(t) by y(t) and assuming it to be small, an equation after linearization in the disturbance terms can be obtained. The perturbation is governed by dy(t) = Jy(t) (2) dt where, in general, J = Dx F(x(t); M) is a n × n matrix with time-dependent coefficients. If we consider an initial deviation y(0), its evolution is described by y(t) = Φ(t)y(0)

(3)

where Φ(t) is the fundamental (transition) matrix solution of (2) associated with the trajectory X(t). For an appropriately chosen y(0) in (3), the rate of the exponential expansion or contraction in the direction of y(0) on the trajectory passing through X0 is given by   y(t) 1 λ¯ i = lim ln t→∞ t y(0)

(4)

where the symbol  denotes a vector norm. The asymptotic quantity λ¯ i is then defined as the Lyapunov exponent. There are several different methods to calculate the Lyapunov exponent, such as the whole Lyapunov exponent, global and local Lyapunov exponent, and Lyapunov spectrum. The method of whole Lyapunov exponent also known as the maximum Lyapunov exponent is suitable for the discrete differential system, whereas the Lyapunov spectrum is more suitable for continuous differential systems [11]. The global Lyapunov exponent, on the other hand, gives a measure for the total predictability of a system; whereas the local Lyapunov

90

L. Dai and Lu Han

exponent estimates the local predictability around a given point X0 in phase space. These different methods in calculating for the Lyapunov exponent can be generally classified into two groups: the “exponents” for the discrete differential systems while the “spectrum” for the continuous differential systems. Specifically, to obtain the Lyapunov spectrum for a continuous dynamical system, a set of n linearly independent vectors y1 , y2 , . . . , yn may form the basis for the n-dimensional state space. Choosing an initial deviation along each of these n factors, n Lyapunov exponent λ¯ i (yi ) can be determined. The set of n numbers λ¯ i (yi ) is defined as the Lyapunov spectrum. For system (3), n orthonormal initial vectors yi such that y1 = (1, 0, 0, . . .), y2 = (0, 1, 0, . . .), . . . , yn = (0, 0, 0, . . . , 1) can be assigned. For each of these initial vectors (2) and (3) can be integrated for a finite time T f and a set of vectors y1 (T f ), y2 (T f ), . . . , yn (T f ) can then be obtained. The new set of vectors is orthonormalized using the Gram–Schmidt procedure to produce y1 (T f )  yˆ1 =  y1 (T f ) ... n−1

yn (T f ) − ∑ [yn (T f ).yˆi ]yˆi i=1  yˆn =    n−1   yn (T f ) − ∑ [yn (T f ).yˆi ]yˆi    i=1

(5)

Subsequently, using X(t = Tf ) as an initial condition for (2) and using each of the yˆi as an initial condition for (3), (2) and (3) can be integrated again for a finite time and carry out the Gram–Schmidt procedure to obtain a new set of orthonormal vectors. The norm in the denominator can be denoted by N kj . Thus, after repeating the integrations and the processes of Gram–Schmidt orthonormalization r times, the Lyapunov exponent can be obtained from 1 λˆ i = rT f

r

∑ ln N kj

(6)

k=1

The Lyapunov spectrum can thus be determined.

2.4 Periodicity Ratio It is widely acknowledged that the corresponding Poincar´e map for a steady-state periodic motion of a dynamic system consists of a finite number of visible points [32, 33]. The visible points in the Poincar´e map are then the points overlapping many points that periodically appeared. On the other hand, the points in the Poincar´e map of a chaotic case must distribute in an unpredictable manner. This implies that the overlapping points in the Poincar´e map of a chaotic response

Characteristics Diagnosis of Nonlinear Dynamical Systems

91

are extremely minimal. Quasiperiodic response is another type of phenomenon in nonlinear dynamic systems. A quasiperiodic case may also contain negligibly small number of overlapping points, though some regularity of the system responses can be identified. Based on these findings, Dai and Singh [24] proposed an index named Periodicity Ratio (PR) which counts the ratio of periodic points among all the points in the Poincar´e map. The methodology of periodicity ratio approach is based on the measure of periodicity of a response of a nonlinear system. The more periodic a dynamic system is, the closer the corresponding PR value is to a unit. When the PR approaches zero, the corresponding system has no periodicity at all and therefore represents either chaotic or quasiperiodic response of the system. The most significant advantage of the periodicity ratio method is that the PR value can be used as a single value index in diagnosing the periodicity and therefore the behavior of a dynamic system. Moreover, Periodicity ratio method reveals the fact that there are infinite number of fashions of motion in between chaos and periodic responses for a nonlinear dynamic system. The periodicity ratio is defined as [33]:

γ = lim

n→∞

NPP n

(7)

where NPP is the number of overlapping points and n is designated as the total number of all the points in the Poincar´e map. NPP in (7) can be calculated by NPP = φ (1) +

n

∑







m−1

∏ Q(φ ( j))

φ (m).P

m=2

(8)

j=1

where

φ (l) =



n

∑

P

i=m+q+1

q−l



∏ Q (Xi+h − Xm+l+h)

h=0

l



∏ Q (Xi−k − Xm+l−k)

(9)

k=1

which represents the number of points periodically overlapping the lth point in the Poincar´e map. In the above two equations, q, m, i, and l are all positive integers. Note that q value in the above equation can be different from one group of points to another group of points. In the above equations, two-step functions Q(y), P(z) are introduced. The twostep functions are expressible in the form  Q(y) =

1, if y = 0 , 0, if y = 0

 P(z) =

0, if z = 0 1, if z = 0

(10)

In order to describe the visible and overlapping points in a Poincar´e map, introduce  Xi = xx˙i and denote it as a vector of both displacement and velocity. With this i

92

L. Dai and Lu Han

designation, the determination of whether or not a point in the Poincar´e map is an overlapping point is based on the judgment described by the following equations. Xki = x(t0 + kT ) − x(t0 + iT ) ˙ 0 + kT ) − x(t ˙ 0 + iT ) X˙ki = x(t

(11)

where k is an integer in the range of 1 ≤ k ≤ j, and j represents the finite number of points (known as visible points) appearing in the Poincar´e map corresponding to a dynamic system, and X˙ is the time derivative of X. Points under consideration are overlapping points if and only if the following conditions are satisfied. Xki = 0 X˙ki = 0

(12)

For a system expressible by a nonautonomous nonlinear differential equation, a periodic solution has a period that is either a multiple or integer submultiples of the period 2π /ω , where ω is the frequency of the external source of excitation [11, 32]. Then the successive points on the Poincar´e sections can be denoted by {Xt0 , Xt0 +T , . . . , Xt0 +NT , . . .}. If the system finally leads to a periodic solution, after a long enough period of time, all the points of the Poincar´e map will converge to a finite number of individual points which must have the form {Xm , Xm+1 , . . . , Xm+q }. Thus, any overlapping point X p in a Poincar´e map would be a periodic point, if and only if the following condition is satisfied:  P

q

∑

i=0

 P

q−l



∏ Q (Xi+h − Xm+l+h)

h=0

l

∏ Q (Xi−k − Xm+l−k )

 =1

(13)

k=1

Once the periodic points are determined completely, the periodicity ratio can be determined accurately. If the behavior of a system in a steady state is periodic, the points in the corresponding Poincar´e map must all be overlapping points. Accordingly, the value of the periodicity ratio, γ , should simply be unity. For a chaotic response of a system, on the other hand, the number of periodic points overlapped should be zero or insignificant in comparing with n. This is to say, γ approaches zero for chaos. With the definition of the periodicity ratio, γ is clearly a quantified description of periodicity for a dynamic system. This is to state that γ indicates quantitatively how close the response of a dynamic system is to a perfect periodic motion. For example, a motion with γ equals to 0.9 is more close to a periodic motion in comparing with a motion to which γ equals to 0.8. Contrastively, a motion with γ approaching zero will show no periodic behavior, and therefore is a perfectly nonperiodic motion. When γ takes a value such that 0 < γ < 1, it implies that some points in the Poincar´e map are periodically overlapping points while the others are not. Nonperiodic cases in between chaos and periodic motions may include the intermittent chaos in which chaotic motions occur between periods of regular motion.

Characteristics Diagnosis of Nonlinear Dynamical Systems

93

It should be noted, however, the expression shown in (7) is theoretical, as it requires an infinitely large number of n for a perfect measurement of γ and the time range considered must be t ∈ [0, ∞) such that t will be sufficient for a perfect γ . This implies that the periodicity ratio γ can be precisely calculated only in the cases for which the analytical solutions corresponding to the dynamical systems are available. For most nonlinear dynamic systems, however, the calculation for the periodicity ratio has been done on a numerical basis with the aid of a computer, as analytical solutions for these systems are not available. As Q(y), P(z) in the equations are step functions, the numerical calculation for γ can be conveniently carried out. In numerically determining for γ , therefore, a sufficiently large n should be used in performing the actual numerical calculation for γ in the practice of numerical calculation. In computing the periodicity ratio, errors caused by numerical calculation and by the mathematical models of numerical purpose should also be considered. Furthermore, in numerically calculating for γ , all of the n points must be compared to see whether they are overlapping points or not. Once a point is counted as an overlapping point, it should not be counted again in the numerical calculations. For nonlinear dynamic systems, a motion with periodicity ratio equals to zero may not necessarily be a chaotic motion. By the definition of periodicity ratio, a perfect quasiperiodic motion also has a periodicity ratio of zero. In this case, another technique, Lyapunov exponent approach can be employed.

3 Lyapunov Exponent and Periodicity Ratio Based on the current literature, Lyapunov exponent is probably the most popular criterion used for determining whether a system is convergent or divergent. However, Lyapunov exponent can only describe whether a system is convergent or divergent. This may not necessarily lead to the conclusion whether the system is periodic or chaotic, let alone the other characteristics such as quasiperiodic or nonperiodicnonchaotic behaviors. The former researches of perodicity ratio method [24, 32] concentrated on the derivation of the periodicity ratio together with the development of a methodology for diagnosing the irregular motions from the regular motions of a dynamic system. The characteristics of the periodicity ratio in diagnosing the irregular motions were not discussed. Moreover, for systematically analyzing the behavior of a nonlinear dynamic system with the implementation of the periodicity ratio, a comparison of the periodicity ratio with the other criteria seen in the literature for determining the types of motion of a dynamics system is necessary. A thorough and systematic investigation on the characteristics of the periodicity ratio and Lyapunov exponent approaches together with their implementation in the analyses of nonlinear dynamic systems are therefore necessary and will be described in this section. A detailed comparison of diagnosing the nonlinear system behavior

94

L. Dai and Lu Han

between the periodicity ratio and Lyapunov exponent will also be presented. The applications of Lyapunov exponent and periodicity ratio as criteria will also be discussed. As an example, the generation of the periodic-chaotic region diagrams with the two criteria will be presented and compared for Duffing equation which represents the widely used system in nonlinear dynamics. Based on the approach of Lyapunov exponent in diagnosing the nonlinear behaviors, periodicity, limit circles, and two-torus can be effectively determined and classified as convergent systems. Chaotic systems are then classified as divergent systems. The convergence and divergence of a system can be efficiently and accurately determined with Lyapunov exponent in comparing with the other methods for diagnosing the characteristics of nonlinear systems. Lyapunov exponent describes the essence of evolution tendency of a system and the asymptotic state of an evolution. It also quantifies the sensitivity of a nonlinear dynamic system to the initial conditions. The sensitivity of a system to initial condition reflects one of the most important characteristics of a nonlinear system, chaos. In fact, Lyapunov exponent is suitable for almost all the systems expressible by differential equations including discrete systems. For Lyapunov exponent approach, the calculation of the Lyapunov exponent cannot be carried out analytically in general. In most cases, one must rely on numerical techniques to carry out the integration. In fact, the existing methods for estimating the Lyapunov exponent all suffer from computational difficulties. Specifically, almost all the Lyapunov exponent techniques used for analyzing dynamic systems need to calculate for the composition of Jacobian matrices which is defined in (2) corresponding to the quantitative points along the trajectory considered as shown in (1). In performing the matrix production or computing for the eigenvalues of the Jacobian matrices on a digital computer, the accumulated errors of the calculation must be taken into consideration as they affect the accuracy of the calculated results. This may cause serious concerns when a large number of points along a trajectory over a large time span are considered. The overflow errors of the calculations are the consequence of the exponential expansions in determining the Lyapunov exponent. The errors are also related to the number of points selected along the trajectory and the time span considered in calculating for the Lyapunov exponent. The larger the selected points are and the longer the time span is, the greater could be the errors accumulated. A large enough time span is therefore necessary in determining for the system evolution tendency and to assure the accuracy of Lyapunov exponent to be calculated. Based on the definition of Lyapunov exponent, theoretically, an infinitely large time span is needed. The errors are not avoidable in numerically determining for Lyapunov exponent. Reliable results in the determination depend on the balance of the error and accuracy. The Lyapunov spectrum relies on periodic Gram–Schmidt orthonormalization of the Lyapunov vectors to avoid a misalignment of all the n vectors along the direction of maximal expansion. The reliability and accuracy of the Lyapunov exponent such determined depend on the selection of the small perturbation, stability of the calculation, and the numerical method used in the process for determining the

Characteristics Diagnosis of Nonlinear Dynamical Systems

95

Lyapunov exponent. Moreover, the complexity of governing nonlinear systems may further increase the difficulty in determining for Lyapunov exponent. Mathematically, the readjustment with the Gram–Schmidt orthonormalization or regulation of the trajectory considered can be complex if the dimension of the system is large. Compare to the Lyapunov exponent, the computation of periodicity ratio is merely affected by the complexity of the differential system. Firstly, as per the definition of periodicity ratio method, PR value depends on the number of the periodically overlapping points among all the investigated points in Poincar´e map. The calculation of PR value is always carried out by numerical simulation. Therefore, to accurately determine whether two given points in the Poincar´e map is overlapped is the key factor for obtaining PR value. As the numerical techniques always bring computation errors, the overlapping points in a Poincar´e map can only be very close but usually not perfectly zero as required in theory. In numerically determining the overlapping points, practically, one may have to assume an error range. For a given n dimensional differential dynamic system, once the numerical solution of the system is obtained, the distance between two points in a Poincar´e map can be easily determined by Euclid distance of the n dimensional space. It should be noticed that the Euclid distance can be conveniently determined by simple algebraic computation and it is not limited by the format of governing equations. In comparing with periodicity ratio approach, the Jacobian matrix of Lyapunov exponent approach relies on the format of the governing equations, though it also requires numerical solutions. In periodicity ratio approach, no eigenvalue of matrices needs to be determined and the shortages of Lyapunov exponent approach such as the breakage or error overflow will not occur. The computational efficiency and simplification of the periodicity ratio approach bring obvious advantage over that of Lyapunov exponent. In many cases, the numerically determined Lyapunov exponent values vary with the increasing calculation time. This may affect the conclusion of the Lyapunov exponent approach, in determining whether a system is convergent or divergent. The variation is contributed by the determination of Lyapunov exponent value, as the value is the average of the summation of the log value of the vector norm of the denominator obtained from the integration over a finite time. In each step, the log of vector norm of the denominator as shown in (5) may change from negative to positive or zero, as the vector norm of the denominator only represents the rate of exponential expansion or contraction in the direction of the former vector regardless whether the entire trajectory is divergent or convergent. Thus, the cancellation of the positive values and negative values during the summation may cause instability and affect the resulted Lyapunov exponent. It should be noticed that the cancellation depends on the time span over which the Lyapunov exponent is calculated. The cancellation effect of the Lyapunov exponent caused by different calculating times is not the case for the periodicity ratio approach. For any point in a Poincar´e map, every successive point of this point is inspected to determine whether or not it is overlapped with the given point. Once a point is excluded from periodic points, it will not affect the resulted PR value in any later time. Similarly, once a point is convinced as a periodic point during the given time, it will always be taken as a

96

L. Dai and Lu Han

periodic point over the entire time span. Thus, as long as all the points are inspected in the Poincar´e map through a fairly long time, the result obtained can be authentic. Therefore, the conclusive results of periodicity ratio approach are more stable. In addition, in calculating for the Lyapunov exponent, it requires the investigation on all the points in a phase diagram rather than Poincar´e map. Compared to periodicity ratio approach, Lyapunov exponent approach needs significantly more points to be considered and therefore requires considerably longer calculation time over the same time span considered, in comparison with that of the periodicity ratio approach. Although Lyapunov exponent approach may determine for convergence or divergence of a system, the Lyapunov exponent values usually cannot be directly used to diagnose for all the nonlinear characteristics such as periodic, quasiperiodic, and chaotic or other nonlinear characteristics, which are more significant in physics and engineering applications. Nevertheless, in general, the response of a nonlinear system is considered as periodic when the corresponding Lyapunov exponent are all negative; the response of the system is thought chaotic in the case that all the Lyapunov exponent are positive. It is also widely accepted and true for most cases that a system is chaotic if the maximum Lyapunov exponent or more Lyapunov exponent values are positive; the system is periodic in the cases that the maximum Lyapunov exponent is negative or zero. However, this may not necessarily true for all the cases. Some examples for such cases will be described in the following section. Moreover, Lyapunov exponent cannot be used to diagnose quasiperiodic responses of a system nor some of the nonperiodic cases which are very close to periodic, though the system is convergent by Lyapunov exponent approach. For the systems that are nonperiodic but close to chaos, the Lyapunov exponent approach categorizes the systems as divergent systems though they are not completely chaotic. A significant drawback of the Lyapunov exponent approach is that the approach simply categories the nonlinear systems as either convergent or divergent, rather than characterize them into finer categories such as periodic, quasiperiodic, chaotic, and other nonlinear characteristics which are physically meaningful. In comparison with the Lyapunov exponent approach, periodicity ratio approach characterizes a nonlinear system by the periodicity of the system’s responses. Periodic systems can be conveniently distinguished from all the other systems with employment of periodicity ratio. Periodicity ratio can also be conveniently used to quantitatively describe the behavior and periodicity of the responses which are not falling in any of the characteristics of chaos, periodicity, and quasiperiodicity. The PR values are ranged from 1.0 to 0.0, corresponding to the responses from perfect periodic to perfect nonperiodic. Therefore, periodicity ratio quantifies infinite number of characteristics between periodicity to chaos. However, periodicity ratio approach cannot be used to evaluate the sensitivity of a system and it is not suitable for diagnosing quasiperiodic systems from chaos. Also, periodicity ratio does not provide any information regarding convergence or divergence of a system.

Characteristics Diagnosis of Nonlinear Dynamical Systems

97

Although periodicity ratio is not capable of distinguishing quasiperiodic case from chaos, quasiperiodicity and chaos are all falling in the category of zero PR value. Chaos can be distinguished once quasiperiodic cases are identified in the category. Least square polynomial curve fitting method has been used to distinguish quasiperiodic case from chaos. This approach may miss some of the quasiperiodic case to which the corresponding Poincar´e map does not forming a complete loop or continuous curve. The best solution seems the combination of the two approaches of periodicity ratio and Lyapunov exponent.

4 A Duffing System In order to demonstrate the behaviors of a nonlinear dynamic system with Lyapunov exponent and periodicity ratio approaches, a system governed by the following Duffing system [10, 32] is considered: x¨ + kx˙ + x3 = B cos t

(14)

where B can be considered as the amplitude of external excitation acting on the system and k represents the damping coefficient of the system. With implementation of Periodicity Ratio approach, a periodic–quasiperiodic– chaotic diagram can be plotted. The diagram graphically illustrates the nonlinear behavior of a dynamic system and to be used efficiently identify the responses of the dynamic system corresponding to system parameters and initial conditions. The diagram is hence a powerful tool in analyzing the regular and irregular behaviors of a nonlinear dynamic system [24, 32, 34]. Yet, Fig. 1 provides more detailed information in comparing with that of Ueda, in addition to adding the quasiperiodic regions and correcting the periodic and chaotic regions of the diagram on top of Ueda’s. It should be noted that one of the advantages of utilizing Periodicity Ratio approach for generating such diagram is that neither a Poincar´e map nor a phase diagram of any kind needs to be plotted. In generating the diagram, parameter k is taken the value from 0 to 1 with an increment of 0.01 and B is from 0 to 15 with an increment of 0.1, as shown in the figure. More than 15,000 states under different parameter values are examined with their corresponding PR values. The initial displacement is −2 and initial velocity is 0. Each point in the diagram actually represents a steady-state motion for the periodic system. For each point, the first 50 cycles are omitted to eliminate the perturbation caused by the initial values. And 451 cycles are computed to determine the PR value of each point in the diagram. Diamond indicates chaotic case while dot represents periodic case, and cross is the quasiperiodic case. The while blank area is occupied by those nonregular periodic points whose PR value is between 0 and 1 but not 0 or 1. A similar plot is generated by employing Lyapunov exponent approach with the same parameters and time span, as shown in Fig. 2. The diamonds in the figure indicate divergent cases while the dots represent convergent cases. The Lyapunov

98

L. Dai and Lu Han

Fig. 2 Duffing system status diagnosed by periodicity ratio approach

Fig. 3 System status diagnosed by Lyapunov exponent approach

exponent obtained is based on the Lyapunov spectrum method proposed by Wolf [22]. The time step in calculation is 0.01π and the time span used for the new calculations is 1, 000π which is as the same as the calculation time in PR method diagnosing. The initial values are also made as the same as the PR method. By definition [11, 22, 23], the maximum Lyapunov exponent with value zero indicates that the system is in a steady-state mode as a limit cycle.

Characteristics Diagnosis of Nonlinear Dynamical Systems

99

Fig. 4 Poincar´e map and corresponding phase diagram at B = 0.1, k = 0.9, PR = 0.7349, LE = (0.0000 −0.0094 −0.8906)

As can be seen from the two figures, Fig. 2 provides much detailed information regarding the characteristics of the system. All the responses of chaos, quasiperiodicity, and periodicity of the system are illustrated in the diagram. More significantly, those system responses that are not falling in any of the categories of chaos, quasiperiodicity, and periodicity can be identified with the help of the periodicity ratio approach. Figure 3 is plotted with utilization of Lyapunov exponent approach, only shows the cases of convergence and divergence of the system. It is interesting to notice that most of the convergence areas in Fig. 3 match with the periodicity areas

100

L. Dai and Lu Han

Fig. 5 Poincar´e map and corresponding phase diagram at B = 10, k = 0, PR = 0, LE = (0 −0.0002 −9.9996)

in Fig. 2, and most of the divergence areas in Fig. 3 match with the chaotic areas in Fig. 2, though they are not perfectly matched. Based on the numerical simulations performed, Lyapunov exponent approach shows some limitations. Some comparative cases are selected from Figs. 1 and 2 to demonstrate the characteristics of the Lyapunov exponent and periodicity ratio approaches. Figure 4 plots a Poincar´e map and its phase diagram for a case that is declared by Lyapunov exponent as a convergent system. However, with utilization of periodicity ratio approach, the case is more close to a periodic case. Although Lyapunov

Characteristics Diagnosis of Nonlinear Dynamical Systems

101

Fig. 6 Poincar´e map and corresponding phase diagram at B = 10.3, k = 0.19, PR = 0.4462, LE = (0.0001 −0.0080 −0.1820)

exponent of this case indicates a convergent system, as can be seen from Fig. 4, it is not a perfect periodic case and strong regularity and predictability of the system are actually evident. The periodicity ratio of 0.7349 indeed reflects and quantifies the periodicity and regularity of the system. It should be noticed that the maximum Lyapunov exponent of this case is 0.0000. Conventionally, this is considered as a periodic case [11, 22, 23].

102

L. Dai and Lu Han

A quasiperiodic case is shown in the two figures of Fig. 5 for the system. Although the Poincar´e points form a continuous loop in the figure, but a continuous curve is difficult to be identified by least square fitting method due to the existing singularities. A special case is shown in the Poincar´e map and phase diagram of Fig. 6. This case, as per the Lyapunov exponent approach, is considered as a divergent case. Conventionally, with the Lyapunov exponent values (0.0001 −0.0080 −0.1820), this case is mathematically interpreted as a chaos [22, 23]. In order to assure proper accuracy of the Lyapunov exponent values calculated, extra-long time is considered in performing the calculation. The Lyapunov exponent values are stable and unchanged over the extra time span. However, as can be seen from the figure, certain periodicity and regularity can be found. In some areas in the phase diagram, responses of the system can be fairly accurately predicted. This case can hardly be characterized as a perfect chaos. The PR value of 0.4462 states that this case is neither periodic nor chaotic. The response of the system in this case is somewhere in between perfect periodicity and chaos. Therefore, this case is categorized as nonperiodic.

References 1. Gollub JP, Baker GL (1996) Chaotic dynamics. Cambridge University Press, Cambridge 2. Alligood KT, Sauer T, Yorke JA (1997) Chaos: an introduction to dynamical systems. Springer, New York 3. Devaney RL (2003) An introduction to chaotic dynamical systems, 2nd edn. Westview, Boulder 4. Valsakumar MC, Satyanarayana SVM, Sridhar V (1997) Signature of chaos in power spectrum. Pramana-J Phys 48:69–85 5. Peitgen HO, Richter PH (1986) The beauty of fractals: images of complex dynamical systems. Springer, Berlin 6. Peitgen HO, Saupe D (1988) The science of fractal images. Springer, New York 7. Lauwerier H (1991) Fractals. Princeton University Press, Princeton 8. Kumar A (2003) Chaos, fractals and self-organisation, new perspectives on complexity in nature. National Book Trust, New Delhi 9. Zaslavsky GM (2005) Hamiltonian chaos and fractional dynamics. Oxford University Press, New York 10. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York 11. Nayfeh AH, Balachandran B (2004) Applied nonlinear dynamics: analytical, computational, and experimental methods non-linear oscillation. Wiley, New York 12. Strogatz S (2000) Nonlinear dynamics and chaos. Perseus, Cambridge 13. Kolmogorov AN (1941) Local structure of turbulence in an incompressible fluid for very large Reynolds number. Dokl Akad Nauk SSSR 30(4):301–305 14. Kolmogorov AN (1941) On degeneration of isotropic turbulence in an incompressible viscous liqui. Dokl Akad Nauk SSSR 31(6):538–540 15. Kolmogorov AN (1954) Preservation of conditionally periodic movements with small change in the Hamiltonian functio. Dokl Akad Nauk SSSR 98:527–530 16. William FB (2006) Topology and its applications. Wiley, Hoboken

Characteristics Diagnosis of Nonlinear Dynamical Systems

103

17. Qian B, Rasheed K (2004) Hurst exponent and financial market predictability. In: IASTED conference on “financial engineering and applications”, Cambridge pp 203–209 18. Tufillaro AR (1992) An experimental approach to nonlinear dynamics and chaos. AddisonWesley, New York 19. Hristu-Varsakelis D, Kyrtsou C (2008) Evidence for nonlinear asymmetric causality in US inflation, metal and stock returns. Discrete Dyn Nat Soc 2008. Article ID 138547 20. Sprott J (2003) Chaos and time-series analysis. Oxford University Press, Oxford/New York 21. Choi JJ, O’Keefe KH, Baruah PK (1992) Nonlinear system diagnosis using neural networks and fuzzy logic. In: 1992 IEEE international conference, pp 813–820. Nice, France 22. Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Physica D 16(3):285–317 23. Parks PC (1992) Lyapunov’s stability theory – 100 years on. IMA J Math Control I 9:275–303 24. Dai L, Singh MC (1997) Diagnosis of periodic and chaotic responses in vibratory systems. J Acoust Soc Am 102(6):3361–3371 25. Li TY, Yorke JA (1975) Period three implies chaos. Am Math Mon 82:985–992 26. May R (1976) Simple mathematical models with very complicated dynamics. Nature 261(5560):459–467 27. Mandelbrot BB (1982) The fractal geometry of nature. W.H. Freeman, San Francisco 28. Briggs J (1992) Fractals: the patterns of chaos. Thames and Hudson, London 29. Lakshmanan M, Rajasekar S (2003) Nonlinear dynamics: integrability, chaos, and patterns. Springer, New York 30. Rong H, Meng G, Wang X, Xu W, Fang T (2002) Invariant measures and lyapunov-exponents for stochastic Mathieu system. Nonlinear Dyn 30:313–321 31. Shahverdian AY, Apkarian AV (2007) A difference characteristic for one-dimensional deterministic systems. Commun Nonlinear Sci Numer Simul 12(3):233–242 32. Ueda Y (1980) Steady motions exhibited by Duffing’s equations: a picture book of regular and chaotic motions. In: Holmes PJ (ed) New approaches to nonlinear problems in dynamics. Society for Industrial and Applied Mathematics, Philadelphia, pp 311–322 33. Dai L (2008) Nonlinear dynamics of piecewise constant systems and implementation of piecewise constant arguments. World Scientific, Singapore 34. Zhang M, Yang J (2007) Bifurcations and chaos in Duffing’s equation. Acta Math Appl Sin 23(4):665–684

Synchronization of Two Coupled Phase Oscillators Yongqing Wu, Changpin Li, Weigang Sun, and Yujiang Wu

Abstract In this chapter, synchronization between two coupled populations of nonidentical phase oscillators is investigated. Generalizing the linear reformulation (Roberts Phys Rev E 77:031114, 2008), we could find explicit expressions of the synchronization order parameters, which include the phase coherence within each population and the phase coherence between two populations. Finally, numerical example is given to illustrate the theoretical results.

1 Introduction In recent years, collective synchronization in populations of dynamically interacting units has received increasing attention in a variety of fields, ranging from biology to physics and engineering (see [1, 2] and many references cited therein). The Kuramoto model [3–6] is one of the most widely studied frameworks to analyze the synchronization of an infinite number of interacting units with phase oscillators, and it is also a mathematical model describing the dynamics of a system with weakly

Y. Wu School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China Basic Teaching Department, Liaoning Technical University, Huludao 125105, China e-mail: [email protected] C. Li () Department of Mathematics, Shanghai University, Shanghai 200444, China e-mail: [email protected] W. Sun School of Science, Hangzhou Dianzi University, Hangzhou 310018, China e-mail: [email protected] Y. Wu School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 4, © Springer Science+Business Media, LLC 2012

105

106

Y. Wu et al.

coupled oscillators. Synchronization of two symmetrical (or asymmetrical) coupled populations of identical (or nonidentical) phase oscillators is studied in [7] and [8]. More recently, Roberts [9] introduced a linear reformulation method on studying the Kuramoto model, which makes it possible to derive the explicit synchronization order parameter with any number of oscillators. The above-mentioned work focused on the synchronization inside a population, and we refer to it as “inner synchronization” for brevity; however, recently synchronization was generalized to two coupled networks, as “outer synchronization” (see [10, 11]). In this chapter, we attempt to generalize the previous proposed linear reformulation method to study two coupled populations of phase oscillators. The expressions of the synchronization order parameters within each population and between two populations are derived explicitly, and the values of coupling strength can also be derived. The rest of this chapter is organized as follows. In Sect. 2, we describe the linear reformulation of two coupled populations of phase oscillators, and derive the synchronization order parameters explicitly. Section 3 contains a numerical example to illustrate the theoretical results. Finally, we present our concluding remarks in Sect. 4.

2 Theoretical Analysis Consider two symmetrical coupled populations of nonidentical phase oscillators. The system dynamics is expressed as    N  N (1) (1) (1) (1) (1) (1) (2) (1) θ˙k = ωk + ∑ A jk sin θ j − θk + ∑ C jk sin θ j − θk ,

(1)

   N  N (2) (2) (2) (2) (2) (2) (1) (2) θ˙k = ωk + ∑ A jk sin θ j − θk + ∑ C jk sin θ j − θk ,

(2)

j=k

j=1

j=k

j=1

(1,2)

where k = 1, 2, . . . , N  1. Here, θk (1,2) ωk

∈ R describe the phase of kth oscillator in

∈ R are their natural frequencies, respectively. The population 1 or 2, and population 1 (population 2) is     coupled internally with coupling strength matrix (1) (2) (1) (2) A = A jk , whereas the interpopulation coupling is determined A = A jk      (1) (2) C(2) = C jk . The system (1)–(2) has been studied more by C(1) = C jk recently by Abrams et al. [12] and Laing [13]. Now, we propose a linear reformation [9] of the system (1)–(2),   K (1) (1) (1) (1) (1) ψ˙ k = iωk − γk ψk + N

N

∑ ψj j=k

(1)

+

K (1) N

N

∑ ψj

j=1

(2)

,

(3)

Synchronization of Two Coupled Phase Oscillators

107

  K (2) (2) (2) (2) (2) ψ˙ k = iωk − γk ψk + N (1,2)

where ψk

K (2) /N

N

∑ ψj

(2)

j=k

+

K (2) N

N

∑ ψj

(1)

,

(4)

j=1

∈ C, k = 1, 2, . . . , N  1. Here the positive numbers K (1) /N and (1)

are the coupling constants of the above linear system. γk

decay constants which can be tuned to bring the amplitudes of (1,2)

(2)

and γk

are the

(1) ψk

(2) and ψk to a (1) (2) and ωk , ωk

>0 steady state. Throughout this chapter, we always assume γk are real.   (1) (1) (2) (2) T Let ψ = ψ1 , . . . , ψN , ψ1 , . . . , ψN ; the system (3)–(4) can be rewritten as a vector form,

ψ˙ = Bψ ,  where B =

W (1) B1 B2 W (2)

(5)

 , ⎛

K (1) K (1) N N K (1) K (1) N N

···

K (1) K (1) N N

···

K (1) N

K (2) K (2) N N K (2) K (2) N N

···

K (2) N K (2) N

K (2) K (2) N N

···

⎜ ⎜ ⎜ B1 = ⎜ . ⎜ . ⎝ .

⎛

⎜ ⎜ ⎜ B2 = ⎜ . ⎜ . ⎝ .

··· .. . . . .

··· .. . . . .

K (1) N K (1) N

⎞

⎟ ⎟ ⎟ .. ⎟ ⎟ . ⎠

, N×N

⎞

⎟ ⎟ ⎟ .. ⎟ ⎟ . ⎠

K (2) N

, N×N

⎛ W (1) = diag



(1) (1) iω1 − γ1 , · · ·

(1) (1) , iωN − γN



0

⎜ K (1) ⎜ ⎜ N +⎜ . ⎜ . ⎝ .

K (1) N

0 ··· .. . . . .

K (1) K (1) N N

⎛

W

(2)

0 ⎜ K (2)   ⎜ ⎜ N (2) (2) (2) (2) = diag iω1 − γ1 , · · · , iωN − γN + ⎜ . ⎜ . ⎝ .

···

K (2) N

⎞

⎟ ⎟ ⎟ .. ⎟ . ⎟ ⎠

··· 0 ···

0 ··· .. . . . .

K (2) K (2) N N

K (1) N K (1) N

K (2)

, N×N

⎞

N K (2) N

⎟ ⎟ ⎟ .. ⎟ ⎟ . ⎠

··· 0

. N×N

108

Y. Wu et al.

Assume B is diagonalizable; then the general solution of (5) is

ψ=

2N

∑ a j v j eλ j t ,

(6)

j=1

where a j ∈ C are constants determined by the initial conditions, v j and λ j are the eigenvectors and eigenvalues associated with the matrix B. Here, we adopt the convention of ordering all eigenvalues by their real parts, from the real part of λ1 (least) to that of λ2N (greatest). Distinct eigenvalues with the same real part will arbitrarily be assigned consecutive subscripts within the larger sequence. (1,2) to guarantee that Re(λ2N ) = 0. If We can select suitable constants γk Re(λ2N−1 ) = 0; then we have lim ψ = a2N v2N eiωr t ,

(7)

t→∞

where the collective frequency of the fully locked state ωr is given by

ωr = −iλ2N .

(8)

Here, the analysis is restricted to the full locking state. The phase coherence within each population is described by the complex order parameters r(1,2) eiφ

(1,2)

=

1 N

N

(1,2)

∑ eiθ j

,

(9)

j=1

where φ (1,2) ∈ R are the average phases of the oscillators in respective populations and 0 ≤ r(1,2) ≤ 1 describe the degree of synchronization in each population. When all oscillators are close together, r(1,2) will be close to 1 (a perfectly coherent state), and when they are spread uniformly over the interval [−π , π ], r(1,2) will be 0 (an incoherent state). Furthermore, the synchronization indicator between two populations is denoted by 1 r= N

N

∑e

j=1

, 0 ≤ r ≤ 1.

i(θ j −θ j ) (1)

(2)

(10)

In the sequel, we should calculate the eigenvector v2N associated with λ2N to find explicit expressions for r(1) , r(2) , and r, respectively. Since the system is in the full locking state, we get

Synchronization of Two Coupled Phase Oscillators

109

⎧ K (1) ⎪ ⎪ ⎪ , ⎪ ⎪ (2) (i(ω (1) − ω ) − K (1) − γ (1) ) ⎪ K r ⎪ j j N ⎪ ⎪ ⎪ ⎪ ⎨ 1 ≤ j ≤ N, (v2N ) j = ⎪ 1 ⎪ ⎪ ⎪ , ⎪ (2) (2) (2) ⎪ ⎪ i(ω j−N − ωr ) − KN − γ j−N ⎪ ⎪ ⎪ ⎪ ⎩ N + 1 ≤ j ≤ 2N, (1,2)

where j is the index for the components of v2N . Let ψ j (1,2)

(11)

(1,2)

(t) = R j

(1,2)

(t)eiθ j

(t)

(R j ∈ R), with an arbitrary distribution of ωk over finite N in the long-time limit, we have N N (1) 1 N (1) ψ (v ) 1 1 2N j j iθ j (1) r = ∑e = ∑ (1) = ∑ N j=1 ψ N j=1 (v ) N j=1 2N j j  (1) (1) (1) (ω j − ωr )2 + ( KN + γ j )2 1 N , = ∑ N j=1 i(ω (1) − ωr ) − ( K (1) + γ (1)) j

r(2)

1 = N

N

∑e

j=1

(2) iθ j

N

(12)

j

1 N ψ (2) 1 2N (v2N ) j j = ∑ (2) = ∑ N j=1 ψ N j=N+1 (v ) 2N j j

 (2) N (2) (2) (ω j − ωr )2 + ( KN + γ j )2 1 , = ∑ N j=1 i(ω (2) − ωr ) − ( K (2) + γ (2)) j j N 1 r= N

N

(1) (2) i(θ j −θ j )

∑e

j=1

(13)

(2) 1 N ψ (1) ψ j j = ∑ (1) (2) N j=1 ψ ψ j j

(2) (2) K (2) 1 N i(ω j − ωr ) − ( N + γ j ) = ∑ N j=1 i(ω (1) − ωr ) − ( K (1) + γ (1) ) j j N

  (1) (1) 2  (ω j − ωr )2 + ( KN + γ (1) ) j . ·  (2) (2) (2) (ω j − ωr )2 + ( KN + γ j )2 (14)

Next, we will analyze the relations between the linear reformulation (3)–(4) and (1,2)

the original model (1)–(2). Because of ψ j rewritten as

(1,2)

(t) = R j

(1,2)

(t)eiθ j

(t)

, (3)–(4) can be

110

Y. Wu et al. (1)

(1) N

K (1) (1) θ˙k = ωk + N +

K (1) N

N

∑

  (1) (1) sin θ − θ j k (1)

Rj

∑

j=k

Rk

(2)

  (2) (1) , sin θ − θ j k (1)

Rj

(1) N

K (1) (1) (1) R˙ k = −γk Rk + N +

K (1) N

N

∑ Rj

(2)

(1)

∑ Rj j=k

  (2) (1) cos θ j − θk ,

K (2) N

N

∑

  (2) (2) sin θ − θ j k (2)

Rj

∑

j=k

Rk

(1)

  (1) (2) , sin θ − θ j k (2)

Rj

(2) N

N

∑ Rj

(1)

∑ Rj

(2)

j=k

  (2) (2) cos θ j − θk

  (1) (2) cos θ j − θk .

(18)

j=1

(1,2)

If Re(λ2N ) = 0, Re(λ2N−1 ) = 0, Rk (1) lim R (t) t→∞ k

(17)

j=1 Rk

K (2) (2) (2) R˙ k = −γk Rk + N K (2) N

(16)

(2)

(2) N

+

  (1) (1) cos θ j − θk

j=1

K (2) (2) θ˙k = ωk + N +

(15)

j=1 Rk

will reach a steady state, in fact

= |a2N (v2N )k | =



K (1) |a2N | (1)

(1)

(1)

K (2) (ωk − ωr )2 + ( KN + γk )2 (2) lim Rk (t) = a2N (v2N )N+k

,

(19)

t→∞

|a2N | =  , (2) (2) 2 K (2) 2 (ωk − ωr ) + ( N + γk )

(20)

then (15) and (17) become the forms of system (1)–(2), and the coupling constants read as

Synchronization of Two Coupled Phase Oscillators

111

(1)

K (1) R j (1) A˜ jk = lim (1) t→∞ N Rk   (1) (1) 2 K (1) 2 K (1)   (ωk − ωr ) + ( N + γk ) , = (1) (1) (1) N (ω j − ωr )2 + ( KN + γ j )2

(21)

(2)

K (2) R j (2) A˜ jk = lim (2) t→∞ N Rk   (2) (2) (2) (2) (ωk − ωr )2 + ( KN + γk )2 K   = , (2) (2) (2) N (ω j − ωr )2 + ( KN + γ j )2

(22)

(2)

K (1) R j (1) C˜ jk = lim (1) t→∞ N Rk   (1) (1) (1) (ωk − ωr )2 + ( KN + γk )2 K (2)   = , (2) (2) (2) N (ω − ωr )2 + ( K + γ )2 j

N

(23)

j

(1)

K (2) R j (2) C˜ jk = lim (2) t→∞ N Rk   (2) (2) 2 K (2) (1) 2 K   (ωk − ωr ) + ( N + γk ) . = (1) (1) (1) N (ω − ωr )2 + ( K + γ )2 j

(1,2)

N

(24)

j

(1,2)

It is noted that A˜ jk and C˜ jk are independent of initial conditions because a2N vanishes. Hence, (15) and (17) are equal to systems (1)–(2) when time approaches infinity. From the above theoretical analysis, we could perform a nonlinear transformation on two interacting populations of nonidentical oscillators into the linear reformulation (3)–(4), which can be solved exactly with some algebraic knowledge for any (1,2) number of N by introducing the amplitude Rk .

3 Numerical Example In this section, numerical example is given to verify the theoretical results. The (1,2) number of oscillators within each population is N = 100, and we assume that γk = (1,2)

γ > 0, ωk

(1,2)

= ω (1,2) , and Kp

= K (1,2) , k = 1, . . . , N.

112

Y. Wu et al. 1 0.9

r (1)

0.8

r (2) r

r (1), r (2) and r

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4 t

5

6

7

8

Fig. 1 Evolution of the synchronization order parameters in systems (1)–(2). Squares (circles) represent the synchronization order parameter within population 1 (population 2); the solid line corresponds to the synchronization order parameter between two populations

Here, let ω (1) = 1.0, ω (2) = 1.3, K (1) = 0.8, and K (2) = 1.2; then we have λ2N ≈ 1.9787 − γ + 1.1800i. By setting γ = 1.9787, Re(λ2N ) ≈ 0, Re(λ2N−1 ) < 0, and ωr = −iλ2N ≈ 1.1800. Due to (12)–(14), we compute the synchronization order parameters r(1) = r(2) = r = 1 easily when time approaches infinity. From (21)–(24), the values of the coupling strength in systems (1)–(2) can be easily derived, and the amplitude of the complex order parameters 1 N 1 N (1,2) (1) (2) iθ j i(θ j −θ j ) (1,2) = ∑e r , r = ∑ e N j=1 N j=1 can be used to demonstrate whether synchronization happens. As shown in Fig. 1, systems (1)–(2) give the identical result r(1) = r(2) = r = 1.

4 Conclusion In this chapter, we consider synchronization between two symmetrical coupled populations of nonidentical phase oscillators. Through the linear reformulation, the synchronization order parameters within each population and between two populations are shown explicitly, and we can derive the values of the coupling strength matrices. Finally, numerical simulation is consistent with our theoretical results.

Synchronization of Two Coupled Phase Oscillators

113

Acknowledgments This work was supported in part by the National Natural Science Foundation of China (No 10872119), Key Disciplines of Shanghai Municipality under Grant No. S30104, and National Basic Research Program of China 973 under Grant No. 2011CB706903.

References 1. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442 2. Arenas A, Diaz-Guilera A, Kurths J, Moreno Y, Zhou CS (2008) Synchronization in complex networks. Phys Rep 469:93–153 3. Strogatz SH (2000) From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143:1–20 4. Acebr´on JA, Bonilla LL, Vicente CJP, Ritort F, Spigler R (2005) The kuramoto model: a simple paradigm for synchronization phenomena. Rev Mod Phys 77:137–185 5. Mirollo R, Strogatz SH (2007) The spectrum of the partially locked state for the kuramoto model. J Nonliear Sci 17:309–347 6. Basnarkov L, Urumov V (2008) Kuramoto model with asymmetric distribution of natural frequencies. Phys Rev E 78:011113 7. Okuda K, Kuramoto Y (1991) Mutual entrainment between populations of coupled oscillators. Prog Theor Phys 86:1159–1176 8. Montbri´o E, Kurths J, Blasius B (2004) Synchronization of two interacting populations of oscillators. Phys Rev E 70:056125 9. Roberts DC (2008) Linear reformulation of the kuramoto model of self-synchronizing coupled oscillators. Phys Rev E 77:031114 10. Li CP, Sun WG, Kurths J (2007) Synchronization between two coupled complex networks. Phys Rev E 76:046204 11. Li CP, Xu CX, Sun WG, Xu J, Kurths J (2009) Outer synchronization of coupled discrete-time networks. Chaos 19:013106 12. Abrams DM, Mirollo R, Strogatz SH, Wiley DA (2008) Solvable model for chimera states of coupled oscillators. Phys Rev Lett 101:084103 13. Laing CR (2009) Chimera states in heterogeneous networks. Chaos 19:013113

Chaotic Synchronization of Duffing Oscillator and Pendulum Albert C.J. Luo and Fuhong Min

Abstract The chaotic synchronization of the Duffing oscillator and controlled pendulum is investigated. The analytical conditions for the partial and full synchronizations of the controlled pendulum with chaotic motions in the Duffing oscillator are discussed. The partial and full synchronizations are illustrated to show the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum.

1 Introduction In 1989, Pecora and Carroll [1] discussed the synchronization of two systems connected with common signals by use of the sub-Lyapunov exponents, and the common signals were adopted as constraints for such two systems. Carroll and Pecora [2] implemented the synchronized circuits to simulate such synchronization of chaotic responses. Since then, one used control methods and schemes to investigate the synchronization of two dynamical systems with constraints. In 1992, Pyragas [3] investigated chaos control with a small time continuous perturbation for a synchronization of two chaotic dynamical systems. In 1995, Rulkov et al. [4] discussed a generalized synchronization of chaos in directionally coupled chaotic systems. Kocarev and Parlitz [5] decomposed the given systems into the active and passive systems for chaotically synchronized systems. In 1996, Pyragas [6] discussed the weak and strong synchronizations of chaos through the coupling

A.C.J. Luo () Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026–1805, USA e-mail: [email protected] F. Min Nanjing Normal University, Nanjing, Jiangsu, 210042, China e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 5, © Springer Science+Business Media, LLC 2012

115

116

A.C.J. Luo and F. Min

strength of two dynamical systems. In 1997, Boccaletti et al. [7] presented an adaptive synchronization of chaos for secure communication, and Abarbanel et al. [8] used a small force to control a dynamical system to the given orbits. In 1998, Pyragas [9] discussed the generalized synchronization of chaos. In 1999, Yang and Chua [10] investigated generalized synchronization of two dynamical systems by linear transformations. In 2002, Boccaletti et al. [11] gave a review on the synchronization of chaotic systems, and definitions and concepts of dynamical system synchronization were clarified. In 2004, Campos and Urias [12] investigated multimodal synchronization with chaos through a multi-valued, synchronized function. From such a brief survey, the synchronization of two dynamical systems is based on the two similar dynamical systems. Further the error dynamical systems are constructed and the Lyapunov method can be used to determine the asymptotical stability. In 2009, Luo [13] developed a theory for synchronization of dynamical systems with specific constraints via the theory of discontinuous dynamical systems. Such a theory for discontinuous dynamical systems can be found from [14–16]. In such a theory, the G-function functions were introduced to determine the switchability of a flow from one domain to another in discontinuous dynamical systems. In 2010, Luo and Min [17] used a feedback control to achieve the synchronization of a periodically forced Duffing oscillator with chaotic motions in a periodically forced pendulum. The periodically forced pendulum has librational and rotational motions. For the simple, complete synchronization of the slave and master systems, the periodically forced Duffing oscillator only can synchronize with the librational chaos in the pendulum because the forced Duffing oscillator possesses a finite displacement. In Luo and Min [18], the analytical conditions for the controlled pendulum synchronizing with a periodically forced, damped Duffing oscillator were developed, and the synchronization of the controlled pendulum with periodic motions in the Duffing oscillator was discussed. In this paper, the synchronization of the controlled pendulum with chaotic motions in the Duffing oscillator will be discussed. From the analytical conditions in Luo and Min [18], the controlled pendulum synchronizing with chaotic motions in the periodically forced, Duffing oscillator will be investigated, and the partial and full chaotic synchronizations of the two systems will be discussed. The synchronization scenarios for the two systems will be carried out, and numerical simulations for the partial and full synchronizations of chaotic motions for two dynamical systems will be completed to compare with the periodic synchronization.

2 Discontinuous Description A periodically forced, damped Duffing oscillator with a twin well potential is considered as a master system, i.e., x¨ + d x˙ − a1 x + a2x3 = A0 cos ω t,

(1)

Chaotic Synchronization of Duffing Oscillator and Pendulum

117

and a periodically driven pendulum is considered as a slave system y¨ + a0 sin y = Q0 cos Ωt.

(2)

For simplicity, the state variables are introduced as x = (x1 , x2 )T and y = (y1 , y2 )T

(3)

and the corresponding vector fields are defined as F (x,t) = (x2 , F (x,t))T and F(y,t) = (y2 , F(y,t))T

(4)

So the master system is expressed by

where

x˙ = F (x,t)

(5)

x2 ≡ x˙1 and F (x,t) = −d1 x2 + a1x1 − a2x31 + A0 cos ω t.

(6)

The slave system is written by where

y˙ = F(y,t)

(7)

y˙1 ≡ y2 and F(y,t) = −a0 sin y1 + Q0 cos Ωt.

(8)

To enforce the pendulum to synchronize with the Duffing oscillator, the control laws should be exerted to the pendulum, and the controlled pendulum becomes y˙ = F(y,t) − u(x, y,t)

(9)

where u(x, y,t) = (u1 , u2 )T , u1 = k1 sgn(y1 − x1) and u2 = k2 sgn(y2 − x2 ), F(y,t) − u(x, y,t) = (F1 , F2 )T .

(10)

For the controlled slave system, the two control laws make the controlled slave system possess four regions, and the components of the vector fields in (9) are given as follows. 1. For y1 > x1 and y2 > x2 , F1 (y,t) = y2 − k1 F2 (y,t) = −a0 sin y1 + Q0 cos Ωt − k2

(11)

118

A.C.J. Luo and F. Min

2. For y1 > x1 and y2 < x2 , F1 (y,t) = y2 − k1 F2 (y,t) = −a0 sin y1 + Q0 cos Ωt + k2

(12)

3. For y1 < x1 and y2 < x2 , the vector field components for the controlled slave system in (9) is governed by F1 (y,t) = y2 + k1 F2 (y,t) = −a0 sin y1 + Q0 cos Ωt + k2

(13)

4. For y1 < x1 and y2 > x2 , F1 (y,t) = y2 + k1 F2 (y,t) = −a0 sin y1 + Q0 cos Ωt − k2

(14)

Four domains in phase space of the controlled slave systems are defined as Ω1 = {(y1 , y2 )| y1 − x1 (t) > 0, y2 − x2 (t) > 0} , Ω2 = {(y1 , y2 )| y1 − x1 (t) > 0, y2 − x2 (t) < 0} , Ω3 = {(y1 , y2 )| y1 − x1 (t) < 0, y2 − x2 (t) < 0} , Ω4 = {(y1 , y2 )| y1 − x1 (t) < 0, y2 − x2 (t) > 0} .

(15)

The corresponding boundaries are defined as

∂ Ω12 = {(y1 , y2 )| y2 − x2 (t) = 0, y1 − x1 (t) > 0} , ∂ Ω23 = {(y1 , y2 )| y1 − x1 (t) = 0, y2 − x2 (t) < 0} , ∂ Ω34 = {(y1 , y2 )| y2 − x2 (t) = 0, y1 − x1 (t) < 0} , ∂ Ω14 = {(y1 , y2 )| y1 − x1 (t) = 0, y2 − x2 (t) > 0} .

(16)

From the above definition, the velocity and displacement boundaries are shown in Fig. 1a,b, respectively. The dashed curves are the boundaries. The intersection point of the two boundaries is labeled by a filled circular symbol. From (11)–(14), the controlled slave system is in a vector form of y˙ (α ) = F(α ) (y(α ) ,t) where (α )

(α )

F(α ) = (F1 , F2 )T ; (α )

(α )

(α )

(α )

F1 (y(α ) ,t) = y2 − k1 for α = 1, 2; F1 (y(α ) ,t) = y2 + k1 for α = 3, 4;

(17)

Chaotic Synchronization of Duffing Oscillator and Pendulum

119

Fig. 1 Two boundaries in the absolute coordinates: (a) velocity and (b) displacement

(α )

(α )

(α )

(α )

F2 (y(α ) ,t) = −a0 sin y1 + Q0 cos Ωt − k2 for α = 1, 4; F2 (y(α ) ,t) = −a0 sin y1 + Q0 cos Ωt + k2 for α = 2, 3.

(18)

The corresponding dynamical systems on the boundaries are y˙ (αβ ) = F(αβ ) (y(αβ ) , x(t),t); x˙ = F (x,t)

(19)

where (αβ )

F1

(αβ )

(y(αβ ) ,t) = y2 (t) = x2 (t) and F2

(y(αβ ) ,t) = x˙2 (t)

(20)

120

A.C.J. Luo and F. Min

with (αβ )

= x1 (t) and y2

(αβ )

= x1 (t) + C and y2

y1 y1

(αβ )

= x2 (t) on ∂ Ωαβ for (α , β )= (2, 3), (1, 4) (αβ )

= x2 (t) on ∂ Ωαβ for (α , β )= (1, 2), (3, 4)

(21)

From dynamical systems in the absolute coordinate, it is difficult to develop the analytical conditions because the boundary changes with time. Thus, the relative coordinates are introduced z1 = y1 − x1 and z˙1 ≡ z2 = y2 − x2

(22)

The domains and boundaries in the relative coordinates become Ω1 = {(z1 , z2 )| z1 > 0, z2 > 0} , Ω2 = {(z1 , z2 )| z1 > 0, z2 < 0} , Ω3 = {(z1 , z2 )| z1 < 0, z2 < 0} , Ω4 = {(z1 , z2 )| z1 < 0, z2 > 0} .

(23)

∂ Ω12 = {(z1 , z2 )| z2 = 0, z1 > 0} , ∂ Ω23 = {(z1 , z2 )| z1 = 0, z2 < 0} , ∂ Ω34 = {(z1 , z2 )| z2 = 0, z1 < 0} , ∂ Ω14 = {(z1 , z2 )| z1 = 0, z2 > 0} .

(24)

The velocity and displacement boundaries in the relative coordinates are constant. Based on such boundaries, the analytical conditions can be easily developed, and the velocity and displacement boundaries are sketched in Fig. 2. The controlled slave system in the relative coordinates becomes z˙ (α ) = g(α ) (z(α ) , x,t) with x˙ = F (x,t)

(25)

where the controlled pendulum as a slave system gives (α )

(α )

(α )

(α )

g1 (z(α ) , x,t) = z2 − k1 for α = 1, 2 g1 (z(α ) , x,t) = z2 + k1 for α = 3, 4 (α )

g2 (z(α ) , x,t) = G (z(α ) , x,t) − k2 for α = 1, 4 (α )

g2 (z(α ) , x,t) = G (z(α ) , x,t) + k2 for α = 2, 3

(26)

Chaotic Synchronization of Duffing Oscillator and Pendulum

121

Fig. 2 Two boundaries in the relative coordinates: (a) velocity and (b) displacement

with G (z(α ) , x,t)

  (α ) = −a0 sin z1 + x1 + Q0 cos Ωt + d1 x2 − a1 x1 + a2 x31 − A0 cos ω t. (27)

The motion on the boundary in the relative coordinates is also determined by z˙ (αβ ) = g(αβ ) (z(αβ ) , x,t); with x˙ = F (x,t)

(28)

where (αβ )

g1

(αβ )

(z(αβ ) , x,t) = z2 = 0 and g2

(z(αβ ) ,t) = 0

(29)

122

A.C.J. Luo and F. Min

with (αβ )

= 0 and z2

(αβ )

= C and z2

z1 z1

(αβ )

= 0 on ∂ Ωαβ for (α , β )= (2, 3), (1, 4);

(αβ )

= 0 on ∂ Ωαβ for (α , β )= (1, 2), (3, 4).

(30)

From Luo [14], the synchronization state of the controlled slave system with the master system requires a sliding flow on the boundary (or stick motion in physics) as in Luo [14, 15]. Similarly, the nonsynchronization state (penetration state) at the boundary is a passable flow. The desynchronization requires a source flow state to the boundary. From Luo [14–17] the necessary and sufficient conditions for the synchronization, penetration, and desynchronization of the controlled slave system with the master system were discussed in Luo and Min [1]. The G-functions are introduced in the relative coordinates for α = i, j; α = i, j, (i, j) ∈ {(1, 2), (2, 3), (3, 4), (1, 4)} (α )

G∂ Ω (zm , x,tm± ) = nT∂ Ωi j · [g(α ) (zm , x,tm± ) − g(i j) (zm , x,tm± )] ij

(1,α )

G∂ Ω (zm , x,tm± ) = nT∂ Ωi j · [Dg(α ) (zm , x,tm± ) − Dg(i j) (zm , x,tm± )] ij

(31) (32)

From (24), the normal vectors of the relative boundaries are: n∂ Ω12 = n∂ Ω34 = (0, 1)T and n∂ Ω23 = n∂ Ω14 = (1, 0)T .

(33)

From (25)–(30), the corresponding G-functions in (31) and (32) for a flow at the boundary are (α )

(α )

(α )

G∂ Ω (zm , x,tm± ) = G∂ Ω (zm , x,tm± ) = g2 (zm , x,tm± ), 12

34

(α )

(α )

(α )

G∂ Ω (zm , x,tm± ) = G∂ Ω (zm , x,tm± ) = g1 (zm , x,tm± ); 23

14

(1,α )

(1,α )

12

34

(1,α )

(1,α )

23

14

(34)

(α )

G∂ Ω (zm , x,tm± ) = G∂ Ω (zm , x,tm± ) = Dg2 (zm , x,tm± ), (α )

G∂ Ω (zm , x,tm± ) = G∂ Ω (zm , x,tm± ) = Dg1 (zm , x,tm± );

(35)

where the total derivative functions are given by (α )

(α )

Dg1 (z(α ) , x,t) = g2 (z(α ) , x,t) for α = 1, 2, 3, 4; (α )

Dg2 (z(α ) , x,t) = DG (z(α ) , x,t)     (α ) (α ) = −a0 z2 + x2 cos z1 + x1 − Q0 Ω sin Ωt + d1 F2 (x,t) − a1 x2 + 3a2x21 x2 + ω A0 sin ω t, for α = 1, 2, 3, 4.

(36)

Chaotic Synchronization of Duffing Oscillator and Pendulum

123

The G-functions of a flow in domains with respect to the boundary are defined herein in order to illustrate the flow switchability, i.e., (α )

(α )

(α )

(α )

(α )

(α )

G∂ Ω12 (z(α ) , x,t) = G∂ Ω34 (z(α ) , x,t) = g2 (z(α ) , x,t), G∂ Ω (z(α ) , x,t) = G∂ Ω (z(α ) , x,t) = g1 (z(α ) , x,t). 23

14

(37)

3 Analytical Conditions As in Luo and Min [18], the synchronization of the controlled slave system with the master systems occurs at the intersection of the two separation boundaries (zm = 0), and the corresponding synchronization conditions are ⎫ (1) (1) G∂ Ω (zm , x,tm− ) = g1 (zm , x,tm− ) < 0, ⎬ 14

(1) = g2 (zm , x,tm− ) < 0 ⎭ ⎫ (2) (zm , x,tm− ) = g2 (zm , x,tm− ) > 0, ⎬

(1) G∂ Ω (zm , x,tm− ) 12 (2)

G∂ Ω

12

(2) (2) G∂ Ω (zm , x,tm− ) = g1 (zm , x,tm− ) < 0 ⎭ 23 ⎫ (3) (3) G (zm , x,tm− ) = g1 (zm , x,tm− ) > 0, ⎬

∂ Ω23

(3) (3) G∂ Ω (zm , x,tm− ) = g2 (zm , x,tm− ) > 0 ⎭ 34 ⎫ (4) (4) G (zm , x,tm− ) = g2 (zm , x,tm− ) < 0, ⎬

∂ Ω34

(4) (4) G∂ Ω (zm , x,tm− ) = g1 (zm , x,tm− ) > 0 ⎭

for zm ∈ ∂ Ω12 ∩ ∂ Ω14 on Ω1 ;

for zm ∈ ∂ Ω12 ∩ ∂ Ω23 on Ω2 ;

for zm ∈ ∂ Ω23 ∩ ∂ Ω34 on Ω3 ;

for zm ∈ ∂ Ω34 ∩ ∂ Ω14 on Ω4 . (38)

14

From (26), four basic functions are introduced as (α )

(α )

(α )

(α )

g1 (z(α ) , x,t) ≡ g1 (z(α ) , x,t) = z2 − k1 in Ωα for α = 1, 2; g2 (z(α ) , x,t) ≡ g1 (z(α ) , x,t) = z2 + k1 in Ωα for α = 3, 4; (α )

g3 (z(α ) , x,t) ≡ g2 (z(α ) , x,t) = G (z(α ) , x,t) − k2 in Ωα for α = 1, 4; (α )

g4 (z(α ) , x,t) ≡ g2 (z(α ) , x,t) = G (z(α ) , x,t) + k2 in Ωα for α = 2, 3.

(39)

where G (z(α ) , x,t)

  (α ) = −a0 sin z1 + x1 + Q0 cos Ωt + d1 x2 − a1 x1 + a2 x31 − A0 cos ω t. (40)

124

A.C.J. Luo and F. Min

The synchronization conditions in (38) become g1 (zm , x,tm− ) = z2m − k1 < 0, g2 (zm , x,tm− ) = z2m + k1 > 0, g3 (zm , x,tm− ) = G (zm , x,tm− ) − k2 < 0, g4 (zm , x,tm− ) = G (zm , x,tm− ) + k2 > 0.

(41)

For zm = 0, the synchronization conditions of the controlled slave system are g1 (zm , x,tm− ) ≡ −k1 < 0, g2 (zm , x,tm− ) ≡ +k1 > 0, g3 (zm , x,tm− ) = G (x,tm− ) − k2 < 0, g4 (zm , x,tm− ) = G (x,tm− ) + k2 > 0.

(42)

G (x,t) = −a0 sin x1 + Q0 cos Ωt + d1 x2 − a1 x1 + a2 x31 − A0 cos ω t.

(43)

where

For k1 > 0 and k2 > 0, the first two equations of (42) are satisfied, and the third and fourth equations give the synchronization invariant set, i.e., − k2 < G (x,tm− ) < k2 .

(44)

In a small neighborhood of zm = 0, the attractivity conditions for |z − zm | < ε are 0 ≤ z2 < k1 and G (z, x,t) < k2 for z1 ∈ [0,∞) in Ω1 , 0 ≤ z2 < k1 and − k2 < G (z, x,t) for z1 ∈ [0,∞) in Ω2 , − k1 < z2 ≤ 0 and − k2 < G (z, x,t) for z1 ∈ ( − ∞, 0] in Ω3 , − k1 < z2 ≤ 0 and G (z, x,t) < k2 for z1 ∈ ( − ∞, 0] in Ω4 .

(45)

From the foregoing equation, z∗1 and z∗2 are computed and the initial condition for the controlled slave system is computed by y1 = z∗1 + x1 and y2 = z∗2 + x2 .

(46)

Chaotic Synchronization of Duffing Oscillator and Pendulum

125

The conditions of synchronization vanishing for the controlled slave system with (α ) zm = zm are   (α ) (α ) g1 zm , x,tm∓ = z2m − k1 = 0,     (α ) (α ) Dg1 zm , x,tm∓ = G zm , x,tm∓ > 0,   (β ) (β ) g2 zm , x,tm− = z2m + k1 > 0;

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

for (α , β ) = {(1, 4), (2, 3)}

(47)

for (α , β ) = {(1, 4), (2, 3)}

(48)

from zm+ε = y1 − x1 > 0, and   (α ) (α ) g1 zm , x,tm− = z2m − k1 < 0;

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

  (β ) (β ) g2 zm , x,tm∓ = z2m + k1 = 0,    (β ) (β ) Dg2 zm , x,tm∓ = G zm , x,tm∓

⎪ ⎪ ⎪ ⎪  ⎪ ⎪ < 0; ⎭

from zm+ε = y1 − x1 < 0. The conditions of synchronization vanishing for the controlled slave systems (α ) with zm = zm are     ⎫ (α ) (α ) g3 zm , x,tm∓ = G zm , x,tm∓ − k2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬     (α ) (α ) Dg3 zm , x,tm∓ = DG zm , x,tm∓ > 0; for (α , β ) = {(1, 2), (4, 3)} ⎪ ⎪ ⎪ ⎪     ⎪ ⎪ (β ) (β ) = G z , x,t + k > 0; ⎭ g z , x,t m

4

m−

m

m−

2

(49) from z˙m+ε = y2 − x2 > 0, and     ⎫ (α ) (α ) g3 zm , x,tm− = G zm , x,tm− − k2 < 0; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬     (β ) (β ) g4 zm , x,tm∓ = G zm , x,tm∓ + k2 = 0, for (α , β ) = {(1, 2), (4, 3)} ⎪ ⎪ ⎪ ⎪     ⎪ ⎪ (β ) (β ) g z , x,t = DG z , x,t < 0; ⎭ 4

m

m∓

m

m∓

(50) from z˙m+ε = y2 − x2 < 0.

126

A.C.J. Luo and F. Min

The conditions of synchronization onset for the controlled slave systems with (α ) zm = zm are   (α ) (α ) g1 zm , x,tm± = z2m − k1 = 0,     (α ) (α ) Dg1 zm , x,tm± = G zm , x,tm± > 0,   (β ) (β ) g2 zm , x,tm− = z2m + k1 > 0;

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

for (α , β ) = {(1, 4), (2, 3)}

(51)

for (α , β ) = {(1, 4), (2, 3)}

(52)

from zm−ε = y1 − x1 > 0, and   (α ) (α ) g1 zm , x,tm− = z2m − k1 < 0;

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

  (β ) (β ) g2 zm , x,tm± = z2m + k1 = 0,    (β ) (β ) Dg2 zm , x,tm± = G zm , x,tm±

⎪ ⎪ ⎪ ⎪  ⎪ ⎪ < 0; ⎭

from zm+ε = y1 − x1 < 0. The conditions of synchronization onset for the controlled slave systems with (α ) zm = zm are     ⎫ (α ) (α ) g3 zm , x,tm± = G zm , x,tm± − k2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬     (α ) (α ) Dg3 zm , x,tm± = DG zm , x,tm± > 0; for (α , β ) = {(1, 2), (4, 3)} ⎪ ⎪ ⎪ ⎪     ⎪ ⎪ (β ) (β ) = G z , x,t + k > 0; ⎭ g z , x,t m

4

m−

m

m−

2

(53) from z˙m−ε = y2 − x2 > 0, and     ⎫ (α ) (α ) g3 zm , x,tm− = G zm , x,tm− − k2 < 0; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬     (β ) (β ) g4 zm , x,tm± = G zm , x,tm± + k2 = 0, for (α , β ) = {(1, 2), (4, 3)} ⎪ ⎪ ⎪ ⎪     ⎪ ⎪ (β ) (β ) g z , x,t = DG z , x,t < 0; ⎭ 4

m

m±

m

m±

(54) from z˙m−ε = y2 − x2 < 0.

Chaotic Synchronization of Duffing Oscillator and Pendulum

127

4 Illustrations Consider a chaotic motion in the Duffing oscillator with a set of parameters and initial conditions as an example. Duffing : a1 = a2 = 1.0, d1 = 0.25, A0 = 0.454, ω = 1.0; Pendulum : a0 = 1.0, Q0 = 0.275, Ω = 2.18517; x1 = y1 ≈ −0.4180597 and x2 = y2 ≈ 0.2394332,t0= 0.

(55)

From (55), the synchronization invariant domain for the control parameters of k1 = 1 and k2 = 0.9, as shown in Fig. 3. The synchronization invariant domain 50 Master

Synchronization domain y1=x1 and y2=x2

Velocity, x2

25

∞ 0

−∞ -25

Non-synchronzation area

-50 -4.0

-2.0

0.0

2.0

4.0

Displacement,x1

15.0 Non-synchronization domain

Master

Velocity, x2

7.5

0.0

Synchronization domain

-7.5

-15.0 -3.0

-1.5

0.0

1.5

3.0

Displacement,x1

Fig. 3 Synchronization invariant sets of the master system for the controlled slave system to synchronize (Control parameters: k1 = 1 and k2 = 0.9. Duffing: a1 = a2 = 1.0, d1 = 0.25, A0 = 0.265, ω = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, Ω = 2.18519)

128

A.C.J. Luo and F. Min

a

2.0

Velocity, x2 and y2

S

N

S

S

N

1.0

0.0

y2 x2

−1.0

−2.0 0.0

2.0

4.0

6.0

8.0

10.0

Time, t

b

3.0

G-Functions, gi

1.5

S

N

S

N

S

6.0

8.0

g4 g2

0.0 g3 g1

−1.5

−3.0 0.0

2.0

4.0

10.0

Time t

Fig. 4 Partial and chaotic synchronization of the Duffing oscillator and controlled pendulum: (a) velocity responses, (b) G-functions, (c) phase plane, and (d) switching point section (Control parameters: k1 = 1 and k2 = 0.8. Duffing: a1 = a2 = 1.0, d1 = 0.25, A0 = 0.265, ω = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, Ω = 2.18519) (Initial condition: x1 = y1 ≈ −0.4180597 and x2 = y2 ≈ 0.2394332) (S: Synchronization; N: Nonsynchronization). Hollow and filled circular symbols are synchronization appearance and vanishing, respectively

is shaded, and the nonsynchronization is not shaded. For the controlled pendulum synchronizing with the Duffing oscillator, the trajectory of a chaotic motion in the Duffing oscillator should be in the invariant domain. In the outside of the invariant domain, the controlled pendulum cannot synchronize with the Duffing oscillator. In Fig. 3a, the overview of the synchronization invariant domain is presented, and a zoomed view of the synchronization invariant domain is presented in Fig. 3b. The synchronized portions of the trajectories of motions in Duffing oscillator should lie in the synchronization invariant domain.

Chaotic Synchronization of Duffing Oscillator and Pendulum

c

1.2

129

PS

Velocity, x 2 and y2

0.6

I.C.

0.0

-0.6 Synchronization Invaraint Domain -1.2 -1.5

0.0

1.5

Displacement, x1 and y1

d

1.2

Switching Velocity, x2 and y2

PS 0.6

0.0

-0.6 Synchronization Invaraint Domain -1.2 -2.0

-1.0

0.0

1.0

2.0

Switching Displacement, x1 and y1

Fig. 4 (continued)

The partial synchronization of the controlled pendulum with a chaotic motion in the Duffing oscillator is illustrated in Fig. 4 with control parameter (k1 = 1 and k2 = 0.8). In Fig. 4a, the time-velocity history of velocity for a periodic motion of the Duffing oscillator is presented via the solid curve, but the time-history of velocity for the controlled pendulum is depicted by the dashed curve. The corresponding Gfunctions for the controlled pendulum are shown in Fig. 4b. The shaded portions are for synchronization, and the non-shaded regions are for nonsynchronization. The G-functions for the nonsynchronization are presented via the dashed curve. If the G-function of g4 is a dashed curve, a flow of the controlled pendulum is in domain Ωα (α = 1, 4). If the G-function of g3 is a dashed curve, a flow of the controlled

130

A.C.J. Luo and F. Min

a

1.6 Duffing Pendulum

FS

Velocity, x2 and y2

0.8

0.0

-0.8

-1.6 0.0

2.0

4.0

6.0

8.0

10.0

Time, t

b

5.0 FS

G-Functions, gi

2.5

g4 g2

0.0

-2.5

-5.0 0.0

g1 g3

2.0

4.0

6.0

8.0

10.0

Time, t

Fig. 5 Partial and chaotic synchronization of the Duffing oscillator and controlled pendulum: (a) time-history of velocity, (b) time-histories of G-functions, (c) trajectories in phase plane for full synchronization, and (d) Poincar´e mapping section in the invariant domain (Control parameters: k1 = 1 and k2 = 2, Duffing: a1 = a2 = 1.0, d1 = 0.25, A0 = 0.265, ω = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, Ω = 2.18519) (Initial condition: x1 = y1 ≈ −0.4180597 and x2 = y2 ≈ 0.2394332) (FS: Full synchronization)

system is in domain Ωα (α = 2, 3). For the two cases, the controlled pendulum cannot synchronize with chaos in the Duffing oscillator. In Fig. 4c, the trajectory of chaos in phase plane for the Duffing oscillator is depicted by solid curves, but the trajectory of the controlled pendulum is represented by dashed curves. The shaded area is the synchronization invariant domain. In Fig. 4d, the switching point section for the synchronization appearance and disappearance of the controlled pendulum is

Chaotic Synchronization of Duffing Oscillator and Pendulum

131

c 1.2

Duffing Pendulum

FS

Velocity, x2 and y2

0.6

0.0

-0.6 Synchronization Invariant Domain -1.2 -2.0

d

-1.0

0.0 1.0 Displacement, x1 and y1

2.0

1.2

Velocity, x2 and y2

0.6 FS 0.0

-0.6 Synchronization Invariant Domain -1.2 -2.0

-1.0

0.0

1.0

2.0

Displacement, x1 and y1

Fig. 5 (continued)

presented, and the synchronization invariant domain is superimposed. It is observed that such switching points lie in the invariant domain. The same initial conditions in (55) plus the control parameter (k1 = 1 and k2 = 2) are used for a full, chaotic synchronization, as shown in Fig. 5. The time-histories for velocity and the corresponding G-functions are presented in Fig. 5a,b, respectively. The velocity responses of the Duffing oscillator and controlled pendulum are identical. The G-function responses shows that the controlled pendulum should synchronize with the chaotic motion in the Duffing oscillator because of (g1 < 0 and g2 > 0) for displacement and (g3 < 0 and g4 > 0) for velocity. The trajectories of the controlled pendulum and Duffing oscillator are identical, as shown in Fig. 5c.

132

A.C.J. Luo and F. Min

The solid curves give the responses of the Duffing oscillator, and the hollow circular symbols represent the responses of the controlled pendulum. In Fig. 5d, the Poincar´e mapping section for chaotic motions for 10,000 periods of the Duffing oscillator is presented. The Poincar´e mapping section is fully synchronized for the controlled pendulum in the invariant domain.

5 Conclusion In this chapter, the partial and full synchronizations of the controlled pendulum with chaotic motions in the Duffing oscillator were discussed. Compared with the periodic synchronization, the switching points for chaotic synchronization appearance and vanishing of the partial synchronization are chaotic. The partial and full, chaotic synchronizations for the Duffing oscillator and controlled pendulum are illustrated to show the analytical conditions.

References 1. Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64(8): 821–824 2. Carroll TL Pecaora LM (1991) Synchronized chaotic circuit. IEEE Trans Circ Syst 38(4): 453–456 3. Pyragas K (1992) Continuous control of chaos by self-controlling feedback. Phys Lett A 170:421–428 4. Rulkov NF, Sushchik M, Tsimring LS, Abarbanel HD (1995) Generalized synchronization of chaos in directionally coupled chaotic systems. Phys Rev E 50:1642–1644 5. Kocarev L, Parlitz U (1995) General approach for chaotic synchronization with application to communication. Phys Rev Lett 74:1642–1644 6. Pyragas K (1996) Weak and strong synchronization of chaos. Phys Rev E 54:R4508–R4511 7. Boccaletti S, Farini A, Arecchi FT (1997) Adaptive synchronization of chaos for secure communication. Phys Rev E 55:4979–4981 8. Abarbanel HDI, Korzinov L, Mees AI, Rulkov NF (1997) Small force control of nonlinear systems to given orbits. IEEE Trans Circ Syst I Fundam theory Appl 44:1018–1023 9. Pyragas K (1998), Properties of generalized synchronization of chaos. Nonlinear Anal Model Control Vilnius IMI 3:1–28 10. Yang T, Chua LO (1999) Generalized synchronization of chaos via linear transformations. Int J Bifurcat Chaos 9:215–219 11. Boccaletti S, Kurhts J, Osipov G, Valladars DL, Zhou CS (2002) The synchronization of chaotic systems. Phys Rep 366:1–101 12. Campos E, Urias J (2004) Multimodal synchronization of chaos. Chaos 14:48–53 13. Luo ACJ (2009) A theory for synchronization of dynamical systems. Commun Nonlinear Sci Numer Simul 14:1901–1951 14. Luo ACJ (2008) A theory for flow switchability in discontinuous dynamical systems. Nonlinear Anal Hybrid Syst 2(4):1030–1061 15. Luo ACJ (2008) Global transversality, resonance and chaotic dynamics. World Scientific, New Jersey

Chaotic Synchronization of Duffing Oscillator and Pendulum

133

16. Luo ACJ (2009) Discontinuous dynamical systems on time-varying domains. HEP-Springer, Dordrecht 17. Luo ACJ, Min FH (2011) Synchronization of a periodically forced, damped duffing oscillator with a periodically excited pendulum. Nonlinear Anal Real World Appl 12:1810–1827 18. Luo ACJ, Min FH (2010) On the synchronization of a chaotic pendulum with a periodic motion of a periodically forced, damped Duffing oscillator. In: ASME international mechanical engineering congress and exposition, Vancouver, British Columbia, 12–18 Nov 2010, IMECE2010-39182

The Ring Problem of (N + 1) Bodies: An Overview Tilemahos J. Kalvouridis

Abstract The study of N-body systems and their simulation with various models always excited the scientific interest. Here we present an N-body model that has been under investigation in the last 10 years and is called the ring problem of (N+ 1) bodies, or otherwise the regular polygon problem of (N+ 1) bodies. In what follows, we give an overview of the scientific work that has been done through all these years, as well as the major results obtained so far.

1 Introduction The simulation of N-body systems with various models always was at the front line of the research and constituted one of the most attractive issues that excited the scientific interest. Here we deal with an N-body model that has been investigated during the last 10 years. It is called the ring problem of (N + 1) bodies, or otherwise the regular polygon problem of (N + 1) bodies. The problem studies the motion of a small body S in the force field created by N homogeneous, spherical, major bodies called the primaries. The ν = N − 1 of them have equal masses and are located at the vertices of an imaginary ν -gon. The Nth body has a different mass and is located at the center of mass of this formation (Fig. 1). The problem is characterized by two parameters: the number ν of the peripheral primaries and the mass parameter β which is the ratio of the central mass m0 to a peripheral one named m. A similar geometric formation of the primaries was proposed by Maxwell in 1865 in an essay which won the Adams Prize. Since then, this configuration has often been found to be the center of special scientific interest and many papers have been written aiming to prove its central character and to find homographic solutions, relative equilibria, T.J. Kalvouridis () Department of Mechanics, National Technical University of Athens, School of Applied Mathematics and Physical Sciences, 5, Heroes of Politechnion Ave., Zografou Campus, 157 73, Athens, Greece e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 6, © Springer Science+Business Media, LLC 2012

135

136

T.J. Kalvouridis

Fig. 1 The configuration of the ring problem of (N + 1) bodies

conditions for stability, etc., not only for the simple gravitational case but also for cases that include post-Newtonian potentials. However, the ring problem as a new dynamical system, under the description made at the beginning of this paragraph, first appeared in Scheeres’ Ph.D. [36] and a little later in a joint paper with Vinh [37]. The problem was considered anew by the author of the present article, who devoted more than 10 years (1999–2010) of scientific research in investigating many aspects of it. He reformulated the whole problem and he improved the original model by considering not only gravitational forces but also forces coming either from radiation or from post-Newtonian potentials. He also examined some cases where the small body is a triaxial body or a gyrostat. An extended bibliography is exposed at the end of the paper. The main advantage of the problem is its geometric simplicity and the fact that by changing the basic parameters, one can obtain various already known problems and celestial models, like the Copenhagen case of the restricted three-body problem, as well as the restricted five-body model proposed by Oll¨ongren [30] and the restricted four-body problem proposed by Maranhao and Llibre [26].

2 The Gravitational Version 2.1 Equations of Motion By using a synodic coordinate system Oxyz (Fig. 1) the three-dimensional motion of the small body is described by the following dimensionless second-order differential equations [13]

The Ring Problem of (N + 1) Bodies: An Overview

x¨ − 2y˙ =

∂U ∂U ∂U , y¨ + 2x˙ = , z¨ = ∂x ∂y ∂z

137

(1)

where U is the potential function,   ν  1 β 1 2 1 2 U (x, y, z) = x +y + +∑ 2 Δ r0 i=1 ri   ν 1 + β , ϕ = π /ν Δ = 2 sin3 ϕ ∑ sin(i − 1)ϕ i=2

(2)

and r0 , ri are the distances of body S from the primaries. The above equations admit a Jacobian-type integral of motion   C = 2U − x˙2 + y˙2 + z˙2 .

(3)

The gravitational version of the problem is characterized by two parameters: the number ν of the peripheral primaries and the mass parameter β which is the ratio of the central mass m0 to a peripheral one named m. There are ν axes of symmetry that form angles 2π /ν between them. When ν is odd, all the axes of symmetry are equivalent. If ν is even, two groups are formed; each one of them consists of ν /2 equivalent axes.

2.2 Regions of Motion in Two and Three Dimensions Relation [3] is very helpful in determining the regions where planar (z = z˙ = 0) or three-dimensional motions are permitted. Regarding the zero-velocity surfaces C = C(x, y) for the planar motion, the third axis measures the values of the Jacobian constant C and the particle is free to move inside the funnels and beneath the surface (Fig. 2a,b). As β increases, the central funnel enlarges and the closed area around P0 on the xy-plane enlarges as well (Fig. 2).

2.3 Equilibrium Positions The equilibrium positions are distributed on imaginary circles that are called equilibrium zones and are concentric with the imaginary circle of the primaries [13]. The equilibrium positions on each zone are characterized by the same value of the Jacobian constant C and their number equals the number of the peripheral

138

T.J. Kalvouridis

Fig. 2 (a), (b) Networks of zero-velocity curves for ν = 7, β = 2 and ν = 7, β = 10, respectively. (c), (d) zero-velocity surfaces for the same values

primaries. In the general case, these zones are symbolized with A1 , A2 , B, C2 , C1 , as they appear from the center outward. The number of the zones (5 or 3) for a particular number ν of peripheral primaries depends on a critical value of the mass parameter β = lν . This critical value increases with ν . As β increases beyond lν , the three zones A1 , C2 , and C1 approach asymptotically the imaginary circle of the primaries. There are no equilibrium positions outside the plane xy of the primaries. All the equilibria are unstable for any ν and β .

The Ring Problem of (N + 1) Bodies: An Overview

139

Fig. 3 Families of simple periodic orbits for ν = 10, β = 2

2.4 Planar Periodic Motions Regarding the planar motions and their stability, we note that there are families that evolve inside the funnels in the (x0 ,C) diagrams and family orbits outside them (Fig. 3). The orbits of the former families belong to satellite or planetary classes of motions, while the orbits of the latter families are either interplanetary motions or orbits around an equilibrium point. In most families, except for those that evolve inside the funnels, the orbits enlarge as C increases and so do their periods. When C decreases, the orbits shrink and the particle comes very close to one or more primaries. The majority of the orbits are unstable and their stability parameter takes very large values. However, stable orbits exist and we can find them among the members of the families that evolve inside the funnels of the (x0 ,C) diagram (for more details see Kalvouridis [14–16, 19, 21], Croustalloudi and Kalvouridis [21]. Regarding the influence of the mass parameter, as β increases, then, the characteristic curves are “pushed” toward the imaginary circle of the primaries (Fig. 5), the periods of the orbits of the same Jacobian constant increase, while the absolute values of the velocities at t = T (T is the half period) decrease, the part of the orbits described outside the imaginary ring of the peripheral primaries shrinks while the one described inside the ring extends [33]. Symmetric periodic orbits of various types are given in Fig. 4.

2.5 Zero-Velocity Surfaces for Three-Dimensional Motions Their evolution depends on the order of values of the Jacobian constants of the existing equilibrium zones [17]. Figure 6 shows the evolution of the zero-velocity

140

T.J. Kalvouridis

a

b

Family T16C 4

0.3

Family S8 r

y

y C=9.298

2

0 -1.5

-3

0

1.5

L 5A1

x

P1 x

P0

-1

3

-0.8

C=9.309

C=9.32

-2

C= -1.550000187 x0=-1.381585937

-4

-0.3

c

d

Family S32D

0 -0.4

-0.6

1.5

Family S8y

y

C=9.399

y

6

P9 4

0.5

2

P17 -6

P0

0 -4

-2

0

P1 2

4

x

P0

P5

6

-1.5

-0.5

P1 0.5

x 1.5

-2

-0.5 -4

-6

C= 2.499691417 x0=-2.404035889

C=8.966 -1.5

Fig. 4 (a) Triple periodic orbit for ν = 16, β = 2, (b) simple periodic orbits around the unstable equilibrium L5 A1 for ν = 8 and β = 2, (c) double symmetric simple periodic orbit for ν = 16 and β = 2, (d) multiple symmetric orbits for ν = 8 and β = 2

surfaces for a configuration with ν = 10, β = 2. For these values we obtain the inequality CB > CC1 > CC2 > CA1 > CA2 . When C > max(CA1 , CA2 , CB , CC2 , CC1 ), for each value of the Jacobian constant C, two kinds of iso-energetic surfaces coexist: a group of ν small, closed surfaces, which surround each individual primary and a much larger surface that surrounds all these closed surfaces (Fig. 6a). The particle is either trapped inside a small inner shell or is free to move outside the large external surface. As C decreases, the internal closed surfaces touch each other (Fig. 6b) and later form a torus-like closed surface (Fig. 6c). As C decreases even more, the torus-like surface touches the external surface (Fig. 6d). For smaller values of C the surface splits in two parts that shrink as C becomes smaller and smaller, until they finally disappear. The points of zones A1 and C1 are the contact points of the internal closed surfaces around the peripheral primaries Pi and the surface that surrounds the central primary P0 , or the external shell, respectively. The points of zone B are the contact points between the ν internal closed surfaces around the peripheral primaries. At the points of

The Ring Problem of (N + 1) Bodies: An Overview

a

S16N S16M S16Q S16K S16E

141

S16U 1.9

S16R

0.9

x0 -0.1 -2.6

-2.2

-1.8

-1.4

-1

-0.6

-0.2 -1.1

-2.1

b 3

S16K

S16U

2 1

x0 -2.6

0 -2.2

-1.8

-1.4

-1

-0.6

-0.2 -1 -2

S16 S16M S16N S16Q R

-3

Fig. 5 Families of simple periodic orbits for ν = 16. (a) β = 20, (b) β = 100

zones A2 and C2 the internal surfaces or the external shell are detached from plane xy, respectively. In the permitted regions of the zero-velocity surfaces, the particle realizes its motions. Figure 7a shows the characteristic curves of three-dimensional periodic motions. These curves start from points of vertical critical stability on the planar families and terminate on different points of the same or different planar families [11, 12]. Figure 7b shows some three-dimensional orbits of the family 7C1 (ν = 7, β = 2).

3 The Effect of Radiation Pressure In this version we assume that one or more primaries are radiation sources and we apply the simplified theory suggested by Radzievski in order to study the effect of radiation pressure on the motion of the particle. The existence of strong radiation sources in the universe has repeatedly been confirmed from a very early period and the photo-gravitational problems of two or more bodies have attracted much attention during the last decades. The potential function in (1) takes the form [18, 23, 25]

142

T.J. Kalvouridis

Fig. 6 Evolution of zero-velocity surfaces for ν = 10, β = 2. (a) C = 16 (C > CB ), (b) C = CB = 15.29176, (c) C = 14 (CB > C > CC1 ), (d) C = CC1 = 13.98642

Fig. 7 Three-dimensional simple periodic orbits for ν = 7, β = 2; (a) evolution of the families, (b) orbits of the family 7C1

The Ring Problem of (N + 1) Bodies: An Overview

143

Diagram (x0, C) for 5 cases (ν=7,β=2)

C 13

9 1 2 4

5

5

3

1 P0 -3.4

-2.4

-1.4

-0.4

1. Gravitational 4. b0b1b2b3b6b7=0.5 2. b0=0.5 5. All=0.5 3. b0b1b2b7=0.5

x0

P1

0.6

1.6

2.6

-3

Fig. 8 Zero-velocity curves of the diagram (x, C) for ν = 7, β = 2 and five cases of radiating primaries

U (x, y, z) =

1 2

 2  x + y2 + Δ1

qi = 1 − bi, bi =

Fri Fgi



β q0 r0

ν

+∑

i=1

qi ri

(4)

, i = 0, 1, 2, .., ν

Therefore, the problem, besides the two parameters ν and β , is determined by N additional parameters, namely, the N radiation coefficients b0 , b1 , b2 , .., bν of the N = ν + 1 primaries. The symmetry of the created combined field is preserved in three cases: (1) when the central body is a radiation source (whatever the value of the radiation coefficient b0 is), (2) all peripheral bodies are radiation sources and have the same radiation coefficients, and (3) all the primaries are radiation sources and the peripheral bodies have the same radiation coefficients. In all these cases the equilibrium points are still arranged on concentric circular zones. Otherwise, the equilibria that belong to various sets are generally reduced as the total radiation increases. However, their stability does not change. For all the cases we have studied, the value for the Jacobian constant C of a particular equilibrium point reduces, as new radiation sources are added to the system. The consequences of the radiation action may be significant when the bodies are strong radiation emitters (bi , b0 > 0.1). We have noted that, as radiation increases, the zero-velocity boundary curves of the (x, C) diagram shift to lower values of C (Fig. 8), the funnels formed between them and the asymptotes through the primaries become narrower, and the characteristic curves of the families of periodic orbits are either translated toward or away from the origin, or incline with relation to the x-axis (Fig. 9a,b). Figure 9c,d show the evolution of period (2T ) with the Jacobian constant C for the families SX and SY and various cases when ν = 7, β = 2.

144

T.J. Kalvouridis

a

b

SA

SJ C

C

1

1

x0

2

-3

1

3

5

2 0

-2.5

-2

3

-1.5

3

-1

1 Gravitational 2 b0 3 All primaries

-2

4

-1.3

SY 1 Gravitational 2 b0 3 All primaries

3 3

3

3

-1

-3

2T

2T

1

2

2 2

1

2

-2.5

-1.7

d

SX 1 Gravitational 2 b0 3 All primaries

-2.1

1 Gravitational 2 b0 3 All primaries

-3

c

1

x0 -2.5

1 C

C

1 -0.5

1.5

0 4.5

3.5

5.5

6.5

7.5

8.5

9.5

Fig. 9 (a), (b) Variations of the characteristic curves of the families SA and SJ for ν = 7, β = 2 and various cases, (c), (d) period (2T ) versus C for families SX and SY for various cases when ν = 7, β = 2

a

SS- Parametric variation of orbits 0.4

b

SX-Parametric variation of orbits 1.5

y 1

1 Gravitational 2 b0 3 All primaries

-0.2

0

-0.2

-0.4

3

0.5

x

P0

0 -0.4

P2

2

0.2

2

y

0.2

0.4

P0 -1.5

P1

-0.5

3

0.5

x 1.5

-0.5

1 Gravitational 2 b0 3 All primaries

1

P7

-1.5

Fig. 10 Radiation effect on simple periodic orbits for ν = 7, β = 2, of the families (a) SS, (b) SX

Figure 10a shows how radiation affects the planetary-type simple periodic orbits of the family SS (ν = 7, β = 2), described by the particle around the central primary P0 . Figure 10b shows the radiation effect on the simple periodic orbits of interplanetary-type of the family SX [22].

The Ring Problem of (N + 1) Bodies: An Overview

145

4 Post-Newtonian Potentials 4.1 Potential Function and Zero-Velocity Curves and Surfaces Here we assume that the potential created by the central body P0 is a Manev’s-type potential of the form A/r + B/r2, where A and B are constants. For simplicity we shall consider that A = 1 and B = eα , where α is the side of the regular polygon. Then the potential function after normalization of the physical quantities takes the form 

 3   1 1 1 1 2 e 2 , (5) +∑ x +y + U (x, y, z) = β + 2 Δ r0 r02 r i=1 i where a non-Newtonian term appears in the expression of the central body’s potential, and a third parameter e is added to the two already known parameters of the simple gravitational case. Quantity Δ depends on the three parameters ν , β , and e. For example, if ν = 3 this quantity equals to

√ Δ = 3 1 + β 3 + 6β e . When e > 0, the results are similar to the ones of the Newtonian case. This means that the form of the networks of the zero-velocity curves, the zero-velocity surfaces, and the evolution of the permitted and non-permitted regions of motion, as well as the number of the equilibrium zones and their distribution on the xy-plane are qualitative similar to those of the pure Newtonian case. The most significant effect of parameter e is that it changes slightly the value of the mass parameter β at which the number of the equilibrium zones changes from five to three. However, when e < 0, a “folding” of the “chimney” that surrounds the central primary P0 starts to create (see Fig. 11b). As a consequence, a closed area of non-permitted motion in the immediate neighborhood of P0 is created. This region is surrounded by another narrow annular region of permitted motion (see Fig. 11a).

4.2 Focal Points in the (x,C) Diagrams It has been proved [20] that in all regular polygon configurations and for Newtonian potentials, all the zero-velocity curves C = C(x) drawn for y = 0, which concern configurations with the same number of peripheral primaries but various mass parameters β , pass through two different focal points, the position of which does not depend on the value of this parameter. This property can be extended in the case where post-Newtonian potentials exist. Figure 12a shows the case. Furthermore, there are two more focal points when C = C(x) curves are drawn for a particular value of β but for various values of e (see Fig. 12b).

146

T.J. Kalvouridis

Fig. 11 (a) Network of zero-velocity curves for ν = 3, β = 50, e = −0.07, (b) zero-velocity surface for the same values of the parameters

5 The Small Mass as a Rigid Body or a Gyrostat In these versions, the point-like particle S is replaced by a small triaxial body or a small gyrostat. Under this consideration, we form two sets of equations that described the translational and the rotational motion of S, respectively (see for details [24, 38]). We have studied some classes of stationary solutions of S and their linear stability. Figure 13 shows some of these attitudes when ν = 2, β = 0 (Copenhagen case of the restricted three-body problem) and S is an axisymmetric gyrostat.

6 A Summary and Comments Goudas [10] studied the case where the primaries are magnetic dipoles and the small body is a charged particle. Salo and Yoder [35], Roberts [34], Bang and Elmabsout [3], and Vanderbei and Kolemen [39] investigated the relative equilibria of the primaries in ring configurations with Newtonian potentials. Mioc and Stavinschi [28, 29] studied the same issue with post-Newtonian potentials, while Arribas et al. [2] investigated the case where all primaries create generalized central forces. Arribas and Elipe [1] studied some aspects of the particle’s dynamics when the central primary creates a post-Newtonian potential. Pinotsis [31] and Barrio et al. [4, 5] recalculated simple periodic orbits in various configurations. Elipe et al. [9] studied the particle motion when the central primary is a prolate rigid body.

The Ring Problem of (N + 1) Bodies: An Overview

147

a

C

e = - 0.0001

900

β = 500 β = 20 β=5 β=2 β=1 β = 0.5

300

β = 0.2 β = 0.1

x -0.0015

-0.001

-0.0005

0

0.0005

focal point

0.001

β = 0.05

0.0015

focal point -300

b

10

e=-0.01

focal point

e=-0.3

C

5

e=-0.05 e=-0.07

focal point

e=-0.1

x

0 -0.5

-0.3

-0.1

0.1

0.3

0.5

Fig. 12 (a) Focal points for a given e and various β , (b) focal points for a given β and various e

Bountis and Papadakis [6] investigated the Sitnikov orbits of the particle, while Papadakis [32] studied its asymptotic orbits in a general ring configuration. Two very interesting versions of the problem appeared in the last few years. In the first one [40], the authors consider kν primaries in k nested regular polygons. In

148

T.J. Kalvouridis

Fig. 13 Three cases of equilibrium positions of a small gyrostat in a Copenhagen configuration (ν = 2, β = 0)

the second variation, the authors consider a solid circular disc, instead of discrete peripheral bodies [7].

References 1. Arribas M, Elipe A (2004) Bifurcations and equilibria in the extended N-body problem. Mech Res Commun 31:1–8 2. Arribas M, Elipe A, Palacios M (2008) Linear stability of ring systems with generalized central forces. Astron Astrophys 489(2):819–824

The Ring Problem of (N + 1) Bodies: An Overview

149

3. Bang D, Elmabsout B (2004) Restricted N + 1-Body problem: existence and stability of relative equilibria. Celest Mech Dyn Astron 89:305–318 4. Barrio R, Blesa F, Serrano S (2008) Qualitative analysis of the (N + 1)-body ring problem. Chaos Solitons Fractals 36:1067–1088 5. Barrio R, Blesa FS, Serrano S (2009) Periodic, escape and chaotic orbits in the Copenhagen and the (n + 1)-body ring problems. Commun Nonlinear Sci Numer Simul 14:2229–2238 6. Bountis T, Papadakis K (2009) The stability of vertical motion in the N-body circular Sitnikov problem. Celest Mech Dyn Astron 104:205–225 7. Broucke R, Elipe A (2005) The dynamics of orbits in a potential field of a solid ring. R + C Dyn 10:1–15 8. Croustaloudi MN, Kalvouridis TJ (2008) Periodic motions of a small body in the Newtonian field of a regular polygonal configuration of ν + 1 bodies. Astrophys Space Sci 314:7–18 9. Elipe A, Arribas M, Kalvouridis TJ (2007) Periodic solutions and their parametric evolution in the planar case of the (n + 1) ring problem with oblateness. J Guid Control Dyn 30(6):1640– 1648 10. Goudas C (1991) The N-dipole problem and the rings of Saturn. Predictability, stability and chaos in N-body dynamical systems. NATO ASI Ser B Phys 272:371–385 11. Hadjifotinou KG, Kalvouridis TJ (2005) Numerical investigation of periodic motion in the three-dimensional ring problem of N bodies. Int J Bifurcat Chaos 15(8):2681–88 12. Hadjifotinou KG, Kalvouridis TJ, Gousidou-Koutita M (2006) Numerical study of the parametric evolution of bifurcations in the three-dimensional ring problem of N bodies. Mech Res Commun 33:830–836 13. Kalvouridis TJ (1997) A planar case of the n + 1 body problem: the ‘ring’ problem. Astrophys Space Sci 260(3):309–325 14. Kalvouridis TJ (1999a) Periodic solutions in the ring problem. Astrophys Space Sci 266(4):467–494 15. Kalvouridis TJ (1999b) Motion of a small satellite in a planar multi-body surrounding. Mech Res Com 26(4):489–497 16. Kalvouridis TJ (2001a) Multiple periodic orbits in the ring problem: families of triple periodic orbits. Astrophys Space Sci 277(4):579–614 17. Kalvouridis TJ (2001b) Zero-velocity surfaces in the three-dimensional ring problem of N + 1 bodies. Celest Mech Dyn Astron 80:133–144 18. Kalvouridis TJ (2001c) The effect of radiation pressure on the particle dynamics in ring type N-body configurations. Earth Moon Planets 87(2):87–102 19. Kalvouridis TJ (2003) Retrograde orbits in ring configurations of N bodies. Astrophys Space Sci 284(3):1013–1033 20. Kalvouridis TJ (2004) On a property of zero-velocity curves in N-body ring type system. Planet Space Sci 52:909–914 21. Kalvouridis TJ (2008) Particle motions in Maxwell’s ring dynamical systems. Celest Mech Dyn Astron 102(1–3):191–206 22. Kalvouridis TJ, Bratsolis E, Kazazakis D (2008) Radiation effect on a particle’s periodic orbits in a regular polygon configuration of N bodies. Earth Moon Planets 103(3–4):143–159 23. Kalvouridis TJ, Hadjifotinou KG (2008) Bifurcations from planar to three-dimensional periodic orbits in the photo-gravitational restricted four-body problem. Int J Bifurcat Chaos 18(2):465–479 24. Kalvouridis TJ, Tsogas V (2002) Rigid body dynamics in the restricted ring problem of N + 1 bodies. Astrophys Space Sci 282(4):751–765 25. Kazazakis D, Kalvouridis TJ (2004) Deformation of the gravitational field in ring-type N-body systems due to the presence of many radiation sources. Earth Moon Planet 93(2):75–95 26. Mara˜nhao D, Llibre J (1999) Ejection-collision orbits and invariant punctured tori in a restricted four-body problem. Celest Mech Dyn Astron 71:1–14 27. Maxwell JC (1890) On the stability of the motion of Saturn’s rings. In: Scientific papers of James Clerk Maxwell, vol 1. Cambridge University Press, Cambridge, p 228

150

T.J. Kalvouridis

28. Mioc V, Stavinschi M (1998) On the Schwarzschild-type polygonal (n + 1)-body problem and on the associated restricted problem. Baltic Astron 7:637–651 29. Mioc V, Stavinschi M (1999) On Maxwell’s (n + 1)-body problem in the Manev-type field and on the associated restricted problem. Phys Scripta 60:483–490 30. Oll¨ongren A (1988) On a particular restricted five-body problem, an analysis with computer algebra. J Symbolic Comput 6:117–126 31. Pinotsis A (2005) Evolution and stability of the theoretically predicted families of periodic orbits in the N-body ring problem. Astron Astrophys 432:713–729 32. Papadakis KE (2009) Asymptotic orbits in the (N + 1)-body ring problem. Astrophys Space Sci 323:261–272 33. Psarros FE, Kalvouridis TJ (2005) Impact of the mass parameter on particle dynamics in a ring configuration of N bodies. Astrophys Space Sci 298:469–488 34. Roberts G (2000) Linear stability in the (1 + N)-gon relative equilibrium. World Sci Monogr Ser Math 6:303–331 35. Salo H, Yoder CF (1988) The dynamics of co-orbital satellite systems. Astron Astrophys 205:309–327 36. Scheeres DJ (1992) On symmetric central configurations with application to satellite motion about rings. PhD Thesis, The University of Michigan 37. Scheeres DJ, Vinh NX (1993) The restricted P + 2 body problem. Acta Astronaut 29(4): 237–248 38. Tsogas V, Kalvouridis TJ, Mavraganis AG (2005) Equilibrium states of a gyrostat satellite moving in the gravitational field of an annular configuration of N big bodies. Acta Mech 175(1–4):181–195 39. Vanderbei RJ, Kolemen E (2007) Linear stability of ring systems. Astron J 133:656–664 40. Wenzhong L, Tongjie Zh, Bin X (2005) A concise numerical analysis on regular polygon solutions for kN-body problem. J Beijing Normal Univ 41(4):386–388

Part II

Mathematical Methods

Symbolic Computation of Conservation Laws, Generalized Symmetries, and Recursion Operators for Nonlinear Differential–Difference Equations∗ ¨ Unal G¨oktas¸ and Willy Hereman

Abstract Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential–difference equations (DDEs) are presented. The algorithms can be used to test the complete integrability of nonlinear DDEs. The ubiquitous Toda lattice illustrates the steps of the algorithms, which have been implemented in Mathematica. The codes INVARIANTSSYMMETRIES. M and DDERECURSION OPERATOR. M can aid researchers interested in properties of nonlinear DDEs.

1 Introduction A large number of physically important nonlinear models are completely integrable, i.e., they can be linearized via an explicit transformation or can be solved with the Inverse Scattering Transform. Completely integrable continuous and discrete models arise in many branches of the applied sciences and engineering, including classical, quantum, and plasma physics, optics, electrical circuits, to name a few. Mathematically, nonlinear models can be represented by ordinary and partial differential equations (ODEs and PDEs), differential–difference equations (DDEs), or ordinary and partial difference equations (OΔ Es and PΔ Es). This chapter deals with integrable nonlinear DDEs.

∗ This

material is based upon work supported by the National Science Foundation (U.S.A.) under Grant No. CCF-0830783.

¨ G¨oktas¸ () U. ¨ Department of Computer Engineering, Turgut Ozal University, Kec¸i¨oren, Ankara 06010, Turkey e-mail: [email protected] W. Hereman Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401-1887, U.S.A. e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 7, © Springer Science+Business Media, LLC 2012

153

154

¨ G¨oktas¸ and W. Hereman U.

Completely integrable equations have nice analytic and geometric properties reflecting their rich mathematical structure. For instance, completely integrable PDEs and DDEs possess infinitely many conserved quantities and generalized (higher-order) symmetries of successive orders. The existence of an infinite set of generalized symmetries can be established by explicitly constructing recursion operators which connect such symmetries. Finding generalized symmetries and recursion operators is a nontrivial task, in particular, if attempted by hand. For example, in G¨oktas¸ [7] and Hereman and G¨oktas¸ [13] an algorithm is presented to compute recursion operators for completely integrable PDEs, which was only recently implemented in Mathematica [1]. Based on our earlier work in G¨oktas¸ [7] and G¨oktas¸ and Hereman [9, 10], we present in this chapter algorithms for the symbolic computation of conserved densities, generalized symmetries, and recursion operators of nonlinear systems of DDEs. Such systems must be polynomial and of evolution type, i.e., the DDEs must be of first order in (continuous) time. The number of equations in the system, degree of nonlinearity, and order (shift levels) are arbitrary. Furthermore, the current algorithms only cover polynomial densities, symmetries, and recursion operators. We use the dilation (scaling) invariance of the system of DDEs to determine the candidate density, symmetry, or recursion operator. Indeed, these candidates are linear combinations with undetermined coefficients of scaling invariant terms. Upon substitution of the candidates into the corresponding defining equations, one has to solve a linear system for the undetermined coefficients. After doing so, the coefficients are substituted into the density, symmetry, or recursion operator. If so desired, the results can be tested one more time, in particular, by applying the recursion operators to generate the successive symmetries. If the system of DDEs contains constant parameters, the eliminant of the linear system for the undetermined coefficients gives the necessary conditions for the parameters, so that the given DDEs admit the required density or symmetry. In analogy with the PDE case in G¨oktas¸ and Hereman [8], the algorithms can thus be used to classify DDEs with parameters according to their complete integrability as illustrated in G¨oktas¸ and Hereman [9, 10]. As shown in Fokas [4], once the generalized symmetries are known, it is often possible to find the recursion operator by inspection. If the recursion operator is hereditary, as defined in Fuchssteiner et al. [6], then the equation will possess infinitely many symmetries. If, in addition, the recursion operator is factorizable, then the equation has infinitely many conserved quantities. Computer algebra systems can greatly help with the search for conservation laws, symmetries, and recursion operators. The algorithms in this chapter have been implemented in Mathematica. The computer codes (see [12]) can be used to test the complete integrability of systems of nonlinear DDEs, provided they are polynomial and of first order (or can be written in that form after a suitable transformation). With INVARIANTSSYMMETRIES. M, in G¨oktas¸ [7], G¨oktas¸ and Hereman [9, 10], G¨oktas¸ and Hereman computed polynomial conserved densities and generalized

Conservation Laws, Generalized Symmetries, and Recursion Operators of DDEs

155

symmetries of many well-known systems of DDEs, including various Volterra and Toda lattices as well as the Ablowitz–Ladik lattice (for additional results and references, see, e.g., [16]). The existence of, say, a half dozen conserved densities or generalized symmetries is a predictor for complete integrability. Finding a recursion operator then becomes within reach. An existence proof (showing that there are indeed infinitely many densities or generalized symmetries) must be done analytically, e.g., by explicitly constructing the recursion operator which allows one to generate the generalized symmetries order by order. Numerous explicit examples have been reported in the literature but novices could start with the book by Olver [18] to learn about recursion operators for PDEs. To alleviate the burden of trying to find a recursion operator by trial and error, we present a new Mathematica program, DDERECURSIONO PERATOR. M, based on the algorithm in Sect. 5. Like INVARIANTSSYMMETRIES .M, after thorough testing, DDERECURSIONOPERATOR. M will be available from Hereman [12]. If one cannot find a sufficiently large number of densities or symmetries (let alone, a recursion operator), then it is unlikely that the DDE system is completely integrable, at least in that coordinate representation. However, our software does not allow one to conclude that a DDE is not completely integrable merely based on the fact that polynomial conserved densities and generalized symmetries could not be found. Polynomial DDEs that lack the latter may accidentally have non-polynomial densities or symmetries, or a complicated recursion operator, which is outside the scope of the algorithm described in Sect. 5. Currently, our algorithm fails to find recursion operators for the Belov–Chaltikian lattices [2,20,21] and lattices due to Blaszak and Marciniak [3,20,21,24]. In the near future we plan to generalize the recursion operator algorithm so that it can cover a broader class of nonlinear DDEs. The chapter is organized as follows. Basic definitions are given in Sect. 2. In Sect. 3, we show the algorithm for conservation laws, using the Toda lattice as an example. Using the same example, Sects. 4 and 5 cover the algorithms for generalized symmetries and recursion operators, respectively. In Sect. 6, we draw some conclusions and briefly discuss future research.

2 Key Definitions Consider a system of nonlinear DDEs of first order: u˙ n = F(un− , . . . , un−1 , un , un+1 , . . . , un+m ),

(1)

where un and F are vector-valued functions with N components. This chapter only covers DDEs with one discrete variable, denoted by integer n, which often corresponds to the discretization of a space variable. The dot stands for differentiation with respect to the continuous variable (often time t). Each component of F is assumed to be a polynomial with constant coefficients. If parameters are present in

¨ G¨oktas¸ and W. Hereman U.

156

(1), they will be denoted by lower-case Greek letters. F depends on un and a finite number of forward and backward shifts of un . We denote by  (m, respectively), the furthest negative (positive, respectively) shift of any variable in the system. Restrictions are neither imposed on the degree of nonlinearity of F, nor on the integers l and m, which measure the degree of nonlocality in (1).

2.1 Leading Example: The Toda Lattice One of the earliest and most famous examples of completely integrable DDEs is the Toda lattice, discussed in, for instance, [22]: y¨n = exp (yn−1 − yn ) − exp(yn − yn+1 ),

(2)

where yn is the displacement from equilibrium of the nth particle with unit mass under an exponential decaying interaction force between nearest neighbors. In new variables (un , vn ), defined by un = y˙n , vn = exp (yn − yn+1 ), lattice (2) can be written in polynomial form u˙n = vn−1 − vn , v˙n = vn (un − un+1).

(3)

The Toda lattice (3) will be used to illustrate the various algorithms presented in subsequent sections of this chapter.

2.2 Dilation Invariance A DDE is dilation invariant if it is invariant under a dilation (scaling) symmetry. Lattice (3) is invariant under scaling symmetry (t, un , vn ) → (λ −1 t, λ 1 un , λ 2 vn ).

(4)

2.3 Uniformity in Rank We define the weight, w, of a variable as the exponent of the scaling parameter (λ ) which multiplies that variable.   Since λ can be selected at will, t will always be replaced by λt and, thus, w dtd = w(Dt ) = 1.

Conservation Laws, Generalized Symmetries, and Recursion Operators of DDEs

157

Weights of dependent variables are nonnegative, rational, and independent of n. For example, w(un−3 ) = · · · = w(un ) = · · · = w(un+2 ). The rank, denoted by R, of a monomial is defined as the total weight of the monomial. An expression is uniform in rank if all of its terms have the same rank. Dilation symmetries, which are special Lie-point symmetries, are common to many DDEs. Polynomial DDEs that do not admit a dilation symmetry can be made scaling invariant by extending the set of dependent variables with auxiliary parameters with appropriate scales as discussed in G¨oktas¸ and Hereman [9, 10]. In view of (4), we have w(un ) = 1 and w(vn ) = 2 for the Toda lattice. In the first equation of (3), all the monomials have rank 2; in the second equation all the monomials have rank 3. Conversely, requiring uniformity in rank for each equation in (3) allows one to compute the weights of the dependent variables (and, thus, the scaling symmetry) with simple linear algebra. Balancing the weights of the various terms of each equation in (3) yields w(un ) + 1 = w(vn ), w(vn ) + 1 = w(un ) + w(vn ).

(5)

Hence, w(un ) = 1,

w(vn ) = 2,

(6)

which confirms (4).

2.4 Upshift and Downshift Operator We define the shift operator D by Dun = un+1 . The operator D is often called the upshift operator or forward- or right-shift operator. The inverse, D−1 , is the downshift operator or backward- or left-shift operator, D−1 un = un−1 . Shift operators apply to functions by their action on the arguments of the functions. For example, DF(un− , · · · , un−1 , un , un+1 , · · · , un+m ) = F(Dun− , · · · , Dun−1 , Dun , Dun+1 , . . . , Dun+m ) = F(un−+1 , . . . , un , un+1 , un+2 , · · · , un+m+1 ).

(7)

2.5 Conservation Law A conservation law of (1),

Dt ρ + Δ J = 0,

(8)

¨ G¨oktas¸ and W. Hereman U.

158

connects a conserved density ρ to an associated flux J, where both are scalar functions depending on un and its shifts. In (8), which must hold on solutions of (1), Dt is the total derivative with respect to time, Δ = D − I is the forward difference operator, and I is the identity operator. For readability (in particular, in the examples), the components of un will be denoted by un , vn , wn , etc. In what follows we consider only autonomous functions, i.e., F, ρ , and J do not explicitly depend on t and n. A density is trivial if there exists a function ψ so that ρ = Δ ψ . We say that two densities, ρ (1) and ρ (2) , are equivalent if and only if ρ (1) + kρ (2) = Δ ψ , for some ψ and some nonzero scalar k. It is paramount that the density is free of equivalent terms for if such terms were present, they could be moved into the flux J. Compositions of D or D−1 define an equivalence relation (≡) on monomial terms. Simply stated, all shifted terms are equivalent, e.g., un−1 vn+1 ≡ un vn+2 ≡ un+2 vn+4 ≡ un−3 vn−1 since un−1vn+1 = un vn+2 − Δ (un−1vn+1 ) = un+2 vn+4 − Δ (un+1 vn+3 + un vn+2 + un−1vn+1 ) = un−3 vn−1 + Δ (un−2 vn + un−3vn−1 ).

(9)

This equivalence relation also holds for any function of the dependent variables, but for the construction of conserved densities we will apply it only to monomial terms (ti ) in the same density, thereby achieving high computational efficiency. In the algorithm used in Sect. 3, we will use the following equivalence criterion: two monomial terms, t1 and t2 , are equivalent, t1 ≡ t2 , if and only if t1 = Dr t2 for some integer r. If t1 ≡ t2 , then t1 = t2 + Δ J for some J dependent on un and its shifts. For example, un−2 un ≡ un−1 un+1 because un−2 un = D−1 un−1 un+1 . Hence, un−2un = un−1 un+1 + [−un−1un+1 + un−2un ] = un−1 un+1 + Δ J with J = −un−2un . For efficiency, we need a criterion to choose a unique representative from each equivalence class. There are a number of ways to do this. We define the canonical representative as that member that has (a) no negative shifts and (b) a nontrivial dependence on the local (i.e., zero-shifted) variable. For example, un un+2 is the canonical representative of the class {· · · , un−2 un , un−1 un+1 , un un+2 , un+1 un+3 , · · · }. In the case of, e.g., two variables (un and vn ), un+2 vn is the canonical representative of the class {· · · , un−1 vn−3 , un vn−2 , un+1 vn−1 , un+2 vn , un+3 vn+1 , · · · }. Alternatively, one could choose a variable ordering and then choose the member that depends on the zero-shifted variable of lowest lexicographical order. The code

Conservation Laws, Generalized Symmetries, and Recursion Operators of DDEs

159

in Hereman [12] uses lexicographical ordering of the variables, i.e., un ≺ vn ≺ wn , etc. Thus, un vn−2 (instead of un+2 vn ) is chosen as the canonical representative of {· · · , un−1 vn−3 , un vn−2 , un+1 vn−1 , un+2 vn , un+3 vn+1 , · · · }. It was shown in Hickman [17] that if ρ is a density then Dk ρ is also a density. Hence, using an appropriate “upshift” all negative shifts in a density can be removed. Without loss of generality, we thus assume that a density that depends on q shifts has canonical form ρ (un , un+1 , · · · , un+q ). Lattice (3) has infinitely many conservation laws (see, e.g., [11]). Here we list the densities of rank R ≤ 4 :

ρ (1) = un ,

(10)

ρ (2) =

1 2 u + vn , 2 n

(11)

ρ (3) =

1 3 u + un(vn−1 + vn ), 3 n

(12)

ρ (4) =

1 4 1 u + u2n(vn−1 + vn ) + un un+1 vn + v2n + vn vn+1 . 4 n 2

(13)

The first two density-flux pairs are easily computed by hand, and so is (0)

ρn = ln(vn ),

(14)

which is the only non-polynomial density (of rank 0).

2.6 Generalized Symmetry A vector function G(un ) is called a generalized symmetry of (1) if the infinitesimal transformation un → un + ε G leaves (1) invariant up to order ε . As shown by Olver [18], G must then satisfy Dt G = F (un )[G]

(15)

on solutions of (1), where F (un )[G] is the Fr´echet derivative of F in the direction of G. For the scalar case (N = 1), the Fr´echet derivative is F  (un )[G] =

∂ ∂F k F(un + ε G)|ε =0 = ∑ D G, ∂ε ∂ un+k k

(16)

¨ G¨oktas¸ and W. Hereman U.

160

which, in turn, defines the Fr´echet derivative operator F  (un ) = ∑ k

∂F k D. ∂ un+k

(17)

In the vector case with, say, components un and vn , the Fr´echet derivative operator is a matrix operator: ⎛

∂ F1 k ∂ un+k D

1 Dk ∑k ∂ vn+k

∂ F2 Dk ∑k ∂ un+k

∂ F2 Dk ∑k ∂ vn+k

∑k

⎜ F (un ) = ⎝

∂F

⎞ ⎟ ⎠.

(18)

Applied to G = (G1 G2 )T , where T is transpose, one obtains Fi  (un )[G] = ∑ k

∂ Fi k ∂ Fi k D G1 + ∑ D G2 , ∂ un+k ∂ vn+k k

(19)

with i = 1, 2. In (16) and (19) summation is over all positive and negative shifts (including k = 0). The generalization of (18) to a N−component system is straightforward. As computed in Hereman et al. [14], the first two nontrivial symmetries of (3) are

 vn − vn−1 G(1) = , (20) vn (un+1 − un ) G(2) =



vn (un + un+1) − vn−1(un−1 + un) vn (u2n+1 − u2n + vn+1 − vn−1 )

 .

(21)

2.7 Recursion Operator A recursion operator R connects symmetries G( j+s) = R G( j) ,

(22)

where j = 1, 2, · · · , and s is the gap length. The symmetries are linked consecutively if s = 1. This happens in most (but not all) cases. For N-component systems, R is an N × N matrix operator.

Conservation Laws, Generalized Symmetries, and Recursion Operators of DDEs

161

With reference to Olver [18] and Wang [23], the defining equation for R is Dt R + [R, F(un )] =

∂R + R  [F] + R ◦ F (un ) − F (un ) ◦ R = 0, ∂t

(23)

where [ , ] denotes the commutator and ◦ the composition of operators. The operator F (un ) was defined in (18). R  [F] is the Fr´echet derivative of R in the direction of F. For the scalar case, the operator R is often of the form   R = U(un ) O (D − I)−1 , D−1 , I, D V (un ), and then R  [F] = ∑(Dk F) k

(24)

∂U ∂V O V + ∑ UO(Dk F) . ∂ un+k ∂ un+k k

(25)

For the vector case, the elements of the N × N operator matrix R are often of the form   Ri j = Ui j (un ) Oi j (D − I)−1 , D−1 , I, D Vi j (un ). (26) Hence, for the two-component case R  [F]i j = ∑ (Dk F1 ) k

∂ Ui j ∂ Ui j Oi j Vi j + ∑ (Dk F2 ) Oi j Vi j ∂ un+k ∂ vn+k k

+ ∑ Ui j Oi j (Dk F1 ) k

∂ Vi j ∂ Vi j + Ui j Oi j (Dk F2 ) . ∂ un+k ∑ ∂ vn+k k

(27)

2.7.1 Example The recursion operator of (3) is ⎛ ⎜ R=⎝

un I

D−1 + I + (vn − vn−1 )(D − I)−1 v1n I

vn I + vn D un+1 I + vn (un+1 − un)(D − I)−1 v1n I

⎞ ⎟ ⎠.

(28)

It is straightforward to verify that R G(1) = G(2) with G(1) in (20) and G(2) in (21).

3 Algorithm for Conservation Laws As an example, we will compute the density ρ (3) (of rank R = 3) given in (12).

¨ G¨oktas¸ and W. Hereman U.

162

3.1 Density Start from V = {un, vn }, the set of dependent variables with weights. List all monomials in u and v of rank R = 3 or less: M = {u3n , u2n , un vn , un , vn }. Next, for each monomial in M , introduce the correct number of t-derivatives so that each term has rank 3. Using (3), compute d0 u3n = u3n , dt0 d0 un vn = un vn , dt0 du2n = 2unu˙n = 2un vn−1 − 2un vn , dt dvn = v˙n = un vn − un+1vn , dt d 2 un d(vn−1 − vn ) du˙n = = 2 dt dt dt = un−1 vn−1 − un vn−1 − unvn + un+1vn .

(29)

Gather the terms in the right hand sides in (29) to get S = {u3n , un vn−1 , un vn , un−1 vn−1 , un+1 vn }. Identify members belonging to the same equivalence classes and replace them by their canonical representatives. For example, un vn−1 ≡ un+1 vn . Adhering to lexicographical ordering, use un vn−1 instead of un+1 vn . Doing so, replace S by T = {u3n , un vn−1 , un vn }, which has the building blocks of the density. Linearly combine the monomials in T with undetermined coefficients ci to get the candidate density of rank 3 : ρ = c1 u3n + c2 un vn−1 + c3 un vn . (30)

3.2 Undetermined Coefficients ci Compute Dt ρ and use (3) to eliminate u˙n and v˙n and their shifts. Next, introduce the main representatives to get E = (3c1 − c2 )u2n vn−1 + (c3 − 3c1 )u2n vn + (c3 − c2 )vn vn+1 + (c2 − c3 )un un+1 vn + (c2 − c3 )v2n + Δ J,

(31)

Conservation Laws, Generalized Symmetries, and Recursion Operators of DDEs

163

with J = (c3 − c2 )vn−1 vn + c2 un−1 un vn−1 + c2 v2n−1 .

(32)

Set E − Δ J ≡ 0 to get the linear system 3c1 − c2 = 0,

c3 − 3c1 = 0,

c2 − c3 = 0.

(33)

Select c1 = 13 and substitute the solution c1 = 13 , c2 = c3 = 1 into (30) and (32) to obtain ρ (3) in (12) with matching flux J (3) = un−1 un vn−1 + v2n−1 .

4 Algorithm for Symmetries (2)

As an example, we will now compute the symmetry G(2) = (G1 rank G = (3 4)T given in (21).

(2)

G2 )T with

4.1 Symmetry Listing all monomials in un and vn of ranks 3 and 4, or less: L1 = {u3n , u2n , un vn , un , vn }, L2 = {u4n , u3n , u2n vn , u2n , un vn , un , v2n , vn }. Next, for each monomial in L1 and L2 , introduce the necessary t-derivatives so that each term exactly has ranks 3 and 4, respectively. At the same time, use (3) to remove all t-derivatives. Doing so, based on L1 , d0 3 (un ) = u3n , dt0 d0 (un vn ) = un vn , dt0 d 2 (u ) = 2un u˙n = 2un vn−1 − 2un vn , dt n d (vn ) = v˙n = un vn − un+1vn , dt d d d2 (u ) = (u˙n ) = (vn−1 − vn ) 2 n dt dt dt = un−1 vn−1 − un vn−1 − un vn + un+1vn .

(34)

¨ G¨oktas¸ and W. Hereman U.

164

Put the terms from the right-hand side of (34) into a set: W1 = {u3n , un−1 vn−1 , un vn−1 , un vn , un+1 vn }. Similarly, based on the monomials in L2 , construct W2 = {u4n , u2n−1 vn−1 , un−1 un vn−1 , u2n vn−1 , vn−2 vn−1 , v2n−1 , u2n vn , un un+1 vn , u2n+1 vn , vn−1 vn , v2n , vn vn+1 }. Linearly combine the monomials in W1 and W2 with undetermined coefficients ci to get the form of the components of the candidate symmetry: (2)

G1 = c1 u3n + c2 un−1 vn−1 + c3 un vn−1 + c4 un vn + c5 un+1 vn , (2)

G2 = c6 u4n + c7 u2n−1 vn−1 + c8 un−1 un vn−1 + c9 u2n vn−1 + c10 vn−2 vn−1 + c11 v2n−1 + c12 u2n vn + c13 un un+1 vn + c14 u2n+1 vn + c15 vn−1 vn + c16 v2n + c17 vn vn+1 .

(35)

4.2 Undetermined Coefficients ci To determine the coefficients ci , require that (15) hold on any solution of (1). Compute Dt G and use (1) to remove all u˙ n−1 , u˙ n , u˙ n+1 , etc. Compute the Fr´echet derivative (19) and, in view of (15), equate the resulting expressions. Treat as independent all the monomials in un and their shifts, to obtain the linear system that determines the coefficients ci . Apply the strategy to (3) with (35), to get c1 = c6 = c7 = c8 = c9 = c10 = c11 = c13 = c16 = 0, −c2 = −c3 = c4 = c5 = −c12 = c14 = −c15 = c17 . (2)

(2)

Set c17 = 1 and substitute (36) into (35) to get G(2) = (G1 G2 )T , as given in (21). To show how our algorithm filters out completely integrable cases among parameterized systems of DDEs, consider u˙n = α vn−1 − vn , v˙n = vn (β un − un+1),

(36)

where α and β are nonzero constant parameters. Ramani et al. [19] have shown that (36) is completely integrable if and only if α = β = 1.

Conservation Laws, Generalized Symmetries, and Recursion Operators of DDEs

165

Using our algorithm, one can easily compute the compatibility conditions for α and β so that (36) admits a polynomial symmetry, say, of rank (3, 4). The steps are as above; however, the linear system for the ci is parameterized by α and β and must be analyzed carefully (with, e.g., Gr¨obner basis methods). This analysis leads to the condition α = β = 1. Details are given in G¨oktas¸ and Hereman [9, 10].

5 Algorithm for Recursion Operators We will now construct the recursion operator (28) for (3). In this case all the terms in (23) are 2 × 2 matrix operators.

5.1 Rank of the Recursion Operator The difference in the ranks of symmetries is used to compute the rank of the elements of the recursion operator. Use (6), (20), and (21) to compute rank G(1) =

 2 , 3

rank G(2) =

 3 . 4

(37)

Assume that R G(1) = G(2) and use the formula (k+1)

rank Ri j = rank Gi

(k)

− rankG j ,

(38)

to compute a rank matrix associated to the operator R :

rank R =

1 0 2 1

 .

(39)

5.2 Recursion Operator We assume that R = R0 + R1 , where R0 is a sum of terms involving D−1 , I, and D (the form of R1 will be discussed below). The coefficients of these terms are admissible power combinations of un , un+1 , vn , and vn−1 (which come from the terms on the right-hand side of (3)), so that all the terms have the correct rank. The maximum upshift and downshift operator that should be included can be determined by comparing two consecutive symmetries. Indeed, if the maximum upshift in the first symmetry is un+p and the maximum upshift in the next symmetry is un+p+r , then the associated piece that goes into R0 must have D, D2 , . . . , Dr . The same argument determines the minimum downshift operator to be included. For (3), get

¨ G¨oktas¸ and W. Hereman U.

166

R0 =

(R0 )11 (R0 )21

(R0 )12 (R0 )22

 ,

(40)

with (R0 )11 = (c1 un + c2 un+1 ) I, (R0 )12 = c3 D−1 + c4 I, (R0 )21 = (c5 u2n + c6 un un+1 + c7 u2n+1 + c8 vn−1 + c9 vn ) I + (c10 u2n + c11un un+1 + c12u2n+1 + c13vn−1 + c14vn ) D, (R0 )22 = (c15 un + c16un+1 ) I.

(41)

As shown for the continuous case [13], R1 is a linear combination (with undetermined coefficients c˜ jk ) of all suitable products of symmetries and covariants, i.e., Fr´echet derivatives of densities, sandwiching (D − I)−1. Hence, (k)

∑ ∑ c˜ jk G( j)(D − I)−1 ⊗ ρn j

,

(42)

k

where ⊗ denotes the matrix outer product, defined as ⎛ ⎜ ⎝

( j)

G1

( j)

G2

⎞

⎟ (k) (k) −1 ⎠ (D − I) ⊗ ρn,1 ρn,2 ⎛ ⎜ =⎝

( j)

(k)

G1 (D − I)−1ρn,1 ( j)

(k)

G2 (D − I)−1ρn,1

( j)

(k)

G1 (D − I)−1ρn,2 ( j)

(k)

G2 (D − I)−1ρn,2

⎞ ⎟ ⎠.

(43)



(0) Only the pair G(1) , ρn can be used, otherwise the ranks in (39) would be exceeded. Use (14) and (19) to compute (0)

ρn

=

0

1 vn I

,

(44)

From (42), after renaming c˜10 to c17 , obtain ⎛ ⎜ R1 = ⎝

0

c17 (vn−1 − vn )(D − I)−1 v1n I

0

c17 vn (un − un+1 )(D − I)−1 v1n

⎞ ⎟ ⎠. I

(45)

Conservation Laws, Generalized Symmetries, and Recursion Operators of DDEs

167

Add (40) and (45) to get

R = R0 + R1 =

R11 R21

R12 R22

 ,

(46)

with R11 = (c1 un + c2 un+1 ) I, R12 = c3 D−1 + c4 I + c17(vn−1 − vn )(D − I)−1

1 I, vn

R21 = (c5 u2n + c6 un un+1 + c7u2n+1 + c8 vn−1 + c9 vn ) I + (c10 u2n + c11un un+1 + c12u2n+1 + c13vn−1 + c14vn )D, R22 = (c15 un + c16un+1 )I + c17vn (un − un+1)(D − I)−1

1 I. vn

(47)

5.3 Unknown Coefficients Compute all the terms in (23) to find the ci . Refer to Hereman [15] for the details of the computation, resulting in c2 = c5 = c6 = c7 = c8 = c10 = c11 = c12 = c13 = c15 = 0, and c1 = c3 = c4 = c9 = c14 = c16 = 1, and c17 = −1. Substitute the constants into (46) to get (28).

6 Conclusions In this chapter we presented algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear DDEs. We used the Toda lattice to illustrate the steps of the algorithms. The algorithms have been implemented in Mathematica and can be used to test the complete integrability of nonlinear DDEs. Although our algorithm successfully finds conservation laws, generalized symmetries, and recursion operators for various Volterra and Toda lattices as well as the Ablowitz–Ladik lattice, the current recursion operator algorithm fails on nonlinear DDEs due to Belov and Chaltikian and Blaszak and Marciniak. In future research we intend to generalize the recursion operator algorithm so that it can cover a broader class of lattices. Acknowledgments J.A. Sanders, J.-P. Wang, M. Hickman, and B. Deconinck are gratefully acknowledged for valuable discussions.

168

¨ G¨oktas¸ and W. Hereman U.

References 1. Baldwin DE, Hereman W (2010) A symbolic algorithm for computing recursion operators of nonlinear partial differential equations. Int J Comput Math 87:1094–1119 2. Belov AA, Chaltikian KD (1993) Lattice analogues of W -algebras and classical integrable equations. Phys Lett B 309:268–274 3. Blaszak M, Marciniak K (1994) R-matrix approach to lattice integrable systems. J Math Phys 35:4661–4682 4. Fokas AS (1980) A symmetry approach to exactly solvable evolution equations. J Math Phys 21:1318–1325 5. Fokas AS (1987) Symmetries and integrability. Stud Appl Math 77:253–299 6. Fuchssteiner B, Oevel W, Wiwianka W (1987) Computer-algebra methods for investigation of hereditary operators of higher order soliton equations. Comput Phys Commun 44:47–55 ¨ (1998) Algorithmic computation of symmetries, invariants and recursion operators 7. G¨oktas¸ U for systems of nonlinear evolution and differential-difference equations. Ph.D. Thesis, Colorado School of Mines, Golden, 1998 ¨ Hereman W (1997) Symbolic computation of conserved densities for systems of 8. G¨oktas¸ U, nonlinear evolution equations. J Symb Comput 24:591–621 ¨ Hereman W (1998) Computation of conservation laws for nonlinear lattices. Phys 9. G¨oktas¸ U, D 132:425–436 ¨ Hereman W (1999) Algorithmic computation of higher-order symmetries for 10. G¨oktas¸ U, nonlinear evolution and lattice equations. Adv Comput Math 11:55–80 11. H´enon M (1974) Integrals of the Toda lattice. Phys Rev B 9:1921–1923 12. Hereman W (2011) Software. http://inside.mines.edu/$\sim$whereman/ (accessed July 27, 2010) ¨ (1999) Integrability tests for nonlinear evolution equations. In: Wester 13. Hereman W, G¨oktas¸ U M (ed) Computer algebra systems: a practical guide. Wiley, New York, pp 211–232 ¨ Colagrosso MD, Miller AJ (1998) Algorithmic integrability tests for 14. Hereman W, G¨oktas¸ U, nonlinear differential and lattice equations. Comput Phys Comm 115:428–446 15. Hereman W, Sanders JA, Sayers J, Wang J-P (2004) Symbolic computation of polynomial conserved densities, generalized symmetries, and recursion operators for nonlinear differentialdifference equations. In: Winternitz P et al. (ed) Group theory and numerical analysis, CRM Proc. & Lect. Ser., vol 39. AMS, Providence, pp 267–282 16. Hereman W, Adams PJ, Eklund HL, Hickman MS, Herbst BM (2008) Direct methods and symbolic software for conservation laws of nonlinear equations. In: Yan Z (ed) Advances in nonlinear waves and symbolic computation. Nova Scienc Publishers, New York, pp 18–78 17. Hickman M (2008) Leading order integrability conditions for differential-difference equations. J Nonlinear Math Phys 15:66–86 18. Olver PJ (1993) Applications of Lie groups to differential equations, 2nd edn. Springer, New York 19. Ramani A, Grammaticos B, Tamizhmani KM (1992) An integrability test for differentialdifference systems. J Phys A: Math Gen 25:L883–L886 20. Sahadevan R, Khousalya S (2001) Similarity reductions, generalized symmetries and integrability of Belov-Chaltikian and Blaszak-Marciniak lattice equations. J Math Phys 42:3854–3879 21. Sahadevan R, Khousalya S (2003) Belov-Chaltikian and Blaszak-Marciniak lattice equations. J Math Phys 44:882–898 22. Toda M (1981) Theory of nonlinear lattices. Springer, Berlin 23. Wang J-P (1998) Symmetries and conservation laws of evolution equations. Ph.D. Thesis, Thomas Stieltjes Institute for Mathematics, Amsterdam 24. Wu Y, Geng X (1996) A new integrable symplectic map associated with lattice soliton equations. J Math Phys 37:2338–2345

Approximate Polynomial Solution of a Nonlinear Differential Equation with Applications Constantin Bota, Bogdan C˘aruntu, and Liviu Bereteu

Abstract This chapter proposes an approximate polynomial solution for a nonlinear problem of the type: x = F(x , x,t) on the [a,b] interval with initial conditions of the type: x (a) = x1 , x(a) = x0 , where F is a continuously differentiable real function. The approximate solution is expressed in terms of Taylor polynomials, whose coefficients are determined by solving a nonlinear system associated to the problem. The performance of the method is illustrated by two numerical examples.

1 Introduction For many differential equations which model physical phenomena it is usually difficult, or even impossible to find an exact solution. In such a situation, an approximate solution must be found. An often used category of approximate solution consists of polynomial solutions, which presents certain advantages over other types of approximate solutions: compact expression, facile manipulation in subsequent computations, etc. In order to obtain approximate polynomial solutions, various methods were used, such as: methods based on Taylor polynomials [13, 14], Bernstein polynomials [5] and Chebyshev polynomials [7], variational methods [23], homotopy methods [4, 16], Adomian decomposition methods [3, 21], collocation-type methods [8, 22], differential transform methods [1, 15], and harmonic analysis methods [6].

C. Bota () and B. C˘aruntu Department of Mathematics, “Politehnica” University of Timis¸oara, P-t¸a Victoriei, 2, Timis¸oara, 300006, Romania e-mail: [email protected]; [email protected] L. Bereteu Faculty of Mechanics, “Politehnica” University of Timis¸oara, Blv. Mihai Viteazu, Timis¸oara, 300222, Romania e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 8, © Springer Science+Business Media, LLC 2012

169

170

C. Bota et al.

In the present chapter we propose an approximate polynomial solution for a differential equation of the type: x = F(x , x,t)

(1)

under the condition x (a) = x1 , x(a) = x0 , where F is a continuously differentiable real function defined on the [a,b] interval and x0 , x1 , a, b are real constants. Equations of this type have multiple applications in various fields such as nonlinear mechanics, electrodynamics, nonlinear dynamical systems, astrophysics, and quantum mechanics. Our approximate polynomial solution for (1) is expressed in terms of Taylor polynomials and thus it has its roots in the regular differential transform method [2, 9–12, 17, 19], but the coefficients are determined by using an original method, presented in the following.

2 Solution We consider the approximate solution of (1) expressed in terms of Taylor polynomials as: x(t) ˜ =

n

n

k=0

k=0

1

= ∑ x(k) (0) · t k ∑ ck · t k ∼ k!

(2)

where the coefficients ck will be determined in an iterative manner, different from the well-known Taylor matrix method [13, 14]. The parameters c0 , c1 , . . . , cn are calculated using the steps outlined in the following. n

By substituting the approximate solution x(t) ˜ = ∑ ck · t k in (1) we obtain the k=0

following expression:

R(u,t) ˜ = ˙ x˜  (t) − F(x˜ , x,t) ˜

(3)

˜ = 0 for any t in the [a,b] If we could find the constants c0 , c1 , . . . , cn such that R(x,t) interval, then by substituting c0 , c1 , . . . , cn in (2) we obtain the exact solution of (1). In general this situation is rarely encountered. The best approximate solution of the problem (1), (2) is thus the one which minimizes the value of |R(t, x)|. ˜ We remark that, due to the fact that R is a continuous function on the [a,b] interval, the following equivalence is easy to prove: |R(t, x)| ˜ = 0 if and only if

b a

R2 (t, x) ˜ dt = 0

(4)

Approximate Polynomial Solution of a Nonlinear Differential Equation...

171

The initial conditions x(a) = x0 and x (a) = x1 lead to the system consisting of the equations x(a) ˜ = x0 , x˜ (a) = x1 , thus allowing us to calculate the expressions of the coefficients c0 and c1 as functions of the other coefficients c2 , . . . , cn . In order to calculate these other n − 1 coefficients c2 , . . . , cn , we observe that by considering the equidistant partition of the [a,b] interval: a = b0 < b1 < b2 < . . .bn = b, the equality b 2 R (t, x) ˜ dt = 0 is equivalent with the following nonlinear system of equations in a

the unknowns c2 , . . . , cn :

⎧ b 1 2 ⎪ ⎪ ⎪ R (t, x) ˜ dt = 0 ⎪ ⎪ a ⎪ ⎪ ⎪ ⎪ b2 ⎪ ⎪ ˜ dt = 0 ⎨ R2 (t, x) b1

(5)

.. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ bn 2 ⎪ ⎪ ⎪ R (t, x) ˜ dt = 0 ⎩ bn−1

An approximate solution of this system, together with the expressions of the coefficients c0 and c1 as functions of c2 , . . . , cn will give us, by means of the relation n

(2), an approximate polynomial solution of (1), namely, x(t) ˜ = ∑ ck t k . k=0

If the approximate solution of the system (5) is computed using a numerical method which allows us to estimate the error ε , (such as, for example, a Newtontype method), then, if c2 , . . . , cn is an approximate numerical solution of (5) and c02 , . . . , c0n is its corresponding exact solution, we have ci − ε ≤ c0i ≤ ci + ε , i = 1, . . . , n. On the other hand, the error function E(t, c2 , . . . cn ) obtained by replacing the n

exact solution x(t) with the approximate solution x(t) ˜ = ∑ ck t k can be estimated k=0

as E(t, c2 , . . . cn ) = |R(t, x)|. ˜ The maximal error E in this case can be computed as E = max E(t, s2 , . . . sn ). t∈[a,b], si ∈[ci ε , ci +ε ], i=2,...,n

The method described above has the advantage, in comparison with other methods, that it can be applied not only for weakly nonlinear equations but also for strong nonlinear equations. For each of the examples considered in this chapter we will give an estimate of the error, computed as: ˜ = x˜  (t) − F(x˜ , x,t) R(t) ˜

(6)

R˜ represents the error obtained by replacing the exact solution x with the approximate solution x. ˜

172

C. Bota et al.

3 Numerical Illustrations 3.1 Duffing Oscillator We consider the initial value problem corresponding to the Duffing differential equation [20]: ⎧ ⎨ x¨ + 2kx˙ + w2 x = −ε x3 (7) x(0) = 1 ⎩ x(0) ˙ =0 on [0,1], where k = √12 , ε = w = 1. The Duffing equation, which describes the motion of a forced oscillator, is a nonlinear second-order equation of the type (1). It does not admit an exact analytical solution, and various approximate methods were employed in its study. Using our method, the cubic approximate analytical solution is: xapp (t) = 1 − 0.908452t 2 + 0.396015t 3. The following plot contains the graphical representation of this polynomial (solid line) together with the corresponding numerical solution of (7) computed using Mathematica 6 (dashed line) (Fig. 1). Next we present the graphical representation of the error R in Fig. 2. We can observe that the maximal value of the error in this case is less than 0.2. The fifth degree polynomial approximate analytical solution is: xapp (t) = 1 − 1.01309t 2 + 0.557159t 3 − 0.0629464t 4 − 0.0492236t 5 . The following plot contains the graphical representation of this polynomial (solid line) together with the corresponding exact solution of (7) (dashed line) (Fig. 3). 1.0 0.9 0.8 0.7 0.6

0.2

0.4

0.6

0.8

1.0

Fig. 1 Comparison between the numerical solution (dashed line) and the cubic polynomial approximation (solid line)

Approximate Polynomial Solution of a Nonlinear Differential Equation...

173

1.0

0.5

0.2

0.4

0.6

0.8

1.0

−0.5

−1.0

Fig. 2 Graphical representation of the error for the case of a cubic polynomial approximation 1.0 0.9 0.8 0.7 0.6

0.2

0.4

0.6

0.8

1.0

Fig. 3 Comparison between the exact solution (dashed line) and the fifth degree polynomial approximation (solid line)

It can be observed again that the graphical representations are practically overlapping. The graphical representation of the error R is given in Fig. 4. We can observe that the maximal value of the error in this case is less than 0.03.

3.2 van der Pol oscillator We consider the initial value problem on the [0,1] interval corresponding to the Van der Pol differential equation [18]: ⎧ ⎨ x¨ + x − ε (1 − x2 ) x˙ = 0 (8) x(0) = 1 ⎩ x(0) ˙ =0

174

C. Bota et al. 1.0

0.5

0.2

0.4

0.6

0.8

1.0

−0.5

1.0

Fig. 4 Graphical representation of the error for the case of a fifth degree polynomial approximation 1.0

0.9

0.8

0.7

0.6

0.2

0.4

0.6

0.8

1.0

Fig. 5 Comparison between the numerical solution (dashed line) and the cubic polynomial approximation (solid line)

The well-known Van der Pol equation models a damped oscillator and various approximate analytical solutions were proposed, but unfortunately most of these approximate solutions only work for small values of the parameter ε . In contrast, our method does not depend on ε ; we applied our method for the value ε = 1. The cubic approximate analytical solution is: xapp (t) = 1 − 0.460506t 2 + 0.0394723t 3. The following plot contains the graphical representation of this polynomial (solid line) together with the corresponding numerical solution of (8) computed using Mathematica 6 (dashed line) (Fig. 5).

Approximate Polynomial Solution of a Nonlinear Differential Equation...

175

1.0

0.5

0.2

0.4

0.6

0.8

1.0

−0.5

−1.0

Fig. 6 Graphical representation of the error for the case of a cubic polynomial approximation

Fig. 7 Comparison between the exact solution (dashed line) and the fifth degree polynomial approximation (solid line)

Next we present the graphical representation of the error R in Fig. 6. We can observe that the maximal value of the error in this case is less than 0.1. The fifth degree polynomial approximate analytical solution is: xapp (t) = 1 − 0.717481t 2 + 0.599461t 3 − 0.5108754t 4 + 0.12751t 5 . The following plot contains the graphical representation of this polynomial (solid line) together with the corresponding exact solution of (8) (dashed line) (Fig. 7). It can be observed again that the graphical representations are practically overlapping.

176

C. Bota et al.

0.002

0.2

0.4

0.6

0.8

1.0

−0.002

−0.004

−0.006

Fig. 8 Graphical representation of the error for the case of a fifth degree polynomial approximation

The graphical representation of the error R is given in Fig. 8. We can observe that the maximal value of the error in this case is less than 0.005.

4 Conclusions The examples presented here, with applications in nonlinear mechanics, show that our method performs well for equations such as the Duffing equation and Van der Pol equation. A smaller value of the approximation error can be obtained by using a larger degree of the approximating polynomial. In this paper we used cubic and fifth degree, but if a higher precision is needed, polynomials of higher degree can be computed. Moreover, this method can be easily extended for other types of equations and systems of equations, such as nonlinear systems, nonlinear differential systems, integral equations, and as such it can be considered a powerful tool for the computation of approximate solutions for nonlinear problems.

References 1. Abazari N, Abazari R (2009) Solution of nonlinear second-order pantograph equations via differential transformation method. World Acad Sci Eng Technol 58:1052–1056 2. Ahmad JH (2009) Efficiency of differential transformation method for Genesio system. J Math Stat 5(2):93–96 3. Al-Hayani W, Casasus L (2006) On the applicability of the Adomian method to initial value problems with discontinuities. Appl Math Lett 19(1):22–31

Approximate Polynomial Solution of a Nonlinear Differential Equation...

177

4. Belendez A, Hernandez A, Belendez T (2007) Application of He’s homotopy perturbation method to the Duffing harmonic oscillator. Int J Nonlinear Sci Numer Simul 8:79–88 5. Bhatti MI, Bracken P (2007) Solutions of differential equations in a Bernstein polynomial basis. J Comput Appl Math 205:272–280 6. Bota C, Cˇaruntu B, Babescu M (2008) Analytic approximate periodic solutions based on harmonic analysis. In: Proceedings of the 10th international symposium on symbolic and numeric algorithms for scientific computing SYNASC 2008, Washington, DC, pp 177–182 7. Chena B, Garcia-Bolos R, Jodar L, Rosello MD (2005) Chebyshev polynomial approximations for nonlinear differential initial value problems. Nonlinear Anal TMA 64:629–673 8. D’Ambrosio R, Ferro M, Paternostera B (2009) Two-step hybrid collocation methods for y = f(x, y). Appl Math Lett 22(7):1076–1080 9. Ebaid AE, Ali E (2009) On a new aftertreatment technique for differential transformation method and its application to nonlinear oscillatory systems. Int J Nonlinear Sci 8(4):488–497 10. Ert¨urk VS (2007) Solving nonlinear fifth-order boundary value problems by differential transformation method. Selc¸uk J Appl Math 8(1):45–49 11. Ert¨urk VS (2007) Application of differential transformation method to linear sixth-order boundary value problems. Appl Math Sci 1(2):51–58 12. Ert¨urk VS (2007) Differential transformation method for solving differential equations of Lane-Emden type. Math Comput Appl 12(3):135–139 13. Funaro D (1992) Polynomial approximations of differential equations. Springer, Berlin 14. Gulsu M, Sezer M (2006) A Taylor polynomial approach for solving differential-difference equations. J Comput Appl Math 186:349–364 15. Hassan IHA-H, Ert¨urk VS (2007) Applying differential transformation method to the onedimensional planar Bratu problem. Int J Contemp Math Sci 2(30):1493–1504 16. He JH (2008) Recent development of the homotopy perturbation method. Topological methods in nonlinear analysis. J Juliusz Schauder Cent 31:205–209 17. Iscan F, Ongun MY (2009) A numerical method for solving a model for HIV infection of CD4+ cells. In: Proceedings of the 12th symposium of mathematics and applications, “Politehnica” University of Timis¸oara, 5–7 Nov 2009, pp 381–385 18. Liao SJ (2004) An analytic approximate solution for free oscillation of self-excited systems. Int J Nonlinear Mech 39:271–280 19. Odibat ZM, Bertelle C, Aziz-Alaoui MA, Duchamp GHE (2010) A multi-step differential transform method and application to non-chaotic and chaotic systems. Comput Math Appl 50(4):1462–1472 20. Sahmsul Alam M, Abul Kalam Azad M, Hoque MA (2006) A general Struble’s technique for solving a nth order weakly non-linear differential system with damping. Int J Nonlinear Mech 41:905–918 21. Shawagfeh N, Kaya D (2004) Comparing numerical methods for the solutions of systems of ordinary differential equations. Appl Math Lett 17(3):323–328 22. Von Stryk O (1993) Numerical solutions of optimal control problems by direct collocation. In: Optimal control, international series in numerical mathematics, vol 111. Birkhauser, Basel, pp 129–143 23. Zhan-Hua Y (2008) Variational iteration method for solving the multi-pantograph delay equation. Phys Lett A 372:6475–6479

Dynamical Symmetries of Second Order ODE M.I. Timoshin

Abstract The chapter considers the dynamical symmetry usage to the integration of ODE. Symmetries with invariants guaranteeing the lowering of the ODE order are suggested. These symmetries include the whole class of point symmetries. The procedure for the dynamical symmetries finding is demonstrated. Concrete examples of the using of dynamical symmetries are given. The chapter considers the application of the obtained solutions to the investigation of the nonlinear heat conduction equation.

1 Introduction The concept of dynamic symmetry is given, for example, [1]. The differential equation, in this case, is replaced by ODE system of the first order:   dy dz = z, = f (x, y, z) . y = f x, y, y ⇔ dx dx

(1)

Then it is possible to consider the question of an infinitesimal transformation X = ξ (x, y, z)

∂ ∂ ∂ + η (x, y, z) + μ (x, y, z) , ∂x ∂y ∂z

(2)

transforming a solution of the system (1) again into the solution of the same system. Thus the operator (2) should satisfy the condition [X, A] = λ (x, y, z) A, where A =

∂ ∂x

(3)

+ z ∂∂y + f (x, y, z) ∂∂z .

M.I. Timoshin () Ulyanovsk State Technical University, Ulyanovsk, Russian Federation e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 9, © Springer Science+Business Media, LLC 2012

179

180

M.I. Timoshin

It is important to note that the components of the operator (2) do not satisfy the prolongation formula suggested by S. Lie

μ=

dξ dη −z . dx dx

The components ξ , η , μ of the dynamic symmetries operator X are defined only by the condition (3). On application of point symmetries of ODEs, the invariants are constructed with the found operator. These invariants are applied for the lowering of the original ODE order. For this aim it is necessary to solve ODEs of the first order. It is known that in some cases the problem of the invariants finding is equivalent to the problem of the original DE integration . In the case of dynamic symmetries in the general case it would be necessary to solve a dynamic system. In the work [2] it is suggested to begin the procedure of finding the symmetries with the invariants, where the components of the relevant operator can be written out directly with the help of differentiation and arithmetical operations. The disadvantage of such an approach consists in the fact that the linear in the classic case defining system of DEs becomes nonlinear. However such an approach in the case of dynamic symmetries allows to take care of properties of the searched symmetries beforehand. In particular it is natural to require that if the functions u = u (x, y, y ) , v = v (x, y, y ) are the invariants of the operator (2), the expression      du ux + uy y + uy y =  dv vx + vy y + vy y

would also be invariant of once prolonged dynamic symmetry (2). It is more convenient to write the components of the dynamic symmetry operator in the form X =ξ

  ∂ ∂ ∂ ∂ , +η + μ  + μ1 x, y, y , y ∂x ∂y ∂y ∂ y

(4)

and to determine them by solving the system of equations X u = 0, X v = 0, X

du = 0. dv

(5)

At the finding of the symmetries for ODE of the second order   F x, y, y , y = 0, the invariance criterion XF |F=0 ≡ 0,

(6)

Dynamical Symmetries of Second Order ODE

181

can be used. In the general case, this criterion (6), as well as the invariance criterion (3), does not allow the splitting. The advantage of invariance criterion (6) consists in the fact that it allows to find the dynamic symmetry (4) satisfying (5) up to the constant functional factor. Note, that all point symmetries can be considered as ansatze of dynamic ones when u=

βx + βy y , v = β (x, y) , αx + αy y

where α (x, y) , β (x, y) are arbitrary functions. Thereupon with intending to preserve the splitting property for the criterion (6), it is natural to state the question about dynamic symmetries as the question about the extension of the point symmetries set restricting oneself by the functions of two variables. Consider now the question about dynamic symmetries spanned on three functions of two variables containing the whole set of point symmetries. First of all note the point symmetry operator X = ξ (x, y)

∂ ∂ + η (x, y) , ∂x ∂y

having the invariant v = τ (x, y) , in variables (v, y) , takes the form X = χ (v, y)

∂ . ∂y

Representing the function χ (v, y) in the form χ (v, y) =

1 αy ,

the first differential

invariant can be written out u = αv + αy dy dv . Thus taking the first differential invariant dy in the form u = α + β dv , it is possible to extend the point symmetry set with the

help of three functions τ (x, y) , α (v, y) , β (v, y) . The practical finding of the considered type of dynamic symmetries is convenient with the help of several steps: 1. Making the point variables change t = τ (x, y) , y = y, it is necessary to pass from the equation   dy d 2 y F t, y, , 2 = 0 dt dt to the equation   dy d 2 y Φ x, y, , 2 = 0. dx dx

182

M.I. Timoshin

2. On the invariants dy , dx solving the system (5), to construct the dynamic symmetry operator. 3. Splitting by y the invariance criterion v = x, u = α (x, y) + β (x, y)

X Φ |Φ =0 ≡ 0 to write the determining system of differential equations. 4. To resolve the determining system. In the work [2], by the example of the equation y = y + f (y) ,

(7)

the realization of 2, 3, and 4 steps is given. It is shown that the ODE (7) allows the dynamic symmetries, when the function f (y) has the form f (y) = Ay +

B B2 − . y y3

Explicit solutions of corresponding equations are given. In the following section, by the example of (7) the realization of all four steps is considered.

2 Dynamic Symmetries of Second Order ODE Take the (7) in the form d 2 y dy + f (y). = dt 2 dt Making the point change of variables t = τ (x, y), come to the differential equation   3 2 2  τx y − f (y)τx − τxx + τx + 3 f (y)τx τy y     2 2 2 3 3   − 2τxy + 2τx τy + 3τx τy f (y) y − τyy + τy + f (y)τy y = 0. With the invariants v = x,

u = α (x, y) + β (x, y)y ,

  du = αx + βx + αy y + βy y 2 + β y dv

construct the dynamic symmetry operator (4) with components

(8)

Dynamical Symmetries of Second Order ODE

183

ξ (x, y, y ) = 0, η (x, y, y ) = 1, μ (x, y, y ) = − μ1 (x, y, y , y ) = − +

αy + βy y , β

 β + 2β  2  β + 3α  β  + β  β  − β  β −αyy βy  −βyy y 2 y y x y xy y + y + y β β2 β2  β + β  α  + α  2 −αxy x y y . β2

Splitting the invariance criterion X Φ |Φ =0 ≡ 0 by y , get the determining system of differential equations   3 2 3      2    τx + τxy τx + 3 f  (y)τx τy + 3 f (y)τx τy τxy + 3 f (y)τx τyy − τxy τxx −β 2 τxxy   2 2    + τx 4αy β τy + 6αy β f (y)τy − αyy β + 3βy αy − βxy β + βx βy = 0, +4αy β τx τxy   2 2 2     2   2 β 2 −2τxyy τx − 2τyy τx − 3 f  (y)τx τy − 6 f (y)τx τy τyy + 2τxy   2 3   + 3αy τy + 3αy f (y)τy + 2βy τxy +β τx 3αy τyy   2 2 2  +τx 2β βy τy + 3 f (y)β βy τy − βyy β + 2βy = 0,   3 3     2        2 β τxy τyy + τxy τy + τxy f (y)τy − τyyy τx − 2τx τy τyy − f  (y)τy τx − 3 f (y)τyy τx τy   2 3  +2τx βy τyy + τy + f (y)τy = 0,   + αy β τxx + αy β τx + 3αy β τx f (y)τy − f (y)β βy τx −β 2 τx f  (y) − 2τx β 2 f (y)τxy 3

2

2

2

3

 −αxy β τx + αy βx τx + τx αy = 0, 2

which is the PDEs system of four equations relative to three unknown functions α (x, y), β (x, y), τ (x, y). We will try solutions of this system with the group analysis method for PDEs, presented, for example, in [3, 4]. Considering the determining system as the system relative to the function τ (x, y), see that it has the symmetry X=

∂ ∂ +a , ∂x ∂τ

when the functions α (x, y), β (x, y) take the form

α (x, y) = α1 (x) + α2 (y)β1 (x),

β (x, y) = β1 (x)β2 (y).

The obtained symmetry prescribes the finding of the invariant solution in the form

τ (x, y) = ax + τ1 (y).

184

M.I. Timoshin

Using the written representations for the functions τ (x, y), α (x, y), β (x, y), transform the determining system to the form   2 2 −3aβ22 f  (y)τ1 + 3β2 f (y)τ1 aβ2 τ1 − 2aβ2τ1 + α2 τ1       2 2 +2a β2 − β22 τ1 + β2β2 τ1 + β2 3α2 τ1 + τ1 − aβ2 = 0,  2 2β2 τ1 − 3β2τ1 − β2 τ1 − 2β2τ1 τ1 + 2β2 τ1 + 2β2 τ1 = 0,     2 −3aβ22 f  (y)τ1 − 3β2 f (y) aβ2 τ1 − 2α2 τ1 − α2 β2 − 3α2 β2 − 4α2 τ1 β2 = 0,     a2 β2 2 f  (y) + aβ2 f (y) aβ2 − 3α2 τ1 − α2 α2 + aβ2 = 0.

−β2 f  (y)τ1 + f (y)τ1 3

2

Analyzing the obtained ODE system, find the functions f (y) , for which the equation d 2 y dy = + f (y) dt 2 dt allows the dynamic symmetry of the required type. Let us note here several parametrically given functions f (y): 1.

1

f (p) = −

e− 2p+1 (2p + 3)2 , 8 f1 (2p + 1)

1

y(p) =

4e− 2p+1 (p + 1) + f2; f1 (2p + 1)

2. 1

− e 2p+1 ((3 f1 +2 f1 p)2 −(2p+1)2 ) f2

1

,

y(p) =

− 4e 2p+1 (p+1) f 2 f 1 (2p+1)

− e 2p+1 ((3 f1 +2 f1 p)2 +(2p+1)2 ) f2 − , 8 f 1 3 (2p+1)

y(p) =

− 4e 2p+1 (p+1) f 2 f 1 (2p+1)

f (p) = −

8 f1

3 (2p+1)

+ f4 ;

3. 1

f (p) =

1

+ f3 ;

4. f (p) =

pk1 (−k1 p2 +p+2pk1 +2 f 1 f 2 ) , 2 f 2 (p−1)

y(p) = −

f (p) =

pk1 (2p2 f 3 +2pk1 +p−1−k1 ) , 2 f 2 (p−1)

y(p) = −

(2k1 +1)2 (k1 −pk1 +1)pk1 2 f 2 k1 (k1 +1)

+ f3 ;

(2k1 +1)2 (k1 −pk1 +1)pk1 2 f 2 k1 (k1 +1)

+ f4 ;

5.

6.

f1 (p + 2)2 , f (p) = (p + 1)

 y(p) = −4 f1

 p 1 + f3 ; ln + p+1 p+1

Dynamical Symmetries of Second Order ODE

7. f (p) = 8. f (p) =

f 1 (ph+2+p)(ph−2−p) , (p+1)(h2 −1) f 1 ( p2 h2 +p2 +4p+4) (p+1)(h2 +1)

,

9. f (p) = f1 (p + 1) − f2 ,

185

y(p) = y(p) = −

4 f1 (h2 −1) 4 f1 (h2 −1)

 y(p) = f 2 ln



p ln p+1 +



p ln p+1 +

1 p+1

1 p+1

 

+ f3 ; + f3 ;

 p 1 + f3 ; + p+1 p+1

where p, g -parameters, f1 , f2 , f3 , f4 , k1 , h are arbitrary constants. It is interesting to note that variants 4 and 5, for some values of arbitrary constants, allow the explicit representation of the functions f (y). Thus it is possible to obtain both known and new cases of (7) integration. The usage of the dynamic symmetry together with general solutions for the indicated nine cases are given in the following section.

3 Integration of Second Order ODE Consider the procedure of dynamic symmetries usage for (7) with the first type function f (y). Taking into account that the function τ1 (y) here is representative in the parametric form 2 , τ1 (p) = − 2p + 1

4e− 2p+1 (p + 1) y(p) = + f2 , f1 (2p + 1) 1

pass in (8) from the function y(x) to the function p(x). After transformation, the following equation is obtained   3 2 32a(2p + 1)2 p + 64p + 48a(2p + 1)2 p + 4a2 12p2 + 36p + 19 (2p + 1)2 p +a3 (2p + 3)2 (2p + 1)4 = 0.

(9)

Invariant u takes the form u=

2a f1 2 (2p + 1)2 − 2a2 p + 8 f12 p − a2 . a (2p + 1)

(10)

Using this invariant as a new variable, lower the order of equation (9), obtaining the equation       2 32 f1 4 u + au3 + a 8 f1 2 + 3a u2 + a a + 4 f12 3a + 4 f12 u + a2 a + 4 f12 = 0. (11)

186

M.I. Timoshin

Integrating (11), it is not difficult to write out the general solution of (9) e

p= 2

− g2

 e− g2 (g−1) g2

dg + C2

1 − , 2

x=

2 − 2g + 2g lng + C1 . ag

(12)

Taking into account the following connection between variables (p, x) and (y,t) y=

4e

1 − 2p+1

(p + 1) + f2 , f1 (2p + 1)

t = ax −

2 , 2p + 1 2

it is possible to obtain a general solution of the equation ddt 2y = dy dt + f (y) with the function f (y) of the first type. Given here are general solutions with functions f (y) of other types: 1.

1

4e− 2p+1 (p + 1) f2 y= + f4 , f1 (2p + 1)

t = x−

2 , 2p + 1

where p = − 12 +

g f1 C2 −

2.

 g f1 (g+ f1 g+1− f1 ) g(g−1)

2

x = − f1 ln g − ln (g−1) + C1 . g

, dg

1

y=

4e− 2p+1 (p + 1) f2 + f3 , f1 (2p + 1)

t = x−

2 , 2p + 1

where p = − 12 +

3. y=−

e−2 f1 arctan g  −2 f arctan g 2 C2 +2( f1 +1) e 1 dg ( f1 −g)(1+g2 )

,

x = ln

(2k1 + 1)2 (k1 − pk1 + 1) pk1 + f3 , 2 f2 k1 (k1 + 1)

(1+g2 ) (g− f 1 )2

+ 2 f1 arctan g + C1 .

t = x + (2k1 + 1) ln p,

by denoting m = 2k1 + 1, this solution takes the form e8 f 2

p= 8m f2 2

2 2  mg−4 f2 dg gB

 mg−4 f2 2 dg 8 f2 2

 e

gB

B

, dg + C2

x = 32 f2 4 m



dg + C1 , gB

where B = g2 m2 (m + 1 + 4 f 1 f2 ) − 8m f22 g − 16 f24 (m − 1).

Dynamical Symmetries of Second Order ODE

4. y=−

187

(2k1 + 1)2 (k1 − pk1 + 1) pk1 + f4 , 2 f2 k1 (k1 + 1)

t = x + (2k1 + 1) ln p,

by denoting m = 2k1 + 1, this solution takes the form ⎛ p = − ⎝8m f2 2 x = −32 f2 4 m





e

−8 f2 2

 mg+4 f 2 2 dg gB

B

⎞ dg + C2⎠ × e8 f2

2 2  mg+4 f2 dg gB

,

dg + C1 , gB

where B = g2 m2 (−m + 1 − 4 f3) + 8m f22 g + 16 f24 (m + 1).  y = −4 f1 ln

5.

 p 1 + f3 , + p+1 p+1

t = x + ln

p , p+1

where p = −1 +

8 f1 2 eg ,   geg (1 − g) 8 f1 2 2 dg + C2

x = C1 − g − ln(1 − g).

(g−1)

  p 1 4 f1 ln + f3 , + y= 2 (h − 1) p+1 p+1

6.

t = x + ln

p , p+1

where h+1

16hg 2h f1 2 p = −1 −  ,  g 1−h 2h (1+g) 2 2 B 8 f1 (h − 1) dg + C2 B2 x = C1 + 7. y=−

(1 − h2 )

 1+g

2h

gB

dg

, B = gh − h − 1 − g.

  4 f1 p 1 ln + f3 , + (h2 + 1) p+1 p+1

t = x + ln

p , p+1

188

M.I. Timoshin

where p = −1 +

 arctan g 8 f 1 2 h 1 + g2 e− h ,  (1 − gh) 8(h2 + 1) f1 2 B + C2

2 arctan g + h ln(1 + g2 ) − 2h ln(gh − 1) , 2h  arctan g 1 + g2 e− h dg. B= h2 g4 − 2hg3 + g2 − 2hg + 1 + h2g2 x = C1 +

 y = f2 ln

8.

 1 p + f3 , + p+1 p+1

t = x + ln

p , p+1

where f1

p=−

g f2 

1+ f g

−1 − f g1 e 2

1+ f1 g f1 − (1+ f1 g)g f2 e f2 g f 2 g3

,

x = C1 +

dg − C2

1 + f1 g(1 − lng) . f2 g

The possibilities of practical applications of indicated solutions are given in the following section.

4 Traveling Wave in Nonlinear Heat Conduction In the work [5], it is indicated that (7) is connected with the Kolmogorov-PetrovskyPiskunov diffusion equation and with the Semyonov equation (Fitz-Hugo-Nagumo) in the theory of the chain chemical reaction. Consider the nonlinear heat conduction equations

∂ u ∂ 2u − − F(u) = 0. ∂ τ ∂ x2

(13)

If the function F(u) satisfies the conditions F(0) = F(1) = 0,

F  (0) = α > 0,

F  (u) < α ,

0 < u < 1,

(14)

we can say about the Kolmogorov-Petrovsky-Piskunov (K-P-P) equation. If the function F(u) satisfies the conditions F(0) = F(1) = 0, F  (0) ≤

0,

F(a1 ) = 0,

F  (1) <

then it is said to be the Semyonov equation.

0,

0 < a1 < 1,

F  (a

1)

> 0,

(15)

Dynamical Symmetries of Second Order ODE

189

0.02

0.015

0.01

0.005

0

0.2

0.4

0.6

0.8

1

Fig. 1 Example of function F(u) for K-P-P equation

Let us find the traveling wave – type solutions for (13) u(x, τ ) = y(ρ ),

ρ = x + bτ .

transferring to the equation d2y dy b − 2 − F(y) = 0. dρ dρ

(16)

After the change of variables t = bρ , one has d 2 y dy F(y) − 2 . = dt 2 dt b Thus we obtain the equation of the type (7) under the condition  f (y) = −F(y) b2 . Let us show now that the fifth type of the function f (y) allows to write out the exact solutions of the K-P-P and Semyonov equations. 3 1 Setting the values of the constants k1 = 1, f2 = 96, f3 = 10 , f4 = 500 , we obtain f (p) =

p(3p2 + 15p − 10) , 960(p − 1)

y(p) = −

1 3 3 2 p+ p + . 64 128 500

The relevant type of the functions F(y) up to the expansion on the axis F has the form Fig. 1.

190

M.I. Timoshin

0.04

0.03

0.02

0.01

0

0.2

0.4

0.6

0.8

1

–0.01

Fig. 2 Example of function F(u) for Semyonov equation

Setting the values of the constants k1 = 1, f2 = 10, f3 = obtain f (p) =

p(197p2 + 150p − 100) , 1,000(p − 1)

y(p) = −

197 100 ,

f4 =

149 1,000 ,

we

9 9 149 p + p2 + . 20 40 1,000

The relevant type of the functions F(y) up to the expansion on the axis F has the form Fig. 2. Note that the fifth type of the function f (y) representation by means of the constants change allows describing a wide range of solutions for the KolmogorovPetrovsky-Piskunov and Semyonov equations.

References 1. Stephani H (1989) Differential equations: their solution using symmetries. Cambridge University Press, Cambridge 2. Timoshin MI (2009) Dynamic symmetries of ODEs. Ufim Math J 1(3):132–138 3. Ibragimov NH (1985) Transformation groups applied to mathematical physics. Reidel, Dordrecht 4. Ovsjannikov LV (1982) Group analysis of differential equations. Academic, New York 5. Berkovich LM (1992) Factorization as the method of the finding of exact invariant solutions for Kolmogorov-Petrovsky-Piskunov equation and connected with it Semyonov and Zeldovich equation. DAN 322(5):823–827

Invex Energies on Riemannian Manifolds Constantin Udris¸te and Andreea Bejenaru

Abstract The problems tackled in this chapter are directly or indirectly concerned with new trends in multitime optimal control. More precisely, we analyze the Riemannian convexity of energy functionals connected to the volumetric energy and the kinetic energy. The tools we use are those not only of Riemannian geometry, but also of variational calculus and geometric dynamics. Some of the Lagrangians we discuss about are obtained by considering Euler–Lagrange prolongations of a PDE system of order one, such that the solutions of the prolongation are ultrapotential maps. The problems solved here include: Riemannian convexity of kinetic energy functional, convex functions generated by convex functionals, invexity of kinetic and volumetric energy functionals, invexity for least square Lagrangians, etc. A major result of this theory consists in relating the convexity of the energy functionals with the geometry of the underlying manifolds. The main key for our original approach is the new notion of sub-Killing vector fields on Riemannian manifolds.

1 Introduction Surely the theory of minimal submanifolds [3–5] and the theory of harmonic maps [2] are among the simplest and yet general intrinsic variational problems of Riemannian geometry. Being involved in the subject, our research group in University Politehnica of Bucharest changed recently the direction of research from static variational calculus to optimal dynamics, formulating and studying

C. Udris¸te () · A. Bejenaru Faculty of Applied Sciences, Department of Mathematics-Informatics I, University Politehnica of Bucharest, Splaiul Independentei 313, Bucharest, 060042, Romania e-mail: [email protected]; [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 10, © Springer Science+Business Media, LLC 2012

191

192

C. Udris¸te and A. Bejenaru

multitime optimal control problems whose solutions are minimal submanifolds or harmonic (potential) maps (see [14–26]). This direction gives new interpretations and geometric descriptions for the optimal solutions and creates generalizations. Another dynamic approach of minimal submanifolds appeared in [7, 9]. This chapter provides a general principle how to define energies for manifoldvalued mappings between Riemannian manifolds and for least squares approximations of maps. It has three roots: minimal submanifolds theory, harmonic or potential maps theory and geometric dynamics [12, 13, 18, 23, 25]. The first idea, in Sect. 2, is to define and study the smallest volumetric energy submanifolds. Passing from the volume functional to the volumetric functional has practical reasons, since the latter refers to a smooth Lagrangian which is important in the areas of molecular engineering and materials sciences due to anticipated nanotechnology applications. The second idea refers to the kinetic energies and some energy functionals used to extend PDE systems of order one to Euler–Lagrange PDE systems of order two. The idea of considering Euler– Lagrange prolongations of PDE systems of first order was announced in [10, 12] and analyzed afterwards in [8, 12, 25]. We reconsider this problem, giving original proofs. Moreover, we also consider a new extension and we prove that there are certain geometric configurations such that a solution of the constraint PDE system minimizes the total energy. Section 1 contains some historical and bibliographical notes. Section 2 defines and studies the volumetric energy, the kinetic energy, and least squares type energies. Section 3 uses geodesic deformations to study the Riemannian convexity of functionals. Section 4 analyzes the Riemannian invexity of energies. Section 5 contains open problems regarding thin-plate spline energy, pairs of type (Lagrangian, Hamiltonian), and energy–momentum tensor field. The invexity of the functionals is strongly related to the invexity of the Lagrangians. For energies, the invexity requires the existence of the sub-Killing vector fields, while convexity needs stronger sufficient conditions.

2 Energies on Riemannian Manifolds In this section, we use a compact m-dimensional Riemannian manifold (N, h) and an n-dimensional Riemannian manifold (M, g). Lemma 1. Let M1 and M2 be two differentiable manifolds. If T ∈ T p0 (M2 ) is a tensor field on M2 , x : M1 → M2 is a differentiable map, and ϕ : M1 × (−δ , δ ) → M2 is a deformation of x(·), then d ∗ ϕ T |ε =0 = x∗ (Y (T )), dε ε

(1)

Invex Energies on Riemannian Manifolds

193

where ϕε : M1 → M2 , ϕε (t) = ϕ (t, ε ), Y = ϕ∗ ∂∂ε and Y (T ) denotes the Lie derivative of the tensor field T with respect to the vector field Y . In particular, the previous result is valid for differential forms. Lemma 2. With the same hypotheses as above, the vector fields Y = ϕ∗ ∂∂ε and Zγ = ϕ∗ ∂∂t γ satisfy the relations [Y, Zγ ] ◦ x(·) = 0, ∀γ ∈ 1, m.

(2)

Remark 1. The immediate consequence of the previous two results is that d ∗ [ϕ T ]α1 ...α p (t)|ε =0 = Y (T (Zα1 , ... , Zα p ))(x(t)), ∀t ∈ M1 . dε ε The basic energy functionals we shall repeatedly use next are contained in the following two definitions. Definition 1. Let x : N → M be a fixed m-sheet. 1. The multiple integral J[x(·)] =

1 2

 N

(det(x∗ g))(t)dt

(3)

is called the volumetric energy of the submanifold map x(·). 2. The multiple integral 1 J[x(·)] = 2

 N

 Trh (x∗ g)(t) h(t)dt

(4)

is called the kinetic energy of the submanifold map x(·). New important energies are used to build Euler–Lagrange prolongations of a PDE system of order one or least squares approximations of maps (see [8,12,13,18, 24]). To describe them, we start with a distinguished tensor field X = Xαi (t, x) ∂∂xi ⊗ dt α on the product manifold N × M, satisfying the complete integrability conditions i i ∂ Xαi ∂ Xαi j ∂ Xβ ∂ Xβ j + X = + Xα , ∂xj β ∂ tα ∂xj ∂ tβ

and we consider the PDE system of order one

∂ xi (t) = Xαi (t, x(t)), ∂ tα where x(·) denotes an m-sheet, that is, x : N → M.

(5)

194

C. Udris¸te and A. Bejenaru

From now on, we treat N and M as the horizontal, respectively, the vertical components of the product manifold N × M and we identify the lifts of vector fields with the original vector fields, using the notations U,V,..., for the horizontal ones, and Y, Z,..., for the vertical ones. We also denote by ∇ the Levi–Civita connection on N × M. We introduce a (1,2)-tensor field F : X (N) × X (M) → X (M), g(F(U)(Y ), Z) = g(∇Y (X(U)), Z) − g(∇Z (X(U)),Y ),

(6)

∀U ∈ X (N), ∀Y, Z ∈ X (M). If x(·) is a solution of the PDE system (5), we introduce the map x : N → N × M, x(t) = (t, x(t)). Our purpose is to extend the system (5) to a PDE system of order two representing the Euler–Lagrange equations associated to an appropriate action. The first step is to consider the PDEs (5) as a relation between tensor fields along x(·) on N × M, that is: x∗ ◦ x(·) = X ◦ x(·), where x∗ is the differential of the map x(·). If U ∈ X (N), then x∗ (U) = (U, x∗ (U)) and when we differentiate the previous relation with respect to U, we obtain 0 = g(∇U [(x∗ − X)(V )],Y ) − g((x∗ − X)(∇U V ),Y ) +g(∇x∗U x∗V,Y ) − g(F(V )(x∗U),Y ) − g(∇Y (X(V )), x∗U).

(7)

Successively, we modify the last two terms in the PDE system (7), obtaining three extensions: 0 = g(∇U [(x∗ − X)(V )],Y ) − g((x∗ − X)(∇U V ),Y ) +g(∇x∗U x∗V,Y ) − g(F(V )(x∗U),Y ) − g(∇Y (X(V )), X(U));

(8)

0 = g(∇U [(x∗ − X)(V )],Y ) − g((x∗ − X)(∇U V ),Y ) +g(∇x∗U x∗V,Y ) − g(F(V )(X(U)),Y ) − g(∇Y (X(V )), X(U));

(9)

0 = g(∇U [(x∗ − X)(V )],Y ) − g((x∗ − X)(∇U V ),Y ) +g(∇x∗U x∗V,Y ) − g(F(V )(X(U)),Y ) − g(∇Y (X(V )), x∗ (U)).

(10)

Let E denote the set of differentiable m-sheets from N to M and J : E → IR, x(·) → J[x(·)] be a functional. If x(·) ∈ E and ϕ : N × (−δ , δ ) → M is a deformation of x(·), let Y ∈ Xx(·) (M) be the infinitesimal deformation of ϕ along x(·). Definition 2. The operator dJx(·) : Xx(·) (M) → IR, dJx(·) [Y ] = is called the differential of the functional J at x(·).

d J[ϕ (·, ε )]|ε =0 dε

Invex Energies on Riemannian Manifolds

195

Theorem 1. The solutions of the PDE system (8) are critical points for the functional J[x(·)] =

1 2

 N

 j j hαβ (t)gi j (x(t))[xiα (t) − Xαi (t, x(t))][xβ (t) − Xβ (t, x(t))] h(t)dt. (11)

Proof. If ϕ : N × (−δ , δ ) → M is a deformation of x(·), then ϕ : N × (−δ , δ ) → N × M, ϕ (t, ε ) = (t, ϕ (t, ε )) is a deformation of x. It follows that   ∂ ∂ Y = ϕ∗ = 0, ϕ∗ = (0,Y (ϕ (t, ε ))), ∂ε ∂ε that is, Y can be identified with the vector field tangent to the deformation ϕ on M. We also consider the vector fields Zα = ϕ∗

∂ ∂tα

and

∂ = (∂α , Zα ). ∂ tα If we take U = ∂α , V = ∂β and Y = ∂i in (8), then we obtain Zα = ϕ ∗

γ

0 = g(∇∂α (Zβ − Xβ ), ∂i ) − Hαβ g(Zγ − Xγ , ∂i ) + g(∇Zα (Zβ − Xβ ), ∂i ) +g(∇∂i Xβ , Zα − Xα ). Now, since the critical points of the functional J satisfy the condition dJx(·) [Y ] = 0, ∀Y ∈ Xx(·) (M), we need to compute this differential. Using Lemmas 1 and 2 and the remark derived from them, we obtain  √ 1 d hαβ h[(x∗ g)αβ − 2x∗ (g(·, Xβ ))α + g(Xα , Xβ ) ◦ x]dt|ε =0 2 dε N  √ 1 = hαβ h[Y (g(Zα , Zβ )) − 2Y (g(Zα , Xβ )) + Y (g(Xα , Xβ ))]dt 2 N  √ 1 hαβ h[Y (g(Zα − Xα , Zβ − Xβ ))]dt = 2 N  √ = hαβ h[g(∇Y (Zα − Xα ), Zβ − Xβ )]dt

dJx(·) [Y ] =

N

= =



N



N

√ hαβ h[−g(∇Y Xα , Zβ − Xβ ) + g(∇Zα Y, Zβ − Xβ )]dt √ hαβ h[−g(∇Y Xα , Zβ − Xβ ) + Dα (g(Y, Zβ − Xβ ))

−g(Y, ∇Zα (Zβ − Xβ )) − g(Y, ∇∂ α (Zβ − Xβ ))]dt

196

C. Udris¸te and A. Bejenaru

=

 N

√ {hαβ h[−g(∇Y Xα , Zβ − Xβ ) − g(Y, ∇Zα (Zβ − Xβ ))

√ −g(Y, ∇∂ α (Zβ − Xβ )] + Dα [hαβ hg(Y, Zβ − Xβ )] √ −Dα (hαβ h)g(Y, Zβ − Xβ )}dt  √ hαβ h[−g(∇Y Xα , Zβ − Xβ ) − g(Y, ∇Zα (Zβ − Xβ )) = N

γ

−g(Y, ∇∂ α (Zβ − Xβ )) + Hαβ g(Y, Zγ − Xγ )]dt = 0, ∀Y ∈ Xx(·) (M). Therefore, the Euler–Lagrange PDEs write 0 = hαβ [g(∇∂i Xα , Zβ − Xβ ) + g(∂i, ∇∂α (Zβ − Xβ )) γ

+g(∂i , ∇Zα (Zβ − Xβ )) − Hαβ g(∂i , Zγ − Xγ )].



Remark 2. 1. If the complete integrability conditions of the PDE system (5) are not satisfied (i.e., the system has no solution), then the Euler–Lagrange PDEs (8) are not prolongations. 2. The expression of the functional in Theorem 1 suggests the idea of considering a similar multiple integral defined using the determinant, that is, a volumetric type energy functional J[x(·)] =

1 2

 N

det(g˜αβ (t))dt,

(12)

where g˜αβ (t) = gi j (x(t))[xiα (t) − Xαi (t, x(t))][xβj (t) − Xβj (t, x(t))]. Theorem 2. If N is a one-dimensional or two-dimensional compact manifold (i.e., m = 1 or m = 2), then there are two Riemannian structures, h on N and g on M, and a family of potential maps Vαβ = Vαβ (t, x) on N × M such that 1. gradg (Vαβ ) = [∇∂α +Xα X](∂β ), ∀α , β = 1, m 2. The solutions of the extended system (9) are critical points for the functional J[x(·)] =



h N

αβ



  1 j i (t) gi j (x(t))xα (t)xβ (t) + Vαβ (t, x(t)) h(t)dt. 2

Proof. For the first statement of the theorem, we remark that we need to have + m(m+1) + m2 variables and solutions for a PDE system of first order with n(n+1) 2 2 m2 n conditions. If m = 1 or m = 2, then the number of the variables exceeds the number of constraints, therefore we have solutions. Otherwise, we have to give inferior limits for n; for example, if m = 3, then the dimension n of the manifold M must be at least 15.

Invex Energies on Riemannian Manifolds

197

Rewriting the second extension for U = ∂α , V = ∂β , Y = ∂i , we obtain γ

0 = g(∇∂α (Zβ − Xβ ), ∂i ) − Hαβ g(Zγ − Xγ , ∂i ) + g(∇Zα Zβ , ∂i ) − g(∇Xα Xβ , ∂i ). On the other side, dJx(·) [Y ] = =

 N



N

√ hαβ h[g(∇Y Zα , Zβ ) + g(gradVαβ ,Y )]dt √ √ γ Dα [hαβ hg(Y, Zβ )] + hαβ h[Hαβ g(Y, Zγ ) − g(Y, ∇∂α Zβ ) γ

−g(Y, ∇Zα Zβ ) + g(∇∂α Xβ ,Y ) − Hαβ g(Y, Xγ ) + g(∇Xα Xβ ,Y )]dt =



N

√ γ hαβ h[Hαβ g(Y, Zγ ) − g(Y, ∇∂α Zβ ) − g(Y, ∇Zα Zβ ) + g(∇∂α Xβ ,Y ) γ

−Hαβ g(Y, Xγ ) + g(∇Xα Xβ ,Y )]dt, ∀Y ∈ X (M). Consequently, the Euler–Lagrange PDEs are γ

0 = hαβ [g(∂i , ∇∂α (Zβ − Xβ ))−Hαβ g(∂i , (Zγ − Xγ )) + g(∂i , ∇Zα Zβ ) − g(∂i , ∇Xα Xβ)].

Open problem. Is there a Lagrangian having the Euler–Lagrange PDEs γ

0 = hαβ [g(∇∂α (Zβ − Xβ ), ∂i ) − Hαβ g(Zγ − Xγ , ∂i ) + g(∇Zα Zβ , ∂i ) −g(∇Xα Xβ , ∂i ) − g(∇∂i Xβ , Zα − Xα )] (the trace in relations (10))?

3 Riemannian Convexity of Energy Functionals The theory in this section extends some ideas, from functions to functionals, in the Riemannian language of the papers [1, 11]. For this, we need to define the geodesic deformation map. Definition 3. A deformation map ϕ : N × [0, 1] → M is called geodesic deformation if ϕ (t, ·) is a geodesic in (M, g) for each t ∈ N. Definition 4. A subset F ⊂ E is called totally convex if for all pairs of m-sheets x(·), y(·) ∈ F and all geodesic deformations ϕ : N × [0, 1] → M, ϕ (·, 0) = x(·), ϕ (·, 1) = y(·), we have ϕ (·, ε ) ∈ F, ∀ε ∈ [0, 1].

198

C. Udris¸te and A. Bejenaru

Definition 5. Let F ⊂ E be a totally convex subset of m-sheets. A functional J : F → IR is called Riemannian convex if J[ϕ (·, ε )] ≤ (1 − ε )J[x(·)] + ε J[y(·)],

(13)

for all x(·) and y(·) in F, for all geodesic deformations ϕ : N × [0, 1] → M connecting x(·) and y(·) and for all ε ∈ [0, 1]. Definition 6. The functional J is called Riemannian strictly convex if J[ϕ (·, ε )] < (1 − ε )J[x(·)] + ε J[y(·)],

(14)

for all x(·), y(·), ϕ as above, x(·) = y(·) and ε ∈ (0, 1). Lemma 3. The functional J : F → IR is convex (strictly convex) iff, for each geodesic deformation ϕ : N × [0, 1] → M lying in F, the function Jϕ : [0, 1] → IR, Jϕ (ε ) = J[ϕ (·, ε )] is convex (strictly convex) on the interval [0, 1].

3.1 Riemannian Convexity of Kinetic Energy Functional In this subsection, we formulate a simple sufficient condition for the convexity of the kinetic energy. Lemma 4. If the bilinear form

Ωx(·) = hαβ Hess(g(Zα , Zβ )) ◦ x

(15)

is positive semidefinite on Xx(·) (M), then the kinetic energy functional J is Riemannian convex at x(·). If Ωx(·) is positive definite, then J is strictly convex at x(·). Proof. We consider the function Jϕ : [0, 1] → IR, Jϕ (ε ) = J[ϕ (·, ε )]. The kinetic d2J

energy J is Riemannian convex at x(·) iff d ε 2ϕ |ε =0 ≥ 0. Furthermore, if then J is strictly convex at x(·). By direct computation we obtain 1 d2ϕ (0) = dε 2 2

 N

hαβ [X(X(g(Zα , Zβ )))]dt =

1 2

 N

d 2 Jϕ | > 0, d ε 2 ε =0

hαβ Hess(g(Zα , Zβ ))dt.



Theorem 3. If the curvature tensor field RX ∈ T20 (M), RX (U,V ) = R(X,U, X,V ) is negative semidefinite, ∀X ∈ X (M), then the kinetic energy functional is convex.

Invex Energies on Riemannian Manifolds

199

Proof. For x(·) ∈ E, we consider the family of vector fields Zα = ϕ∗ ∂∂t α , with ϕ a geodesic deformation of x(·). If X ∈ X (M) is the vector field tangent to this geodesic deformation, then Hess(g(Zα , Zβ ))(X, X) = X(X(g(Zα , Zβ ))) = g(∇X ∇X Zα , Zβ ) + 2g(∇X Zα , ∇X Zβ ) + g(Zα , ∇X ∇X Zβ ). On the other hand, since [X, Zα ] = 0, ∀α ∈ 1, m, and ∇X X = 0, it follows g(∇X ∇X Zα , Zβ ) = −R(Zβ , X, Zα , X) Hence Hess(g(Zα , Zβ ))(X, X) = −2R(Zβ , X, Zα , X) + 2g(∇X Zα , ∇X Zβ ). We also have and

hαβ R(Zβ , X, Zα , X) ◦ x = Trh (x∗ RX ) ≤ 0

hαβ g(∇X Zα , ∇X Zβ ) ◦ x = hαβ g(∇Zα X, ∇Zβ X) ◦ x = Trh (x∗ ΩX ),

where ΩX = g(∇X, ∇X) and it follows

Ωx(·) (X, X) ◦ x = −2Trh (x∗ RX ) + 2Trh (x∗ ΩX ). Proving the fact that ΩX is positive semidefinite, ∀X ∈ X (M), is the last step. Indeed, ΩX (Y,Y ) = g(∇Y X, ∇Y X) = ∇Y X2 ≥ 0 and, since RX ≤ 0, we conclude that Ωx(·) is positive semidefinite along x(·) and J is a convex functional.

Remark 3. If we repeat the arguments in Lemma 4 for the volumetric energy functional, we obtain d2ϕ 1 (0) = 2 dε 2



+

1 2

N

det(x∗ g){(x∗ g)αβ Y (Y (g(Zα , Zβ )))dt

 N

det(x∗ g)[(x∗ g)αβ (x∗ g)μν − (x∗ g)α μ (x∗ g)β ν ]

×Y (g(Zα , Zβ ))Y (g(Zμ , Zν ))dt. Therefore, serious difficulties arise when trying to find a connection between the convexity of the volumetric energy and the Riemannian structure of the manifold (M, g). Things become even more complicated when we analyze the convexity of the least squares type energies. Later, we shall prove that simple sufficient conditions equally ensure us about the invexity of all these energy functionals.

200

C. Udris¸te and A. Bejenaru

3.2 Convex Functions Generated by Convex Functionals The following theorem gives us a method for creating convex functions on a complete Riemannian manifold (M, g) using convex functionals (see also [11]). Let us consider N ⊂ Rm , two fixed points t0 and t1 in N, the set E of all differentiable maps from N to M and a functional J : E → IR. If x0 ∈ M is a fixed point, let

Γx = {Φ ∈ E| Φ (t0 ) = x0 , Φ (t1 ) = x}. Theorem 4. If J : E → IR is a convex functional, then f : M → IR, f (x) = inf J[Φ ] Φ ∈Γx

is a Riemannian convex function on M. Proof. Let x, y ∈ M and let ψ : [0, 1] → M be a geodesic such that ψ (0) = x and ψ (1) = y. We also consider the set

Γ = {ϕ : N × [0, 1] → M| ϕ (t, ·) geodesic, ∀t ∈ N, ϕ (t0 , ·) = x0 , ϕ (t1 , 0) = x, ϕ (t1 , 1) = y}. We have f (ψ (ε )) =

inf J[Φ ] =

Φ ∈Γψ (ε )

inf

ϕ ∈Γ ,ϕ (t1 ,·)=ψ (·)

J[ϕ (·, ε )]

≤ (1 − ε ) inf J[ϕ (·, 0)] + ε inf J[ϕ (·, 1)] ϕ

ϕ

= (1 − ε ) inf J[Φ ] + ε inf J[Φ ] = (1 − ε ) f (x) + ε f (y). Φ ∈Γx

Φ ∈Γy



Therefore, f is a convex function.

In the following we give an example, based on the convexity of the kinetic energy functional. Let (M, g) be a complete Riemannian manifold and x0 ∈ M be a fixed point. If RX is negative semidefinite on M, ∀X ∈ X (M), then f : M → IR, f (x) = d 2 (x0 , x) is a convex function (a direct prove can be found in [11]). We prove this statement using the previous theorem. We consider N = [a, b], h = 2dt ⊗ dt, and the kinetic energy functional J corresponding to (N, h) and (M, g). Then, f (x) = inf

 b

γ ∈Γx a

g(γ˙(t), γ˙(t))dt = inf J[γ ]. γ ∈Γx

Since J is a convex functional it follows that f (x) = d 2 (x0 , x) is a convex function.

Invex Energies on Riemannian Manifolds

201

4 Invex Energies This section extends some ideas from functions (see [6]) to functionals. Let J 1 (N, M) denote the jet bundle associated to N and M. A differentiable function L on J 1 (N, M) is called Lagrangian and the functional defined by multiple integral   ∂ xi α i J : E → IR, J[x(·)] = L t , x (t), α (t) dt ∂t N 

is the action associated toL. If x(·) ∈ E, we denote by Φ : N → J 1 (N, M) the i submanifold map Φ (t) = t α , xi (t), ∂∂txα (t) and we substitute J[x(·)] with J[Φ ]. We consider the set 

   ∂ xi 1 α i  F = Φ : N → J (N, M)  Φ (t) = t , x (t), α (t) . ∂t Moreover, if Φ ∈ F, we denote  

 TΦ F = X ∈ X (J 1 (N, M))  X α (Φ (t)) = 0, Xαi (Φ (t)) = Dα X i (Φ (t)) , where Dα denotes the total derivative with respect to t α . Definition 7. A vector map

η : F × F → X (J 1 (N, M)), η (Ψ , Φ ) ∈ TΦ F

(16)

is called pairing map on F. Example 1. If Φ ∈ F and VΦ = {Ψ ∈ F| Ψ (t) ∈ VΦ (t) , ∀t ∈ N}, where VΦ (t) is a neighborhood of Φ (t) such that expΦ (t) : TΦ (t) J 1 (N, M) → VΦ (t) is a diffeomorphism, then we consider the map

η (Φ ) : VΦ → XΦ (J 1 (N, M)), η (Φ ) (Ψ )(t) = exp−1 Φ (t) (Ψ (t)).

(17)

Furthermore, we denote by η0 a pairing map satisfying

η0 (Ψ , Φ ) = η (Φ ) (Ψ ), ∀Ψ ∈ VΦ .

(18)

If η : F × F → X (J 1 (N, M)) is a pairing map and Φ , Ψ ∈ F, then we consider a geodesic deformation γΨ Φη : N × (−δ , δ ) → J 1 (N, M), [0, 1] ⊂ (−δ , δ ), satisfying

γΨ Φη (t, 0) = Φ (t), ∀t ∈ N

202

C. Udris¸te and A. Bejenaru

and

∂ γΨ Φη (t, 0) = η (Ψ , Φ )(t), ∀t ∈ N. ∂ε Definition 8. Let J : F → IR be a functional. The map dJ(Φ ) : TΦ F → IR, dJ(Φ )[X] =

d J[ϕ (·, ε )]|ε =0 , dε

(19)

where ϕ : N × (−δ , δ ) → J 1 (N, M) is a deformation of Φ in F such that XΦ (t) = ∂ϕ ∂ ε (t, 0)

is called the differential of the functional J at Φ .

Definition 9. A map Φ ∈ F is called critical point of the functional J if dJ(Φ )[X] = 0, ∀X ∈ TΦ F. Definition 10. Let η : F × F → X (J 1 (N, M)) be a pairing map on F. A functional J : F → IR is called η -convex at Φ ∈ F if J[Ψ ] − J[Φ ] ≥ dJ(Φ )[η (Ψ , Φ )], ∀Ψ ∈ F.

(20)

The functional J is called strictly η -convex at Φ ∈ F if J[Ψ ] − J[Φ ] > dJ(Φ )[η (Ψ , Φ )], ∀Ψ ∈ F, Ψ = Φ .

(21)

Definition 11. The functional J is called invex if there is a pairing map η : F × F → X (J 1 (N, M)) such that J is η -convex. Remark 4. When taking η = η0 , the η -convexity is the classical Riemannian convexity from the previous section. Theorem 5. A functional J : F → IR is invex iff all its critical points are global minimum points. Proof. Let Φ ∈ F be a critical point for J. Then dJ(Φ )[η (Ψ , Φ )] = 0, ∀Ψ ∈ F, and, since J is invex, it follows that J[Ψ ] ≥ J[Φ ], ∀Ψ ∈ F, therefore Φ is a global minimum point. Conversely, we suppose that each critical point is a global minimum. If Φ ∈ F is a critical point, then dJ(Φ )[X] = 0, ∀X ∈ TΦ F and, for Ψ ∈ F arbitrary, we consider η (Ψ , Φ ) = 0. If Φ is not a critical point, there is a vector field X ∈ TΦ F such that dJ(Φ )[X] = 0 and we consider

η (Ψ , Φ ) =

[J[Ψ ] − J[Φ ]] XΦ . dJ(Φ )[X]

Both times η (Ψ , Φ ) ∈ TΦ F, that is, η is a pairing map and J[Ψ ] − J[Φ ] − dJ(Φ )[η (Ψ , Φ )] ≥ 0.



Invex Energies on Riemannian Manifolds

203

In the following, if Y ∈ X (M) is a nonzero vector field on M, then

∂ ∂Y i j ∂ Y¯ = Y i i + j xα i ∂x ∂x ∂ xα denotes the lift of Y onto J 1 (N, M). This vector field satisfies the condition Y¯ ◦ Φ ∈ TΦ F, ∀Φ ∈ F. Furthermore, for each vector field Y on M and its lift Y¯ on the jet bundle, we define the set

 CritY (L) = (t α , xi , xiα ) ∈ J 1 (N, M)| Y¯ (L)(t α , xi , xiα ) = 0 . Theorem 6. If there is a vector field Y on M such that CritY (L) = Crit(L) and all the critical points of the Lagrangian L are minimum points, then L is an invex function and the action J is an invex functional. Proof. Suppose there is a vector field Y on M such that CritY (L) = Crit(L). Then all the points of CritY (L) are minimum points for L. Now, if (t α , xi , xiα ) ∈ CritY (L), we consider the function fY (t α , yi , yiα , xi , xiα ) = 0. Otherwise, Y¯ (L)(t α , xi , xiα ) = 0 and we define L(t α , yi , yiα ) − L(t α , xi , xiα ) fY (t α , yi , yiα , xi , xiα ) = . Y¯ (L)(t α , xi , xiα ) Let η¯ (t α , yi , yiα , xi , xiα ) = fY (t α , yi , yiα , xi , xiα )Y¯ (t α , xi , xiα ). The pairing map η¯ defined above ensures us about the invexity of the Lagrangian L. Indeed, L(t α , yi , yiα )− L(t α , xi , xiα )−η¯ (t α , yi , yiα , xi , xiα )(L)=L(t α , yi , yiα )−L(t α , xi , xiα )≥0, if (t α , xi , xiα )∈ CritY (L) and L(t α , yi , yiα ) − L(t α , xi , xiα ) − η¯ (t α , yi , yiα , xi , xiα )(L) = 0, otherwise. Furthermore, for Ψ , Φ ∈ F, we define the vector map

η (Ψ , Φ )(t) = [inf fY (Ψ (t), Φ (t))]Y¯ (Φ (t)). t∈N

Since Y¯ ◦ Φ ∈ TΦ F, it follows that η (Ψ , Φ ) ∈ TΦ F, therefore η is a pairing map on F. Moreover, η (Ψ , Φ )(t) ≤ η¯ (Ψ (t), Φ (t)) and J[Ψ ] − J[Φ ] − dJ(Φ )[η (Ψ , Φ )] = ≥

 N

 N

[L(Ψ (t)) − L(Φ (t)) − η (Ψ , Φ )(t)(L)] dt

[L(Ψ (t)) − L(Φ (t)) − η¯ (Ψ (t), Φ (t))(L)] dt ≥ 0.

The above inequality ensures us about the invexity of the functional J.



Definition 12. Let (M, g) be a Riemannian manifold. A vector field X ∈ X (M) for which the Lie derivative LX g of the Riemannian structure g, with respect to X, is positive (definite) semidefinite is called (strictly) sub-Killing vector field. Theorem 7. If the Riemannian manifold (M, g) admits some (strictly) convex function, then there exists a (strictly) sub-Killing vector field on M.

204

C. Udris¸te and A. Bejenaru

Proof. If f : M → R is a convex function on M, let X = grad f . Then LX g = 2 Hess f > 0.

Theorem 8. If X ∈ X (M) is a conformal vector field with strictly positive divergence, then X is a sub-Killing vector field. Proof. The vector field X is conformal iff LX g = 2n (divX) g.



4.1 Invexity of Kinetic Energy Functional Let (N, h) be a compact m-dimensional Riemannian manifold with local coordinates (t 1 , ...,t m ), let (M, g) be a complete n-dimensional Riemannian manifold with local coordinates (x1 , ..., xn ) and let E be the set of all submanifold maps from N to M. We recall proving that a negative semidefinite Riemann curvature tensor field on M ensures us about the η0 -convexity of the kinetic energy functional. The following theorem improves this result by giving a simplified sufficient condition for invexity. Theorem 9. If the Riemannian manifold (M, g) admits a strictly sub-Killing vector field, then the kinetic energy functional is invex. Proof. Let Y be a sub-Killing vector field on (M, g). The Lagrangian (the energy density) associated to this functional is L(t γ , xi , xiγ ) = and Y (L) =

1 αβ h (t)gi j (x)xiα xβj 2

1 αβ h (Y (g))i j xiα xβj . 2

We conclude that CritY (L) = Crit(L) = {(t, x, 0)| t ∈ N, x ∈ M}. Since all the critical points of L are also minimum points, it follows that L is an invex Lagrangian and J is an invex functional.

4.2 Invexity of Volumetric Energy Functional In the following, we use the same techniques as above in order to prove the invexity of the volumetric energy functional. Theorem 10. If the Riemannian manifold (M, g) admits a strictly sub-Killing vector field, then the volumetric energy functional is invex.

Invex Energies on Riemannian Manifolds

205

Proof. Let Y be a sub-Killing vector field on M and Y¯ = Y + of Y on

J 1 (N, M).



∂Y i j x ∂xj α



∂ ∂ xiα

be the lift

We introduce the following differentiable functions on J 1 (N, M):

gαβ (t γ , xi , xiγ ) = gi j (x) xiα xβ ;

g(t γ , xi , xiγ ) = det(gαβ ).

j

The Lagrangian corresponding to the volumetric energy functional is L(t γ , xi , xiγ ) =

1 γ i i g(t , x , xγ ) 2

and Y (L) = =

  1 αβ 1 ∂Y k i j j j g g Y (gi j (x) xiα xβ ) = g gαβ Y (gi j ) xiα xβ + 2gk j x x 2 2 ∂ xi α β 1 αβ j g g (Y (g))i j xiα xβ . 2

Since Y is a strictly sub-Killing vector field, the following equality holds CritY (L) = Crit(L) = Crit(g) = {(t, x, 0)| t ∈ N, x ∈ M} and since all these critical points are also minimum points, it follows that the Lagrangian is invex and, consequently, the functional J is invex.

4.3 Invexity of Least Squares Lagrangian In Sect. 2, we have obtained new functionals (deformations of the volumetric and kinetic energy functionals), when we considered Euler–Lagrange prolongations for a PDE system of first order (see [8, 10–25]). Let X = Xαi (t, x) ∂∂xi ⊗ dt α be a distinguished tensor field on N × M, satisfying the integrability conditions i i ∂ Xαi ∂ Xαi j ∂ Xβ ∂ Xβ j + X = α + Xα . ∂xj β ∂t ∂xj ∂ tβ

We introduce the functionals JX : E → IR, JX [x(·)] = and

1 J X : E → IR, J X [x(·)] = 2

1 2

 N

 N

det(g˜αβ (t))dt

 hαβ (t)g˜αβ (t) h(t)dt,

206

where

C. Udris¸te and A. Bejenaru

   g˜αβ (t) = gi j (x(t)) xiα (t) − Xαi (t, x(t)) xβj (t) − Xβj (t, x(t)) .

Theorem 11. If Xαi (t, x) = Xαi (x) (i.e., X is an autonomous tensor field) and there is γ ∈ {1, ..., m} such that Yγ = Xγi ∂∂xi is a sub-Killing vector field on (M, g), then JX and J X are invex functionals. Proof. Suppose that Yγ is a sub-Killing vector field on (M, g) and let Y γ = Yγ +  ∂ Yγi j  ∂ x be the lift of Yγ on J 1 (N, M). The Lagrangians corresponding to the ∂ x j α ∂ xiα previous two functionals are     j 1 j L(t α , xi , xiα ) = det gi j (x) xiα − Xαi (x) xβ − Xβ (x) 2 and

   j 1 j L(t α , xi , xiα ) = hαβ (t) xiα − Xαi (x) xβ − Xβ (x) . 2 By computation, we obtain  j      ∂ Xβ k 1 αβ Y γ (L) = g˜ g˜ X Yγ (gi j ) xiα − Xαi xβj − Xβj − 2gi j xiα − Xαi 2 ∂ xk γ    i  j  ∂ Xγj k   i 1 αβ  j i i = x Y x g ˜ g ˜ + 2gi j xα − Xα (x) x (g) − X − X γ α β β ij α 2 ∂ xk β and, similarly, Y γ (L) =

  j   1 αβ  j h Yγ (g) i j xiα − Xαi xβ − Xβ . 2

Therefore, Crit Yγ (L) = Crit(L) = {(t, x, Xαi )| t ∈ N, x ∈ M}, Crit Yγ (L) = Crit(L) = {(t, x, Xαi )| t ∈ N, x ∈ M} and since all these points are global minimum points, it follows that both JX and J X are invex functionals.

4.4 Invexity or Convexity? The previous geometric examples suggest that invexity needs weaker sufficient conditions than convexity. The next set of corollaries emphasizes this idea by analyzing some elementary examples.

Invex Energies on Riemannian Manifolds

207

Corollary 1. 1. If N is a bounded subset of Rm and M = Rn , both endowed with the Euclidean structures, then the kinetic energy functional is convex (therefore invex). 2. If N is a bounded subset of Rm , m ≥ 2, and M = Rn , both endowed with the Euclidean structures, then the volumetric energy functional is invex. Hint. 1. The Riemann curvature (0,2)-tensor field is negative semidefinite and we apply Theorem 3. 2. The vector field Y = xi ∂∂xi ∈ X (M) is a strictly sub-Killing vector field and we apply Theorem 10.

Corollary 2. Let (N, h) be a compact Riemannian manifold and M ⊂ S3 be the south hemisphere of the sphere S3 , endowed with the Riemannian structure g induced by the stereographic projection from north pole. Then, both the kinetic and the volumetric energy functionals corresponding to the pair (N,M) are invex. Proof. If (x1 , x2 , x3 ) are the local coordinates on M, then (x1 )2 + (x2 )2 + (x3 )2 < 1. The Riemannian structure on M is gi j =

4δ i j (1 + x2 )2

and if we consider the vector field Y = xi ∂∂xi ∈ X (M), then Y (g) = 2

1−x2 gi j 1+x2

> 0.

The foregoing inequality expresses the fact that Y is a sub-Killing vector field on M. Therefore, the kinetic and the volumetric energies are invex.



5 Open Problems 5.1 Thin-Plate Spline Energy Let x = (xi ) : N → M, t → x(t) and (Hess x)iαγ =

∂ 2 xi j λ − Γαγ (t)xiλ + Γjki (x)xα xkγ . ∂ tα ∂ tγ

The map x converts geodesics of (N, h) into geodesics of (M, g) if and only if (Hess x)iαγ = 0. Analyze the extremals of the second-order Lagrangian L=

1 αβ j h (t)hγδ (t)gi j (x)(Hess x)iαγ (Hess x)β δ , 2

208

C. Udris¸te and A. Bejenaru

which reduces to the thin-plate spline energy in the Euclidean case. How could we define, in general, the invexity of a functional associated to a second-order Lagrangian?

5.2 Pairs of Type (Lagrangian, Hamiltonian) Let L = L(t, x, xα )) be a Lagrangian or a kinetic potential. If the equation pαi = ∂L (t, x, xα ) defines an implicit bijection xα → pα , then the formula ∂ xi α

H = xiα

∂L −L ∂ xiα

(22)

represents a Legendrian duality between a Lagrangian and a Hamiltonian, and produces a duality between the Euler–Lagrange PDEs

∂L ∂L − Dα i = 0 ∂ xi ∂ xα and the Hamilton PDEs

∂ pαi ∂ xi ∂H ∂H (t) = (t, x(t), p(t)), (t) = − i (t, x(t), p(t)). α α α ∂t ∂ pi ∂t ∂x  Making abstraction of det(h), using the formula (22), and preserving the independent variables t, x, xα , we can introduce some pairs of functions (L, H) (see also [24]): 1 j j 1. L = hαβ gi j xiα xβ − hαβ gi j xiα Xβ , 2 1 H = hαβ gi j xiα xβj 2 1 1 j j 2. L = hαβ gi j xiα xβ − hαβ gi j Xαi Xβ , 2 2 1 1 H = hαβ gi j xiα xβj + hαβ gi j Xαi Xβj 2 2 hαβ gi j xiα xβ hγδ gkl Xγk Xδl j

3. L =

n hλ μ gmn xm λ Xμ

, H = −L

Invex Energies on Riemannian Manifolds

209

n 2 4. L = hαβ gi j xiα xβ hγδ gkl Xγk Xδl − (hλ μ gmn xm λ Xμ ) , j

H = −(hαβ gi j xiα xβ )2 j

5. L =

hαβ gi j xiα xβj hγδ gkl Xγk Xδl n 2 (hλ μ gmn xm λ Xμ )

, H = −2L

j

6. L = det(gi j xiα xβ ), H = (m − 1)L   1 7. L = det gi j (xiα Xβj + xiβ Xαj ) , H = (m − 1)L 2 All the Lagrangians in this paper are algebraic functions of the following arguments: u = hαβ gi j xiα xβj , v = hαβ gi j xiα Xβj , w = hαβ gi j Xαi Xβj ,  j

r = det(gi j xiα xβ ), s = det

 1 j j gi j (xiα Xβ + xiβ Xα ) . 2

The pairs (L, H) contain two geometrical ingredients: the length and the angle. Of course, since hαβ , gi j are positive definite metrics, we can obtain inequalities satisfied by L or H. Write the previous Hamiltonians as functions of p = (pαi ). If we use the same independent variables, how we distinguish between a Lagrangian and a Hamiltonian? Which is the relation between the invexity of a functional and the invexity of its corresponding Hamiltonian?

5.3 Least Squares Hamiltonian Our geometric dynamics [8,12,13,18,24] used a least squares Lagrangian L (square of the length) and its associated Hamiltonian (scalar product) √ 1 H = hαβ gi j (xiα − Xαi (t, x))(xβj + Xβj (t, x)) det h. 2 Of course, this is not conserved along the extremals of L or along the solutions of Hamilton PDEs. On  the other hand,  we remark that H corresponds either to the pair  i xα (t), Xαi (t, x(t)) or to the pair xiα (t), −Xαi (t, x(t)) . If we use simultaneously the PDE system ∂ xi (t) = Xαi (t, x(t)) ∂ tα

210

C. Udris¸te and A. Bejenaru

and the PDE system

∂ xi = −Xαi (t, x(t)), ∂ tα

then the manifold N must be star-like with respect to the origin. Also, if Xαi are changed into −Xαi , then the Lagrangian L modifies, but the Hamiltonian H remains invariant.

5.4 Energy–Momentum Tensor Field Since a multitime Hamiltonian is not conserved along the extremals of the associated Lagrangian, reasons from physics ask to introduce the energy–momentum tensor field ∂L T α β = xiβ i − Lδβα . ∂ xα This tensor field describes the density and flux of energy, and momentum in spacemultitime. If the Lagrangian L does not depend explicitly on the multiparameter t, then the energy–momentum tensor field represents a conservation law in the sense that the divergence of the tensor field T α β is zero. Find the energy–momentum tensor field for each of the previous Lagrangians.

5.5 Interaction Between Vector Fields To describe the “interaction” between vector fields or 1-forms, when they are working together in a multitime physical or economical dynamical system, we recommend to use their generated distributions in the sense of the differential geometry, using Span or Pffaff equations. Of course, in this way we overpass the old idea of summing similar vector fields or similar 1-forms and after that of introducing the total vector field or total 1-form in a single-time dynamical system. We pass in fact to multitime dynamical systems on distributions.

5.6 Least Squares Lagrangian Associated to Maxwell PDEs We recall some open problems from the book [24]. Let U be a domain of linear homogeneous isotropic media in the Riemannian manifold (M = R3 , δi j ). Maxwell’s equations (coupled PDEs of first order) div D = ρ , rot H = J + ∂t D, div B = 0, rot E = −∂t B,

Invex Energies on Riemannian Manifolds

211

where ∂t is the time derivative operator, together with the constitutive equations B = μ H, D = ε E, on R × U, reflect the relations between the electromagnetic fields: E H J ε μ B − D −

[V /m] Electric field strength [A/m] Magnetic field strength Electric current density [A/m2 ] [As/V m] Permittivity [V s/Am] Permeability [T ] = [V s/m2 ] Magnetic induction − (magnetic flux density) [C/m2 ] = [As/m2 ] Electric displacement − (electric flux density)

Since div B = 0, the vector field B is source free, hence may be expressed as rot of some vector potential A, that is, B = rot A. Then the electric field strength is E = −gradV − ∂t A. Find interpretations for the extremals of least squares Lagrangians of the type L1 =

1 1 ||rot E + ∂t B||2 + ||rot H − J − ∂t D||2 2 2 1 1 + (div D − ρ )2 + (div B)2 2 2

which are not solutions of Maxwell equations. Can we derive, in this way, the Dirac theory of magnetic monopole? Let us refer to Maxwell theory in terms of differential forms. In this sense, it is well known that E, H are differential 1-forms, J, D, B are differential 2-forms, ρ is a differential 3-form, and star operator from D = ε ∗ E, B = μ ∗ H is the Hodge operator. If d is the exterior derivative operator, and ∂t is the time derivative operator, then the Maxwell’s equations for static media are dE = −∂t B, dH = J + ∂t D, dD = ρ , dB = 0 (coupled PDEs of first order) on R × U. Find interpretations for the extremals of least squares Lagrangians 1 1 1 1 L2 = ||dE + ∂t B||2 + ||dH − J − ∂t D||2 + ||dD − ρ ||2 + ||dB||2 , 2 2 2 2 which are not solutions of Maxwell equations.

212

C. Udris¸te and A. Bejenaru

References 1. Bejenaru A (2011) Generalized convex functionals. Balkan J Geom Appl 16(1):12–26 2. Helein F (2003) Harmonic maps, conservation laws and moving frames. Cambridge University Press, Cambridge 3. Hermann R (1974) Geometric structure of systems-control theory and physics. Interdisciplinary mathematics, part A, vol IX. Math Sci Press, Brookline 4. Hermann R (1977) Differential geometry and the calculus of variations. Interdisciplinary mathematics, vol XVII. Math Sci Press, Brookline 5. Jung Y, Chu KT, Torquato S (2007) A variational level set approach for surface area minimization of triply-periodic surfaces. J Comput Phys 223:711–730 6. Mititelu S (2009) Generalized convexities. Geometry Balkan Press (in Romanian), Bucharest 7. Nabutovsky A, Rotman R (2006) Curvature-free upper bounds for the smallest area of a minimal surface. Geom Funct Anal 16(2):453–475 8. Pitea A, Udris¸te C (2008) Determining a pair of metrics by boundary energy associated to a multitime PDE system. WSEAS Trans Math 7(1):45–50 9. Schief W (2006) On a maximum principle for minimal surfaces and their integrable discrete counterparts. J Geom Phys 56(9):1484–1485 10. Tsagas G, Udris¸te C (2002) Vector fields and their applications (Selected papers). Geom Balkan Press, Bucharest 11. Udris¸te C (1994) Convex functions and optimization methods on Riemannian manifolds. Mathematics and its applications, vol 297. Kluwer Academic, Dordrecht/Boston/London 12. Udris¸te C (2000) Geometric dynamics. Mathematics and its applications, vol 513. Kluwer Academic, Dordrecht/Boston/London 13. Udris¸te C (2003–2004) From integral manifolds and metrics to potential maps. Atti dell’ Academia Peloritana dei Pericolanti, Classe I. di Scienze Fis Mat et Nat 81–82:1–16 14. Udris¸te C (2007) Multi-time maximum principle. Short communication at international congress of mathematicians, Madrid, 22–30 Aug, ICM abstracts, p 47 (2006). Plenary lecture at 6-th WSEAS international conference on circuits, systems, electronics, control and signal processing (CSECS’07), 10–11 (2007) and 12-th WSEAS international conference on applied mathematics, Cairo, 29–31 Dec 2007, p ii 15. Udris¸te C (2007) Multi-time stochastic control theory. Selected topics on circuits, systems, electronics, control and signal processing. In: Proceedings of the 6-th WSEAS international conference on circuits, systems, electronics, control and signal processing (CSECS’07), Cairo, 29–31 Dec 2007, pp 171–176 16. Udris¸te C (2008) Finsler optimal control and geometric dynamics. Mathematics and computers in science and engineering. In: Proceedings of the American conference on applied mathematics, Cambridge, pp 33–38 17. Udris¸te C (2008) Lagrangians constructed from Hamiltonian systems. Mathematics and computers in business and economics. In: Proceedings of the 9th WSEAS international conference on mathematics and computers in business and economics(MCBE-08), Bucharest, 24–26 June 2008, pp 30–33 18. Udris¸te C (2008) Non-classical lagrangian dynamics and potential maps. WSEAS Trans Math 7:12–18 19. Udris¸te C (2009) Simplified multitime maximum principle. Balkan J Geom Appl 14(1):102–119 20. Udris¸te C (2009) Nonholonomic approach of multitime maximum principle. Balkan J Geom Appl 14(2):117–127 21. Udris¸te C (2010) Equivalence of multitime optimal control problems. Balkan J Geom Appl 15(1):155–162 22. Udris¸te C, T¸evy I (2007) Multi-time Euler–Lagrange dynamics. Proceeding of the 7th WSEAS international conference on Systems Theory and Scientific Computation, (ISTASC07), Greece, 24–26 Aug 2007, pp 66–71

Invex Energies on Riemannian Manifolds

213

23. Udris¸te C, Udris¸te A (2006) From flows and metrics to dynamics and winds. Bull Cal Math Soc 98(5):389–394 24. Udris¸te C, Ferrara M, Opris¸ D (2004) Economic geometric dynamics. Monographs and Textbooks, vol 6. Geometry Balkan, Bucharest 25. Udris¸te C, T¸evy I, Arsinte V (2010) Minimal surfaces between two points. J Adv Math Stud 3(2):105–116 26. Udris¸te C, Bejenaru A (2011) Multitime optimal control with area integral costs on boundary. Balkan J Geom Appl 16(2):138–154

Weyl’s Limit Point and Limit Circle for a Dynamic Systems Adil Huseynov

Abstract We show that analogues of classical concepts of the Weyl limit point and limit circle cases can be introduced and investigated for second order linear dynamic equations on time scales. Since dynamical equations on time scales unifies and extends continuous and discrete dynamical equations (i.e., differential and difference equations), in this way, we establish more general theory of the limit point and limit circle cases.

1 Introduction The present chapter deals with the second order linear dynamic equations (differential equations on time scales are called the dynamic equations) on the semiunbounded time scale intervals of the form − [p(t)y∇ (t)]Δ + q(t)y(t) = λ y(t),

t ∈ (a, ∞)T ,

(1)

and develops for such equations an analogue of the classical Weyl limit point and limit circle theory given by him for the usual Sturm–Liouville equation − [p(t)y (t)] + q(t)y(t) = λ y(t),

t ∈ (a, ∞)R ,

(2)

in the first decade of the twentieth century [12]. The limit point and limit circle theory plays an important role in the spectral analysis of differential equations on unbounded intervals (see [5,11,13]). For discrete analogues of (2) (for infinite Jacobi matrices) the concepts of the limit point and limit circle cases were introduced and investigated by Hellinger [8] (see also [1], Chap. 1).

A. Huseynov () Department of Mathematics, Ankara University, 06100 Tandogan, Ankara, Turkey e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 11, © Springer Science+Business Media, LLC 2012

215

216

A. Huseynov

Our consideration of the problem in this chapter for (1) on time scales allows to unify the known continuous and discrete cases (i.e., differential and difference equations) and extend them to more general context of time scales. For a general introduction to the calculus and the dynamic equations on time scales we refer the reader to [3, 4, 9].

2 Limit Point and Limit Circle Let T be a time scale which is bounded from below and unbounded from above so that inf T = a > −∞ and sup T = ∞. By the closedness of T in R, we have a ∈ T. We will denote such T also as [a, ∞)T and call it a semi-infinite (or semi-unbounded ) time scale interval. Consider the equation − [p(t)y∇ (t)]Δ + q(t)y(t) = λ y(t),

t ∈ (a, ∞)T ,

(3)

where p(t) is a real-valued Δ -differentiable function on [a, ∞)T with piecewise continuous Δ -derivative pΔ (t), p(t) = 0 for all t, q(t) is a real-valued piecewise continuous function on [a, ∞)T , λ is a complex parameter (spectral parameter). We define the quasi ∇-derivative y[∇] (t) of y at t by y[∇] (t) = p(t)y∇ (t). For any point t0 ∈ [σ (a), ∞]T and any complex constants c0 , c1 (3) has a unique solution y satisfying the initial conditions y(t0 ) = c0 ,

y[∇] (t0 ) = c1 .

If y1 , y2 : [a, ∞)T → C are two ∇-differentiable on [σ (a), ∞]T functions, where σ denotes the forward jump operator in T, then the Wronskian of y1 and y2 is defined for t ∈ [σ (a), ∞]T by [∇]

[∇]

∇ Wt (y1 , y2 ) = y1 (t)y2 (t) − y1 (t)y2 (t) = p(t)[y1 (t)y∇ 2 (t) − y1 (t)y2 (t)].

(4)

The Wronskian of any two solutions of (3) is independent of t. Two solutions of (3) are linearly independent if and only if their Wronskian is nonzero. Equation (3) has two linearly independent solutions and every solution of (3) is a linear combination of these solutions. We say that y1 and y2 form a fundamental set (or a fundamental system) of solutions for (3) provided their Wronskian is nonzero.

Weyl’s Limit Point and Limit Circle

217

Let us consider the nonhomogeneous equation − [p(t)y∇ (t)]Δ + q(t)y(t) = h(t),

t ∈ (a, ∞)T ,

(5)

where h : T → C is a piecewise continuous function. If y1 and y2 form a fundamental set of solutions of the homogeneous equation (3) and ω = Wt (y1 , y2 ), then the general solution of the corresponding nonhomogeneous equation (5) is given by y(t) = c1 y1 (t) + c2y2 (t) +

1 ω

 t t0

[y1 (t)y2 (s) − y1 (s)y2 (t)]h(s)Δ s,

(6)

where t0 is a fixed point in T, c1 and c2 are arbitrary constants (see [2]). Formula (6) is called the variation of constants formula. Let L denote the linear operator defined by Lx = −(px∇ )Δ + qx. Lemma 1. If Lx(t, λ ) = λ x(t, λ ) and Ly(t, λ  ) = λ  y(t, λ  ), then for any b ∈ (a, ∞)T , (λ  − λ )

 b

σ (a)

xyΔ t = Wσ (a) (x, y) − Wb (x, y),

(7)

where Wt (x, y) is the Wronskian of x and y defined by (4). Proof. We have, using the integration by parts formula  d c

f Δ (t)g(t)Δ t = f (t)g(t) |dc −

 d c

f (t)g∇ (t)∇t

established in [7], that (λ  − λ )

 b σ (a)

xyΔ t =

 b σ (a)

= −xpy∇ |bσ (a) +

(xLy − yLx)Δ t = −

 b σ (a)

 b σ (a)

[x(py∇ )Δ − y(px∇ )Δ ]Δ t

x∇ py∇ ∇t + ypx∇ |bσ (a) −

 b σ (a)

y∇ px∇ ∇t

= −p(xy∇ − x∇ y) |bσ (a) = Wσ (a) (x, y) − Wb(x, y). The proof is complete. Corollary 1. If, in particular, λ = u + iv, λ  = λ = u − iv (u, v ∈ R), then we can take y(t, λ  ) = x(t, λ ) and (7) yields  b

2v

σ (a)

|x(t, λ )|2 Δ t = i{Wσ (a)(x, x) − Wb (x, x)}.

(8)

218

A. Huseynov

Let ϕ (t, λ ), θ (t, λ ) be two solutions of (3) satisfying the initial conditions

ϕ (σ (a), λ ) = sin α , θ (σ (a), λ ) = cos α ,

ϕ [∇] (σ (a), λ ) = − cos α ,

(9)

θ [∇] (σ (a), λ ) = sin α ,

(10)

where 0 ≤ α < π . (Note that the ∇-derivative is not defined at a if a is rightscattered). Then, since the Wronskian of any two solutions of (3) does not depend on t, we get Wt (ϕ , θ ) = Wσ (a) (ϕ , θ ) = sin2 α + cos2 α = 1. Then ϕ , θ are linearly independent solutions of (3), ϕ , ϕ [∇] , θ , θ [∇] are entire functions of λ and continuous in (t, λ ). These solutions are real for real λ . Every solution y of (3) except for ϕ is, up to a constant multiple, of the form y = θ + lϕ

(11)

for some number l which will depend on λ . Take now a point b ∈ (a, ∞)T and consider the boundary condition y(b) cos β + y[∇] (b) sin β = 0

(0 ≤ β < π )

(12)

and ask what must l be like in order that the solution y, (11), satisfy (12). If denote the corresponding value of l by lb (λ ), then we find that lb (λ ) = −

θ (b, λ ) cot β + θ [∇] (b, λ ) . ϕ (b, λ ) cot β + ϕ [∇] (b, λ )

Let us take any complex number z and introduce the function l = lb (λ , z) = −

θ (b, λ )z + θ [∇] (b, λ ) . ϕ (b, λ )z + ϕ [∇](b, λ )

(13)

If b and λ are fixed, and z varies, (13) may be written as l=

Az + B , Cz + D

(14)

where A = −θ (b, λ ), B = −θ [∇] (b, λ ), C = ϕ (b, λ ), D = ϕ [∇] (b, λ ). Since AD − BC = Wb (ϕ , θ ) = 1 = 0, the linear-fractional transformation (14) is one-to-one conformal mapping which transforms circles into circles; straight lines being considered as circles with infinite

Weyl’s Limit Point and Limit Circle

219

radii. Besides, applying (8) to the solution ϕ (t, λ ) and taking into account (9) by virtue of which Wσ (a) (ϕ , ϕ ) = 0, we have  b

2v

σ (a)

|ϕ (t, λ )|2 Δ t = −iϕ (b, λ )ϕ [∇] (b, λ ) + iϕ [∇](b, λ )ϕ (b, λ )

which implies that ϕ (b, λ ) = 0 and ϕ [∇] (b, λ ) = 0 if ℑλ = v = 0. Therefore, if ℑλ = v = 0, then lb (λ , z) varies on a circle Cb (λ ), with a finite radius, in the lplane, as z varies over the real axis of the z-plane. The center and the radius of the circle Cb (λ ) will be defined as follows. The center of the circle is the symmetric point of the point at infinity with respect to the circle. Thus, if we set lb (λ , z ) = ∞ and

lb (λ , z ) = the center

of

Cb (λ ),

then z must be the symmetric point of z with respect to the real axis of the z-plane, namely, z = z . On the other hand,  lb

ϕ [∇] (b, λ ) λ,− ϕ (b, λ )

 = ∞.

Therefore, the center of the circle Cb (λ ) is given by  lb

ϕ [∇] (b, λ ) λ,− ϕ (b, λ )

 =−

Wb (θ , ϕ ) . Wb (ϕ , ϕ )

The radius rb (λ ) of the circle Cb (λ ) is equal to the distance between the center of Cb (λ ) and the point lb (λ , 0) on the circle Cb (λ ). Hence    θ [∇] (b, λ ) W (θ , ϕ )   W (θ , ϕ )     b b . rb (λ ) =  [∇] − =  ϕ (b, λ ) Wb (ϕ , ϕ )   Wb (ϕ , ϕ )  On the other hand, by virtue of (9), (10), Wb (θ , ϕ ) = Wσ (a) (θ , ϕ ) = −1. Further, by virtue of (8) and (9), we have  b

2v a

|ϕ (t, λ )|2 Δ t = iWσ (a) (ϕ , ϕ ) − iWb (ϕ , ϕ ) = −iWb (ϕ , ϕ ),

(15)

where v = ℑλ . Therefore, we obtain rb (λ ) =

2 |v|

b a

1 |ϕ (t, λ )|2 Δ t

,

ℑλ = v = 0.

(16)

220

A. Huseynov

Since θ (b, λ )ϕ [∇] (b, λ ) − ϕ (b, λ )θ [∇] (b, λ ) = Wb (θ , ϕ ) = −1 = 0, the transformation (13) has a unique inverse which is given by z=−

ϕ [∇] (b, λ )l + θ [∇] (b, λ ) . ϕ (b, λ )l + θ (b, λ )

(17)

We shall now prove the following statement. Lemma 2. If v = ℑλ > 0, then the interior of the circle Cb (λ ) is mapped onto the lower half plane of the z-plane by the transformation (8), and, the exterior of the circle Cb (λ ) is mapped onto the upper half plane of the z-plane. Proof. Since the real axis of the z-plane is the image of the circle Cb (λ ) by the transformation (8), the interior of Cb (λ ) is mapped onto either the upper half plane or the lower half plane of the z-plane, and further, the point at infinity of the l-plane is mapped onto the point −ϕ [∇] (b, λ )/ϕ (b, λ ) of the z-plane. On the other hand, by making use of (15), 

ϕ [∇] (b, λ ) ℑ − ϕ (b, λ )



i = 2 =−



ϕ [∇] (b, λ ) ϕ [∇] (b, λ ) − ϕ (b, λ ) ϕ (b, λ )

i Wb (ϕ , ϕ ) v = 2 2 |ϕ (b, λ )| |ϕ (b, λ )|2



 b σ (a)

|ϕ (t, λ )|2 Δ t > 0.

This means that −ϕ [∇] (b, λ )/ϕ (b, λ ) belongs to the upper half plane of the z-plane. Hence the point at infinity which is not contained in the interior of Cb (λ ) is mapped into the upper half plane. This proves the lemma. Lemma 3. If v = ℑλ > 0, then l belongs to the interior of the circle Cb (λ ) if and only if  b ℑl |θ (t, λ ) + l ϕ (t, λ )|2 Δ t < − , v σ (a) and, l lies on the circle Cb (λ ) if and only if  b σ (a)

|θ (t, λ ) + l ϕ (t, λ )|2 Δ t = −

ℑl . v

Proof. In view of Lemma 2, if ℑλ = v > 0, then l belongs to the interior of the circle Cb (λ ) if and only if ℑz < 0, that is, i(z − z) > 0. From (8) it follows that 

[∇]

ϕ [∇] (b, λ )l + θ [∇] (b, λ ) ϕ [∇] (b, λ )l + θ (b, λ ) + i(z − z) = i − ϕ (b, λ )l + θ (b, λ ) ϕ (b, λ )l + θ (b, λ ) =

iWb (θ + l ϕ , θ + l ϕ ) |ϕ (b, λ )l + θ (b, λ )|2

.



Weyl’s Limit Point and Limit Circle

221

Therefore, ℑz < 0 if and only if iWb (θ + l ϕ , θ + l ϕ ) > 0. By formula (8) with x = θ + l ϕ , we have  b

2v

σ (a)

|θ + l ϕ |2 Δ t = i{Wσ (a)(θ + l ϕ , θ + l ϕ ) − Wb (θ + l ϕ , θ + l ϕ )}.

Further, by (9), (10), we have Wσ (a) (θ , ϕ ) = −1, Wσ (a) (ϕ , θ ) = 1, Wσ (a) (θ , θ ) = Wσ (a) (ϕ , ϕ ) = 0. Therefore Wσ (a) (θ +l ϕ , θ +l ϕ )=Wσ (a) (θ , θ )+lWσ (a) (θ , ϕ )+lWσ (a) (ϕ , θ )+|l|2 Wσ (a) (ϕ , ϕ ) = l − l = 2iℑl. Consequently,  b

2v

σ (a)

|θ + l ϕ |2 Δ t = −2ℑl − iWb (θ + l ϕ , θ + l ϕ )

and the statements of the lemma follow. Remark 1. It is easy to see that Lemma 3 also holds when v = ℑλ < 0. In the both cases v > 0 and v < 0 the sign of ℑl is opposite of the sign of v. Lemma 4. If v = ℑλ = 0, and 0 < b < b , then C b (λ ) ⊂ C b (λ ), where C b (λ ) is the set composed of the circle Cb (λ ) and its interior. Proof. If l belongs to the interior of the circle Cb (λ ) or is on Cb (λ ), then taking into account Lemma 3, we have  b σ (a)

|θ + l ϕ |2 Δ t ≤

 b σ (a)

|θ + l ϕ |2 Δ t ≤ −

ℑl . v

Hence the lemma follows by using again Lemma 3. Lemma 4 implies that, if v = ℑλ = 0, then the set ∩b>σ (a)C b (λ ) = C∞ (λ ) is either a point or a closed circle with a nonzero finite radius.

222

A. Huseynov

Definition 1. According as C∞ (λ ) is a point or a circle, the (3) is said to be in the limit point case or the limit circle case. According to this definition, the classification seems to depend on the p(t), q(t), and λ . However, it is independent of λ and depends only on p(t), q(t), as is shown in the next section. Let m = m(λ ) be the limit point C∞ (λ ) or any point on the limit circle C∞ (λ ). Then for any b ∈ (a, ∞)T , we have  b σ (a)

Hence

 ∞ σ (a)

|θ (t, λ ) + m(λ )ϕ (t, λ )|2 Δ t ≤ −

ℑm(λ ) . v

|θ (t, λ ) + m(λ )ϕ (t, λ )|2 Δ t ≤ −

ℑm(λ ) . v

Denote by L2Δ (a, ∞) the space of all complex-valued Δ -measurable (see [6]) functions f : [a, ∞)T → C such that  ∞ a

| f (t)|2 Δ t < ∞.

Similarly, we can introduce the space L2∇ (a, ∞). Thus, we have obtained the following theorem. Theorem 1. For all nonreal values of λ there exists a solution

ψ (t, λ ) = θ (t, λ ) + m(λ )ϕ (t, λ ) of (3) such that ψ ∈ L2Δ (a, ∞). In the limit circle case the radius rb (λ ) tends to a finite nonzero limit as b → ∞. Then (7) implies that in this case also ϕ ∈ L2Δ (a, ∞). Therefore, in the limit circle case all solutions of (3) belong to L2Δ (a, ∞) for ℑλ = 0 because in this case both ϕ (t, λ ) and θ (t, λ ) + m(λ )ϕ (t, λ ) belong to L2Δ (a, ∞), and this identifies the limit circle case. We will see below in Theorem 2 that in the limit circle case all solutions of (3) belong to L2Δ (a, ∞) also for all real values of λ . In the limit point case, rb (λ ) tends to zero as b → ∞, and from (7) this implies that ϕ (t, λ ) is not of class L2Δ (a, ∞). Therefore, in this situation there is only one solution of class L2Δ (a, ∞) for ℑλ = 0. Note that in the limit point case the equation may not have any nontrivial solution of class L2Δ (a, ∞) for real values of λ . For example, (for λ = 0), the equation −y∇Δ = 0 has the general solution y(t) = c1 + c2 t and evidently this solution belongs to L2Δ (a, ∞) only for c1 = c2 = 0.

Weyl’s Limit Point and Limit Circle

223

3 Invariance of the Limit Point and Limit Circle In the previous section the expressions “limit point case” and “limit circle case” were applied to particular values of λ ; but in fact if the limit is a circle for any complex λ , it is a circle for every complex λ . In the present section, we prove this property. Theorem 2. If every solution of Ly = λ0 y is of class L2Δ (a, ∞) for some complex number λ0 , then for arbitrary complex number λ every solution of Ly = λ y is of class L2Δ (a, ∞). Proof. It is given that two linearly independent solutions y1 (t) and y2 (t) of Ly = λ0 y are of class L2Δ (a, ∞). Let χ (t) be any solution of Ly = λ y, which can be written as Ly = λ0 y + (λ − λ0 )y. By multiplying y1 by a constant if necessary (to achieve Wt (y1 , y2 ) = 1) a variation of constants formula (6) yields

χ (t) = c1 y1 (t) + c2 y2 (t) +(λ − λ0 )

 t c

[y1 (t)y2 (s) − y1 (s)y2 (t)]χ (s)Δ s,

(18)

where c1 , c2 are constants and c is any fixed point in (a, ∞)T . Let us introduce the notation

  t

χ c,t =

c

|χ (s)|2 Δ s

1/2

for t ∈ [a, ∞)T with t ≥ c. Next, let M be such that y1 c,t ≤ M, y2 c,t ≤ M for all t ∈ [a, ∞)T with t ≥ c; such a constant M exists because y1 and y2 are of class L2Δ (a, ∞). Then the Cauchy–Schwarz inequality gives  t     [y1 (t)y2 (s) − y1 (s)y2 (t)] χ (s)Δ s  c  ≤ |y1 (t)| + |y2 (t)|

 t c

 t c

|y2 (s)| |χ (s)| Δ s |y1 (s)| |χ (s)| Δ s

≤ M(|y1 (t)| + |y2 (t)|) χ c,t . Using this in (18) yields | χ (t)| ≤ |c1 | |y1 (t)| + |c2 | |y2 (t)| + |λ − λ0 | M(|y1 (t)| + |y2 (t)|) χ c,t .

224

A. Huseynov

Hence applying the Minkowski inequality (note that the usual Cauchy–Schwarz and Minkowski inequalities hold on time scales, see [3]), we get χ c,t ≤ (|c1 | + |c2 |)M + 2 |λ − λ0 | M 2 χ c,t . If c is chosen large enough so that 1 |λ − λ 0 | M 2 < , 4 then

χ c,t ≤ 2(|c1 | + |c2 |)M.

Since the right side of this inequality is independent of t, it follows that χ ∈ L2Δ (a, ∞) and the theorem is proved.

4 A Criterion for the Limit Point In this section, we present a simple criterion for the limit point case. For the usual Sturm–Liouville equation it was established earlier by Putnam [10]. Theorem 3. If p is arbitrary and q ∈ L2Δ (a, ∞), then (3) is in the limit point case. Proof. It is sufficient to show that the equation − [p(t)y∇ (t)]Δ + q(t)y(t) = 0,

t ∈ (a, ∞)T ,

(19)

does not have two linearly independent solutions belonging to L2Δ (a, ∞). If y is such a solution, then, because of the condition q ∈ L2Δ (a, ∞), the function [∇] (y )Δ = qy belongs to L1Δ (a, ∞) by the Cauchy–Schwarz inequality. Therefore the limit  ∞ lim y[∇] (t) = y[∇] (t0 ) + (y[∇] )Δ (s)Δ s t→∞

t0

exists and is finite. Hence the function y[∇] (t) is bounded as t → ∞. Now let y1 , y2 be two linearly independent solutions of (19); then [∇]

[∇]

y1 (t)y2 (t) − y1 (t)y2 (t) = c = 0. [∇]

[∇]

If y1 ∈ L2Δ (a, ∞) and y2 ∈ L2Δ (a, ∞), then y1 and y2 are bounded, and so the [∇] [∇] function y1 y2 − y1 y2 = c = 0 also belongs to L2Δ (a, ∞), which is impossible. The theorem is proved. Acknowledgment This work was supported by the Scientific and Technological Research Council of Turkey (TUBITAK). The author thanks Elgiz Bairamov for useful discussions.

Weyl’s Limit Point and Limit Circle

225

References 1. Akhiezer NI (1965) The classical moment problem and some related questions in analysis. Hafner, New York 2. Atici FM, Guseinov GS (2002) On Green’s functions and positive solutions for boundary value problems on time scales. J Comput Appl Math 141:75–99 3. Bohner M, Peterson A (2001) Dynamic equations on time scales: an introduction with applications. Birkh¨auser, Boston 4. Bohner M, Peterson A (eds) (2003) Advances in dynamic equations on time scales. Birkh¨auser, Boston 5. Coddington EA, Levinson N (1955) Theory of ordinary differential equations. McGraw-Hill, New York 6. Guseinov GS (2003) Integration on time scales. J Math Anal Appl 285:107–127 7. Guseinov GS (2005) Self-adjoint boundary value problems on time scales and symmetric Green’s functions. Turk J Math 29:365–380 8. Hellinger E (1922) Zur stieltjesschen kettenbruchtheorie. Math Ann 86:18–29 9. Hilger S (1990) Analysis on measure chains–a unified approach to continuous and discrete calculus. Results Math 18:18–56 10. Putnam CR (1949) On the spectra of certain boundary value problems. Am J Math 71:109–111 11. Titchmarsh EC (1962) Eigenfunction expansions assosiated with second-order differential equations, Part I, 2nd edn. Oxford University Press, Oxford ¨ 12. Weyl H (1910) Uber gew¨ohnliche differentialgleichungen mit singulrit¨aten und die zugeh¨origen entwicklungen willk¨urlicher funktionen. Math Ann 68:220–269 13. Yosida K (1960) Lectures on differential and integral equations. Interscience, New York

Remarks on Suzuki (C)-Condition Erdal Karapinar

Abstract In this manuscript, first we introduce some new conditions, inspirit of Suzuki’s (C)-condition, on a self-mapping T on a subset K of a Banach space E. Secondly, we obtain some new fixed point theorems under these conditions.

1 Introduction The Banach contraction mapping principle [4] has a crucial role in Fixed Point Theory and has many applications in several branches of mathematics and also in economics. A self-mapping T on a metric space X is called “contraction” if there exists a constant k ∈ [0, 1) such that d(T x, Ty) ≤ d(x, y) for each x, y ∈ X.

(1)

Due to Banach, we know that every contraction on a complete metric space has a unique fixed point. This theorem, known as the Banach contraction mapping principle, is formulated in his thesis in 1920 and published in 1922 [4]. After Banach, many authors attempt to generalize the Banach contraction mapping principle such as Kannan [9], Reich [19], Chatterjea [5], Hardy and Rogers [8], ´ c [6], and many others (see e.g. [1–3, 10–18]). Ciri´ Very recently, Suzuki proved the following fixed point theorem: Theorem 1. (Suzuki [22].) Let (X, d) be a compact metric space and let T be a mapping on X. Assume 12 d(x, T x) < d(x, y) implies d(T x, Ty) < d(x, y) for all x, y ∈ X. Then T has a unique fixed point.

E. Karapinar () Atılım University, Department of Mathematics, 06836, Incek, Ankara, Turkey e-mail: [email protected]; [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 12, © Springer Science+Business Media, LLC 2012

227

228

E. Karapinar

This result is based on the following two theorems: Theorem 2. (Edelstein [7].) Let (X, d) be a compact metric space and let T be a mapping on X. Assume d(T x, Ty) < d(x, y) for all x, y ∈ X with x = y. Then T has a unique fixed point. Theorem 3. (Suzuki [20, 21].) Define a nonincreasing function θ from [0,1) onto (1/2,1] by ⎧ √ ⎪ 1 if 0 ≤ r ≤ ( 5 − 1)/2, ⎪ ⎨ √ θ (r) = (1 − r)r−2 if ( 5 − 1)/2 ≤ r ≤ 2−1/2 , ⎪ ⎪ ⎩ (1 + r)−1 if 2−1/2 ≤ r < 1. Then for a metric space (X, d), the following are equivalent: 1. X is complete. 2. Every mapping T on X satisfying the following has a fixed point: There exists r ∈ [0, 1) such that θ (r)d(x, T x) ≤ d(x, y) implies d(T x, Ty) ≤ rd(x, y) for all x, y ∈ X. A mapping T on a subset K of a Banach space E is called a nonexpansive mapping if T x − Ty ≤ x − y for all x, y ∈ K. Definition 1. ([20, 21]) Let T be a mapping on a subset K of a Banach space E. Then T is said to satisfy (C)-condition if 1 x − T x ≤ x − y implies that T x − Ty ≤ x − y 2 for all x, y ∈ K. Let F(T ) be the set of all fixed points of a mapping T . A mapping T on a subset K of a Banach space E is called a quasi-nonexpansive mapping if T x− z ≤ x− z for all x ∈ K and z ∈ F(T ). Inspirit of Suzuki’s (C)-condition, we give some new definitions as follow. Definition 2. Let T be a mapping on a subset K of a Banach space E. Then T is said to satisfy (for all x, y ∈ K) (RSC)

(RSCS)

Reich–Suzuki–(C) condition (in short, (RSC)-condition) if 1 x − T x ≤ x − y implies that 2  1 T x − Ty ≤ x − y + Tx − x + y − Ty , 3 Reich–Chatterjea–Suzuki–(C) condition (in short, (RCSC)-condition) if 1 x − T x ≤ x − y implies that 2  1 T x − Ty ≤ x − y + Tx − y + x − Ty , 3

Remarks on Suzuki (C)-Condition

(HRSC)

229

Hardy–Rogers–Suzuki–(C) condition (in short, (HRSC)-condition) if 1 x − T x ≤ x − y implies that 2

T x − Ty ≤

 1 x − y + Tx − x + y − Ty + Tx − y + x − Ty . 5

In this manuscript, we give new theorems that can be considered as an extension of the results of Suzuki [21] and also Singh-Mishra [24].

2 Some Basic Observations In this section, we obtain some basic results that will be used in the proof of the main theorem. Proposition 1. If a mapping T satisfies (RSC)-condition and has a fixed point, then it is quasi-nonexpansive mapping. Proof. Let T be a mapping on a subset K of a Banach space E and satisfy (RSC)condition. Suppose T has a fixed point, in other words, z ∈ F(T ). Hence, 0 = 12 z − T z ≤ z − y implies that  1 z − y + Tz − z + y − Ty 3 1 = [z − y + y − Ty] 3  1 ≤ z − y + y − z + z − Ty 3  1 = 2y − z + Tz − Ty 3

T z − Ty ≤

(2)

thus z − Ty = T z − Ty ≤ z − y which completes the proof. Proposition 2. If a mapping T satisfies (RCSC)-condition and has a fixed point, then it is quasi-nonexpansive mapping. Proof. Let T be a mapping on a subset K of a Banach space E and satisfy (RCSC)condition. Suppose T has a fixed point, in other words, z ∈ F(T ). Clearly, 0 = 12 z − T z ≤ z − y implies that  1 z − y + Tz − y + z − Ty 3  1 = 2y − z + Tz − Ty 3

T z − Ty ≤

thus z − Ty = T z − Ty ≤ z − y which completes the proof.

(3)

230

E. Karapinar

Proposition 3. If a mapping T satisfies (HRSC)-condition and has a fixed point, then it is quasi-nonexpansive mapping. Proof. Let T be a mapping on a subset K of a Banach space E and satisfy (HRSC)condition. Suppose T has a fixed point, in other words, z ∈ F(T ). Therefore, 0 = 1 2 z − T z ≤ z − y implies that  1 z − y + Tz − z + y − Ty + Tz − y + z − Ty 5 1 = [2z − y + y − Ty + z − Ty] 5  1 ≤ 2z − y + y − z + z − Ty + z − Ty 5  1 = 3y − z + 2Tz − Ty 5

T z − Ty ≤

(4)

thus z − Ty = T z − Ty ≤ z − y which completes the proof. Proposition 4. Let T be a mapping on a closed subset K of a Banach space E. Assume that T satisfies (RSC)-condition. Then F(T ) is closed. Moreover, if E is strictly convex and K is convex, then F(T ) is also convex. Proof. Let {xn } be a sequence in F(T ) and converge to a point x ∈ K. It is clear that 1 xn − T xn  = 0 ≤ xn − x for n ∈ IN. 2 Thus, we have lim sup xn − T x = lim sup T xn − T x n→∞

n→∞

≤ lim sup n→∞

 1 xn − x + Txn − xn  + x − Tx 3

1 = lim sup [xn − x + x − Tx] n→∞ 3  1 ≤ lim sup 2x − xn + xn − T x . n→∞ 3 Hence,

(5)

2 2 lim sup xn − T x ≤ lim sup x − xn = 0 3 n→∞ 3 n→∞

which implies that {xn } converges to T z. By uniqueness of limit, T z = z. Hence, z ∈ F(T ), that is, F(T ) is closed.

Remarks on Suzuki (C)-Condition

231

Suppose that K is convex and E is strictly convex. Take x, y ∈ F(T ) with x = y and define z = tx + (1 − t)y ∈ K for a fixed t ∈ (0, 1). Since E is strictly convex, then there exits s ∈ [0, 1] such that T z = sx + (1 − s)y. Thus, (1 − s)x − y = T x − T z ≤ = ≤ = =

1 [x − z + Tx − x + Tz − z] 3 1 [x − z + Tz − z] 3 1 [x − z + Tz − T x + T x − z] 3 1 [2x − z + Tz − T x] 3 1 [(1 − s)x − y + 2(1 − t)x − y] 3

(6)

and also sx − y = Ty − T z ≤ = ≤ = =

1 [2y − z + Ty − y + Tz − z] 3 1 [y − z + Tz − z] 3 1 [y − z + Tz − Ty + Ty − z] 3 1 [2y − z + Tz − Ty] 3 1 [sx − y + 2tx − y]. 3

(7)

One conclude from (6) and (7) that (1 − s) ≤ (1 − t) and s ≤ t, and hence s = t. Thus, z ∈ F(T ). Proposition 5. Let T be a mapping on a closed subset K of a Banach space E. Assume that T satisfies (RCSC)-condition. Then F(T ) is closed. Moreover, if E is strictly convex and K is convex, then F(T ) is also convex. Proof. Let {xn } be a sequence in F(T ) which converges to a point x ∈ K. It is clear that 1 xn − T xn  = 0 ≤ xn − x for n ∈ IN. 2

232

E. Karapinar

So, we obtain lim sup xn − T x = lim sup T xn − T x n→∞

n→∞

 1 xn − x + Txn − x + xn − T x n→∞ 3  1 = lim sup 2xn − x + xn − T x . n→∞ 3 ≤ lim sup

Therefore

(8)

2 2 lim sup xn − T x ≤ lim sup x − xn = 0 3 n→∞ 3 n→∞

which implies that {xn } converges to T z. By uniqueness of limit, T z = z. Hence, z ∈ F(T ), that is, F(T ) is closed. Assume that K is convex and E is strictly convex. Take x, y ∈ F(T ) with x = y and define z = tx + (1 − t)y ∈ K for a fixed t ∈ (0, 1). Since E is strictly convex, then there exits s ∈ [0, 1] such that T z = sx + (1 − s)y. Thus, (1 − s)x − y = T x − T z 1 [x − z + Tx − z + Tz − x] 3 1 = [x − z + x − z + Tz − T x] 3 1 = [2x − z + Tz − T x] 3 1 = [(1 − s)x − y + 2(1 − t)x − y] 3 ≤

(9)

and also sx − y = Ty − Tz 1 [y − z + Ty − z + Tz − y]| 3 1 = [2y − z + Tz − y] 3 1 = [sx − y + 2tx − y]. 3

≤

(10)

One conclude from (9) and (10) that (1 − s) ≤ (1 − t) and s ≤ t, and hence s = t. Thus, z ∈ F(T ). Proposition 6. Let T be a mapping on a closed subset K of a Banach space E. Assume that T satisfies (HRSC)-condition. Then F(T ) is closed. Moreover, if E is strictly convex and K is convex, then F(T ) is also convex.

Remarks on Suzuki (C)-Condition

233

Proof. Let {xn } be a sequence in F(T ) and converge to a point x ∈ K. It is clear that 1 xn − T xn  = 0 ≤ xn − x for n ∈ IN. 2 Hence, lim sup xn − T x = lim sup T xn − T x n→∞

n→∞

1 xn − x + Txn − xn  + x − Tx n→∞ 5  +T xn − x + xn − T x

≤ lim sup

 1 2xn − x + x − Tx + xn − T x n→∞ 5  1 ≤ lim sup 2xn − x + x − xn + xn − T x + xn − T x n→∞ 5  1 ≤ lim sup 3xn − x + x − xn + 2xn − T x . (11) n→∞ 5 = lim sup

Consequently, we have 3 3 lim sup xn − T x ≤ lim sup x − xn = 0 5 n→∞ 5 n→∞ which implies that {xn } converges to T z. By uniqueness of limit, T z = z. Hence, z ∈ F(T ), that is, F(T ) is closed. Suppose that K is convex and E is strictly convex. Take x, y ∈ F(T ) with x = y and define z = tx + (1 − t)y ∈ K for a fixed t ∈ (0, 1). Since E is strictly convex, then there exits s ∈ [0, 1] such that T z = sx + (1 − s)y. Hence, (1 − s)x − y = T x − T z ≤ = = ≤ = =

1 [x − z + Tx − x + Tz − z + Tx − z + Tz − x] 5 1 [x − z + x − z + Tz − T x] 5 1 [2x − z + Tz − z + Tz − T x] 5 1 [2x − z + Tz − x + x − z + Tz − T x] 5 1 [3x − z + 2Tz − T x] 5 1 [2(1 − s)x − y + 3(1 − t)x − y] (12) 5

234

E. Karapinar

and also sx − y = Ty − T z ≤ = ≤ = = =

1 [y − z + Ty − y + Tz − z + Ty − z + Tz − y]| 5 1 [2y − z + Tz − z + Tz − y] 5 1 [2y − z + Tz − y + y − z + Tz − y] 5 1 [3y − z + 2Tz − y] 5 1 [3y − z + 2Tz − Ty] 5 1 [2sx − y + 3tx − y]. 5

(13)

One conclude from (12) and (13) that (1 − s) ≤ (1 − t) and s ≤ t, and hence s = t. Thus, z ∈ F(T ). Proposition 7. Let T be a mapping on a closed subset K of a Banach space E. If T satisfies (RSC)-condition, then, for x, y ∈ K, the following hold: (i) T x − T 2 x ≤ x − Tx (ii) Either 12 x − T x ≤ x − y or 12 T x − T 2 x ≤ T x − y   (iii) Either T x − Ty ≤ 13 x − y + T x − x + Ty − y or T 2 x − Ty ≤   1 2 3 T x − y + T x − T x + Ty − y Proof. The first statement follows from (RSC)-condition. Indeed, we always have 1 2 x − T x ≤ x − T x which yields that  1 x − T x + T x − x + T 2 x − T x 3  1 ≤ 2T x − x + T 2 x − T x . 3

T x − T 2 x ≤

Hence, we get (i). It is clear that (iii) is consequence of (ii). To prove (ii), assume the contrary, that is, 1 1 x − Tx > x − y and T x − T 2 x > T x − y 2 2 holds for all x, y ∈ K. Then by triangle inequality and (i), we have x − Tx ≤ x − y + y − Tx <

1 1 x − T x + T x − T 2 x 2 2

Remarks on Suzuki (C)-Condition

≤

235

1 1 x − T x + x − T x = x − T x 2 2

which is a contradiction. Thus, we have (ii). Proposition 8. Let T be a mapping on a closed subset K of a Banach space E and satisfying (RCSC)-condition. Then, for x, y ∈ K, the following hold: (i) T x − T 2 x ≤ x − T x (ii) Either 12 x − T x ≤ x − y or 12 T x − T 2 x ≤ T x − y   (iii) Either T x − Ty ≤ 13 x − y + T x − x + Ty − y or T 2 x − Ty ≤   1 2 3 T x − y + T x − y. + Ty − Tx Proof. The first statement follows from (RCSC)-condition. Indeed, we always have 1 2 x − T x ≤ x − T x which yields that  1 x − Tx + T x − T x + T 2 x − x 3  1 = x − Tx + T 2 x − x 3  1 ≤ x − Tx + T x − x + T 2 x − T x 3  1 ≤ 2T x − x + T 2 x − T x 3

T x − T 2 x ≤

which implies (i). It is clear that (iii) is consequence of (ii). To prove (ii), assume the contrary, that is, 1 1 x − Tx > x − y and T x − T 2 x > T x − y 2 2 holds for all x, y ∈ K. Then by triangle inequality and (i), we have x − Tx ≤ x − y + y − Tx 1 1 x − T x + T x − T 2 x 2 2 1 1 ≤ x − T x + x − T x = x − T x 2 2 <

which is a contradiction. Thus, we have (ii). Proposition 9. Let T be a mapping on a closed subset K of a Banach space E. and satisfy (HRSC)-condition. Then, for x, y ∈ K, the following hold: (i) T x − T 2 x ≤ x − Tx (ii) Either 12 x − T x ≤ x − y or 12 T x − T 2 x ≤ T x − y   (iii) Either T x− Ty ≤ 15 x− y + Tx− x + Ty− y + Tx− y + Ty− x or   T 2 x − Ty ≤ 15 T x − y + T 2 x − T x + Ty − y + T 2 x − y + Ty − Tx

236

E. Karapinar

Proof. The first statement follows from (HRSC)-condition. Indeed, we always have 1 2 x − T x ≤ x − T x which yields that  1 x − T x + Tx − x + T 2 x − T x + T x − T x + T 2 x − x 5  1 = 2x − Tx + T 2 x − T x + T 2 x − x 5  1 ≤ 2x − Tx + T 2 x − T x + Tx − x + T 2 x − T x 5  1 ≤ 3T x − x + 2T 2 x − T x 5

T x − T 2 x ≤

which implies (i). It is clear that (iii) is consequence of (ii). To prove (ii), assume the contrary, that is, 1 1 x − Tx > x − y and T x − T 2 x > T x − y 2 2 holds for all x, y ∈ K. Then by triangle inequality and (i), we have x − Tx ≤ x − y + y − Tx 1 1 x − T x + T x − T 2 x 2 2 1 1 ≤ x − T x + x − T x = x − T x 2 2 <

which is a contraction. Thus, we have (ii).

3 Main Results Proposition 10. Let T be a mapping on a subset K of a Banach space E and satisfy (RSC)-condition. Then x − Ty ≤ 7T x − x + x − y holds for all x, y ∈ K. Proof. Proof is based on Proposition 7 which says that either T x − Ty ≤ x − y or

T 2 x − Ty ≤ T x − y

holds. Consider the first case, then we have x − Ty ≤ x − Tx + T x − Ty 1 ≤ x − Tx + {x − y + Tx − x + Ty − y} 3 4 1 1 ≤ x − T x + x − y + Ty − y 3 3 3 4 1 1 ≤ x − T x + x − y + [Ty − x + x − y]. 3 3 3

Remarks on Suzuki (C)-Condition

237

Consequently, we have 4 2 2 x − Ty ≤ x − T x + x − y 3 3 3 ⇔ x − Ty ≤ 2x − Tx + x − y.

(14)

Take the second case into account. Thus, we obtain that x − Ty ≤ x − T x + T x − T 2 x + T 2 x − Ty 1 ≤ 2x − T x + [T x − y + T 2 x − T x + Ty − y] 3 1 1 7 ≤ x − T x + T x − y + Ty − y 3 3 3 7 1 1 ≤ x − T x + T x − y + [Ty − x + x − y]. 3 3 3 So, we get 7 1 1 x − Ty ≤ x − T x + x − y 3 3 3 ⇔ x − Ty ≤ 7x − Tx + x − y.

(15)

Hence, the result follows by (14) and (15). Regarding the analogy, we omit the proof of the following Corollaries. Corollary 1. Let T be a mapping on a subset K of a Banach space E and satisfy (RCSC)-condition. Then x − Ty ≤ 9T x − x + x − y holds for all x, y ∈ K. Corollary 2. Let T be a mapping on a subset K of a Banach space E and satisfy (HRSC)-condition. Then x − Ty ≤ 15T x − x + x − y holds for all x, y ∈ K. Theorem 4. Let T be a mapping on a compact convex subset K of a Banach space E and satisfies (RSC)-condition. Define a sequence {xn } in K by x1 ∈ K and xn+1 = λ T xn + (1 − λ )xn , for n ∈ IN, where λ lies in [ 12 , 1). Suppose limn→∞ T xn − xn  = 0 holds. Then {xn } converge strongly to a fixed point of T . Proof. Regarding that K is compact, one can conclude that {xn } has an subsequence {xnk } which converges to some number, say z, in K. By Proposition 10, we have xnk − T z ≤ 7T xnk − xnk  + xnk − z, ∀k ∈ IN.

(16)

Notice that limn→∞ T xn − xn  = 0. Taking into account this fact with (16), we conclude that {xnk } converges to T z which implies that T z = z. In other words, z ∈ F(T ). On account of the fact that limn→∞ T xn − xn  = 0, we get

238

E. Karapinar

xn+1 − z ≤ λ T xn − z + (1 − λ )xn − z for n ∈ IN. Thus, {xn } converges to z. Corollary 3. Let T be a mapping on a compact convex subset K of a Banach space E. Define a sequence {xn } in K by x1 ∈ K and xn+1 = λ T xn + (1 − λ )xn , for n ∈ IN, where λ lies in [ 12 , 1). Suppose limn→∞ T xn − xn  = 0 holds. If T satisfies one of the following: 1. (RCSC)-condition. 2. (HRSC)-condition. then {xn } converge strongly to a fixed point of T . Theorem 5. Let E be a Banach space and T, S : K → E such that T (K) ⊂ S(K) and S(K) is compact convex subset of E. Assume that for x, y ∈ K, 1 Sx − Tx ≤ Sx − Sy ⇒ T x − Ty 2 1 ≤ [Sx − Sy + Sx − Tx + Ty − Sy]. 3 Define a sequence {xn } in T (K) by x1 ∈ S(K) and Sxn+1 = λ T xn + (1 − λ )Sxn, for n ∈ IN, where λ lies in [ 12 , 1). Suppose limn→∞ T xn − Sxn  = 0 holds. Then T and S have a coincidence point. Proof. Let R : S(K) → S(K), where Ra = T (S−1 a) for each a ∈ S(K). It is clear that R is well defined. Indeed, take x, y ∈ S−1a such that b = T x and c = Ty. For x ∈ S−1a, we obtain Ra = T x and Ra ⊂ S(K) since T (K) ⊂ S(K). Thus, since Sx = Sy, we get b = c. Thus, R is well defined. Let a, b ∈ S(K) such that 12 a − Ra ≤ a − b. Then for x ∈ S−1 a and y ∈ S−1b, one has 1 1 Sx − Tx = a − Ra ≤ a − b = Sx − Sy 2 2 1 ⇒ T x − Ty ≤ [Sx − Sy + Sx − Tx + Ty − Sy]. 3 Thus, 12 a − Ra ≤ a − b implies that 1 Ra − Rb ≤ [a − b + Ra − b + Rb − a]. 3 Thus, all conditions of Theorem 4 are satisfied. Thus, R has a common fixed point, say t. Then for any z ∈ S−1t, we have T z = Rt = t = Sz. Hence, S and T have a coincidence point.

Remarks on Suzuki (C)-Condition

239

Corollary 4. Let E be a Banach space and T, S : K → E such that T (K) ⊂ S(K) and S(K) is compact convex subset of E. Define a sequence {xn } in T (K) by x1 ∈ S(K) and Sxn+1 = λ T xn + (1 − λ )Sxn , for n ∈ IN, where λ lies in [ 12 , 1). Suppose limn→∞ T xn − Sxn  = 0 holds. If S, T satisfy one of the following: (i)

1 2 Sx − Tx

≤ Sx − Sy implies that 1 T x − Ty ≤ [Sx − Ty + Tx − Sy], 2

(ii)

1 2 Sx − Tx

≤ Sx − Sy implies that

1 T x − Ty ≤ {[Sx − Sy + Sx − Ty + Tx − Sy]}, 3 (iii)

1 2 Sx − Tx

≤ Sx − Sy implies that

T x − Ty ≤

1 {[Sx − Sy + Sx − Tx 5 +Ty − Sy + Sx − Ty + Tx − Sy]},

then T and S have a coincidence point. Definition 3. Let E be a Banach space. E is said to have Opial property [23] if for each weakly convergent sequence {xn } in E with weak limit z lim inf xn − z ≤ lim inf xn − y, for all y ∈ E with y = z. n→∞

n→∞

All Hilbert spaces, all finite dimensional Banach space and Banach sequence spaces  p (1 ≤ p < ∞) have the Opial property (See [21]). Proposition 11. Let T be a mapping on a subset K of a Banach space E with Opial property and satisfy (RSC)-condition. If {xn } converges weakly to z and limn→∞ T xn − xn  = 0, then T z = z. That is, I − T is demiclosed at zero. Proof. Due to Proposition 10, we have xn − T z ≤ 7T xn − xn  + xn − z, for all n ∈ IN. Hence, lim inf xn − T z ≤ lim inf xn − z. n→∞

n→∞

Thus, Opial property implies that T z = z. Corollary 5. Let T be a mapping on a subset K of a Banach space E with Opial property and satisfy one of the following 1. (RSSC)-condition. 2. (HRSC)-condition.

240

E. Karapinar

If {xn } converges weakly to z and limn→∞ T xn − xn  = 0, then T z = z. That is, I − T is demiclosed at zero. Theorem 6. Let T be a mapping on a weakly compact convex subset K of a Banach space E with Opial property and satisfy (RSC)-condition. Define a sequence {xn } in K by x1 ∈ K and xn+1 = λ T xn + (1 − λ )xn, for n ∈ IN, where λ lies in [ 12 , 1). Suppose limn→∞ T xn − xn  = 0 holds. Then {xn } converge weakly to a fixed point of T . Proof. We have limn→∞ T xn − xn  = 0. Since K is weakly compact, one can conclude that {xn } has an subsequence {xnk } which converges weakly to a number, say z, in E. On account of Proposition 11, we observe that z is a fixed point of T . Note that {xn − z} is a nondecreasing sequence. Indeed, xn+1 − z ≤ λ T xn − z + (1 − λ )xn − z. We show that {xn } converges to z. Assume the contrary, that is, {xn } does not converge to z. Then there exists a subsequence {xnm } of {xn } and u ∈ K such that {xnm } converges weakly to u and u = z. By Proposition 11 Tu = u. Since E has a Opial property, lim xn − z = lim xnk − z

n→∞

k→∞

< lim xnk − u = lim xn − u = lim xnm − u k→∞

n→∞

m→∞

< lim xnm − z = lim xn − z m→∞

n→∞

(17)

which is a contradiction. Hence, proof is completed. Corollary 6. Let T be a mapping on a weakly compact convex subset K of a Banach space E with Opial property and satisfy one of the following 1. (RSSC)-condition. 2. (HRSC)-condition. Define a sequence {xn } in K by x1 ∈ K and xn+1 = λ T xn + (1 − λ )xn , for n ∈ IN, where λ lies in [ 12 , 1). Suppose limn→∞ T xn − xn  = 0 holds. Then {xn } converge weakly to a fixed point of T . Theorem 7. Let E be a Banach space and T, S : K → E such that T (K) ⊂ S(K) and S(K) is weakly compact convex subset of E with Opial property. Assume for x, y ∈ K, 12 Sx − Tx ≤ Sx − Sy implies that 1 T x − Ty ≤ [Sx − Sy + Tx − Sy + Sx − Ty]. 3 Define a sequence {xn } in T (K) by x1 ∈ S(K) and Sxn+1 = λ T xn + (1 − λ )Sxn, for n ∈ IN, where λ lies in [ 12 , 1). Suppose limn→∞ T xn − Sxn  = 0 holds. Then T and S have a coincidence point.

Remarks on Suzuki (C)-Condition

241

Regarding the analogy with the proof of Theorem 5, we omit the proof. Corollary 7. Let E be a Banach space and T, S : K → E such that T (K) ⊂ S(K) and S(K) is weakly compact convex subset of E with Opial property. Define a sequence {xn } in T (K) by x1 ∈ S(K) and Sxn+1 = λ T xn + (1 − λ )Sxn , for n ∈ IN, where λ lies in [ 12 , 1). Suppose limn→∞ T xn − Sxn  = 0 holds. If S, T satisfy one of the following: (i)

1 2 Sx − Tx

≤ Sx − Sy implies that

1 T x − Ty ≤ {[Sx − Sy + Sx − Ty + Tx − Sy]}, 2 (ii)

1 2 Sx − Tx

(18)

≤ Sx − Sy implies that T x − Ty ≤

1 [Sx − Sy + Sx − Tx 5 +Ty − Sy + Sx − Ty + Tx − Sy],

(19)

then T and S have a coincidence point. A Banach space E is called strictly convex if x + y < 2 for all x, y ∈ E with x = y = 1 and x = y. A Banach space E is called uniformly convex in every direction (in short, UCED) if for ε ∈ (0, 2] and z ∈ E with z = 1, there exists δ := δ (ε , z) > 0 such that x + y ≤ 2(1 − δ ) for all x, y ∈ E with x ≤ 1, y ≤ 1 and x − y ∈ {tz : t ∈ [−2, −ε ] ∪ [ε , 2]}. Lemma 1. (See [21]) For a Banach space E, the following are equivalent: 1. E is UCED 2. If sequences {un } and {vn } in E satisfy limn→∞ un  = 1 = limn→∞ vn , limn→∞ un + vn  and {un − vn } ⊂ {tw : t ∈ IR} for some w ∈ E with w = 1, then limn→∞ un − vn  = 0 holds. Lemma 2. (See [21]) For a Banach space E, the following are equivalent: 1. E is UCED 2. If {xn } is a bounded sequence in E, then a function f on E defined by f (x) = lim supn→∞ xn − x is strictly quasi-convex, that is, f (tx + (1 − t)y) < max{ f (x), f (y)} for all t ∈ (0, 1) and x, y ∈ E with x = y. Theorem 8. Let T be a mapping on a weakly compact convex subset K of a UCED Banach space E and satisfy (RSC)-condition. Define a sequence {xn } in K by x1 ∈ K and xn+1 = λ T xn + (1 − λ )xn , for n ∈ IN, where λ lies in [ 12 , 1). Suppose limn→∞ T xn − xn  = 0 holds. Then T has a fixed point.

242

E. Karapinar

Proof. Set a sequence {xn } in K in a way that xn+1 = 12 T xn + 12 xn for each n ∈ IN where x1 ∈ K. Notice that lim supn→∞ T xn − xn  = 0. Define a continuous convex function f from K into [0, ∞) by f (x) = lim supn→∞ xn − x, for all x ∈ K. Since K is weakly compact and f is weakly lower semicontinuous, there exists z ∈ K such that f (z) = min{ f (x) : x ∈ K}. Regarding Proposition 10, we have xn − T z ≤ 7T xn − xn  + xn − z and thus f (T z) ≤ f (z). On account of f (z) is the minimum, f (z) = f (T z) holds. To show T z = z, we assume the contrary, that is, T z = z. Since f is strictly quasi-convex, we have  f (z) ≤ f

z+Tz 2

< max{ f (z), f (T z)} = f (z)

which is a contradiction. Thus, we get the desired result. Corollary 8. Let T be a mapping on a weakly compact convex subset K of a UCED Banach space E and satisfy one of the following 1. (RSSC)-condition. 2. (HRSC)-condition. Define a sequence {xn } in K by x1 ∈ K and xn+1 = λ T xn + (1 − λ )xn , for n ∈ IN, where λ lies in [ 12 , 1). Suppose limn→∞ T xn − xn  = 0 holds. Then T has a fixed point. Theorem 9. Let S be a family of commuting mappings on a weakly compact convex subset K of a Banach space E. Suppose each mapping in S satisfy (RSC)condition. Then S has a common fixed point. Proof. Let I = {1, 2, ..., ν } be an index set. Let Ti ∈ S , i ∈ I. Due to Theorem 8, Ti has a fixed point in K, that is, F(Ti ) = 0/ for i ∈ I. Proposition 4 implies that each F(Ti ) is closed and convex. Suppose that F := ∩k−1 i=1 F(Ti ) is nonempty, closed, and convex for some k ∈ IN such that 1 < k ≤ ν . For x ∈ F and i ∈ I with 1 ≤ i < k, Tk x = Tk ◦ Ti x = Ti ◦ Tk x since S is commuting. Thus, Tk x is a fixed point of Ti which yields Tk x ∈ F. So, Tk (F) ⊂ F. In other words, Tk (F) ⊂ F. By Theorem 8, Tk has a fixed point in F, that is, F ∩ F(Tk ) = ∩ki=1 F(Ti ) = 0. / Due to Proposition 4, this set is closed and convex. By induction, we obtain ∩νi=1 F(Ti ) = 0. / This is equivalent to say that {F(T ) : T ∈ S } has the finite intersection property. Since K is weakly compact and F(T ) is weakly closed for every T ∈ S , then ∩T ∈S F(T ) = 0. / Corollary 9. Let S be a family of commuting mappings on a weakly compact convex subset K of a Banach space E. Suppose each mapping in S satisfy one of the following 1. (RSSC)-condition. 2. (HRSC)-condition. Then S has a common fixed point.

Remarks on Suzuki (C)-Condition

243

Acknowledgments I would like to thank to Professor Dimitru BALENAU who encouraged and supported me to attend NSC2010.

References 1. Abdeljawad T, Karapınar E (2009) Quasi-cone metric spaces and generalizations of Caristi Kirk’s theorem. Fixed Point Theory Appl 2009:9. Article ID 574387. doi:10.1155/ 2009/574387 2. Abdeljawad T, Karapınar E (2010) A gap in the paper “A note on cone metric fixed point theory and its equivalence”. [Nonlinear Anal 72(5):2259–2261]. Gazi Univ J Sci 24(2), 233– 234 (2011) 3. Abdeljawad T, Karapınar E (2011) A common fixed point theorem of a Gregus type on convex cone metric spaces. J Comput Anal Appl 13(4):609–621 4. Banach S (1922) Surles operations dans les ensembles abstraits et leur application aux equations itegrales. Fund Math 3:133–181 5. Chatterjea SK (1972) Fixed-point theorems. C R Acad Bulgare Sci 25:727–730 ´ c LB (1974) A generalization of Banach principle. Proc Am Math Soc 45:267–273 6. Ciri´ 7. Edelstein M (1962) On fixed and periodic points under contractive mappings. J Lond Math Soc 37:74–79 8. Hardy GE, Rogers TD (1973) A generalization of a fixed point theorem of Reich. Canad Math Bull 16:201–206 9. Kannan R (1968) Some results on fixed points. Bull Cal Math Soc 60:71–76 10. Karapınar E (2009) Fixed point theorems in cone Banach spaces. Fixed Point Theory Appl 2009:9. Article ID 609281. doi:10.1155/2009/609281 11. Karapınar E (2010) Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput Math Appl 59(12):3656–3668. doi:10.1016/j.camwa.2010.03.062 (SCI) 12. Karapınar E (2010) Some fixed point theorems on the cone Banach spaces. In: Proceedings of 7th ISAAC congress, World Scientific, Singapore/Hackensack/London, pp 606–612 ´ c type on cone metric spaces. 13. Karapınar E (2010) Some nonunique fixed point theorems of Ciri´ Abstr Appl Anal 2010:14. Article ID 123094. doi:10.1155/2010/123094 14. Karapınar E (2011) Couple fixed point on cone metric spaces. Gazi Univ J Sci 21(1):51–58 15. Karapınar E (2011) Fixed point theory for cyclic weak-φ -contraction. Appl Math Lett 24(6):822–825. doi:10.1016/j.aml.2010.12.016 16. Karapınar E (2011) Weak φ -contraction on partial contraction and existence of fixed points in partially ordered sets, Mathematica Aeterna 1(4):237–244 17. Karapınar E, T¨urkolu DA (2010) Best approximations theorem for a couple in cone Banach space. Fixed Point Theory Appl 2010:9. Article ID 784578 18. Karapınar E, Y¨uksel U (2011) On common fixed point theorems without commuting conditions in tvs-cone metric spaces. J Comput Anal Appl 13(6):1115–1122 19. Reich S (1971) Kannan’s fixed point theorem. Boll Un Mat. Ital 4(4):1–11 20. Suzuki K (2008) A generalized Banach contraction principle that characterizes metric completeness. Proc Am Math Soc 136:1861–1869 21. Suzuki K (2008) Fixed point theorems and convergence theorems for some generalized non expansive mappings. J Math Anal Appl 340:1088–1095 22. Suzuki K (2009) A new type of fixed point theorem in metric spaces. Nonlinear Anal Theory Methods Appl 71(11):5313–5317 23. Opial Z (1967) Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull Am Math Soc 73:591–597 24. Singh SL, Mishra SN (2010) Remarks on recent fixed point theorems. Fixed Point Theory Appl 2010:18. Article ID 452905. doi:10.1155/2010/452905

On the Eigenvalues of a Non-Hermitian Hamiltonian Ebru Ergun

Abstract We study a 4 × 4 complex matrix Jacobi (tri-diagonal matrix) arised from a non-Hermitian discrete quantum system. Reality of the eigenvalues of the matrix in question is investigated.

1 Introduction Some important problems of mathematical physics give rise to the consideration of non-Hermitian (non-selfadjoint) operators [3, 4, 7]. For the past 10 years, non-Hermitian Hamiltonians and complex extension of quantum mechanics have received a lot of attention, see review papers [1,10]. It turned out that non-Hermitian operators having real eigenvalues (spectrum) can also be used in quantum mechanics. This motivates construction and investigation of non-Hermitian operators with real spectrum. Recently, the author considered in [5] the following discrete nonHermitian problem: −Δ 2 yn−1 + qnyn = λ ρn yn ,

n ∈ Ω = {−M, . . . , −2, −1} ∪ {2, 3, . . ., N}, (1)

y−1 = y1 ,

Δ y−1 = e2iδ Δ y1 ,

y−M−1 = yN+1 = 0,

(2) (3)

N+1 is a desired solution, where M ≥ 1 and N ≥ 2 are some fixed integers, (yn )n=−M−1 Δ is the forward difference operator defined by

Δ yn = yn+1 − yn

so that

Δ 2 yn−1 = yn−1 − 2yn + yn+1 ,

E. Ergun () Department of Physics, Ankara University, 06100 Tandogan, Ankara, Turkey e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 13, © Springer Science+Business Media, LLC 2012

245

246

E. Ergun

the coefficients qn are real numbers given for n ∈ Ω , δ ∈ [0, π /2), and ρn are given for n ∈ Ω by  2iδ i f n ≤ −1, e (4) ρn = e−2iδ i f n ≥ 2. The problem (1)–(3) can be viewed as a discrete analogue of the real line equivalent of some continuous quantum systems on a complex contour examined in [2, 9]. The main distinguishing features of problem (1)–(3) are that it involves a complex-valued coefficient ρn of the form (4) and that transition conditions (impulse conditions) of the form (2) are presented which also involve a complex coefficient. Such a problem is non-Hermitian with respect to the usual inner product. In spite of this fact, the eigenvalues (spectrum) of problem (1)–(3) may be real. The eigenvalue problem (1)–(3) can be reduced to investigation of the eigenvalues and eigenvectors of a complex Jacobi matrix (tri-diagonal matrix) and this matrix plays the role of a Hamiltonian. The main question related to the problem (1)–(3), in which we are interested, is to find the conditions on qn and δ under which the eigenvalues of this problem are all real. Solution of this problem for arbitrary M and N turns out to be rather complicated. If δ = 0, then the problem (1)–(3) is selfadjoint (see [5]) and hence its eigenvalues are all real in this case. Further we will assume that δ ∈ (0, π /2). In [5], it is shown that if M = 1, N = 2 (2 × 2 matrix case), and δ ∈ (0, π /2), then the eigenvalues of problem (1)–(3) are all real if and only if q−1 = 0,

q2 = −1,

and

 π . δ ∈ 0, 6

Moreover, under these conditions the eigenvalues are positive, distinct for δ ∈ (0, π /6) and equal to each other for δ = π /6. In [6], it is proved that if M = 1, N = 3 (3 × 3 matrix case), then the eigenvalues of problem (1)–(3) are all real if and only if a − 1 = b + c,

(5)

b + c − 1 + 2(bc − 1) cos2δ = 0,

(6)

(b + c)(bc − 1) + 2(bc − c − 1) cos2δ = 0,

(7)

α 2 β 2 − 4β 3 − 27γ 2 − 4α 3 γ + 18αβ γ ≥ 0,

(8)

α = −1 − 2(b + c) cos2δ ,

(9)

where

β = (b + c)2 − bc + 1, a = 2 + q−1,

γ = bc − c − 1,

b = 2 + q2 ,

c = 2 + q3 .

(10) (11)

On the Eigenvalues of a Non-Hermitian Hamiltonian

247

In the present paper, we investigate reality of the eigenvalues of problem (1)–(3) in the case M = 2 and N = 3 (4 × 4 matrix case).

2 The 3×3 non-Hermitian system For the completeness, in this section following [6] we prove that conditions (5)–(8) are necessary and sufficient for the reality of all eigenvalues of the matrix ⎤ 0 1 + (a − 1)e−2iδ −1 A=⎣ −e2iδ be2iδ −e2iδ ⎦ . 0 −e2iδ ce2iδ ⎡

(12)

Note that the eigenvalues of problem (1)–(3) in the case M = 1 and N = 3 coincide with the eigenvalues of the 3 × 3 matrix A given in (12), where a, b, and c are real numbers defined by (11). First we prove the necessity of conditions (5)–(8). Assume that the eigenvalues of the matrix A are all real. The eigenvalues of the matrix A coincide with the roots of the characteristic equation det(A − λ I3) = 0 that is,

λ 3 − [1 + (a − 1)e−2iδ + (b + c)e2iδ ]λ 2 +[(a − 1)(b + c) + (b + c − 1)e2iδ + (bc − 1)e4iδ ]λ −(a − 1)(bc − 1)e2iδ − (bc − c − 1)e4iδ = 0.

(13)

Denote the roots of (13) by λ1 , λ2 , and λ3 . By well-known relations between the roots and the coefficients of a polynomial, we have

λ1 + λ2 + λ3 = 1 + (a − 1)e−2iδ + (b + c)e2iδ , λ1 λ2 + λ1 λ3 + λ2 λ3 = (a − 1)(b + c) + (b + c − 1)e2iδ + (bc − 1)e4iδ , λ1 λ2 λ3 = (a − 1)(bc − 1)e2iδ + (bc − c − 1)e4iδ . If the roots λ1 , λ2 , and λ3 are all real, then left-hand sides in the last three equations are real. Then the right-hand sides must also be real. This yields (1 − a + b + c) sin2δ = 0,

(14)

(b + c − 1) sin2δ + (bc − 1) sin4δ = 0,

(15)

(a − 1)(bc − 1) sin2δ + (bc − c − 1) sin4δ = 0.

(16)

248

E. Ergun

Since sin 2δ = 0 for δ ∈ (0, π /2) and sin 4δ = 2 sin 2δ cos 2δ , we get from (14)–(16), 1 − a + b + c = 0, b + c − 1 + 2(bc − 1) cos2δ = 0, (a − 1)(bc − 1) + 2(bc − c − 1) cos2δ = 0. Therefore, the necessity of conditions (5)–(7) is proved. We will need the following known result (see, for example, [8]). Proposition 1. Consider the cubic equation

μ 3 + pμ + q = 0

(17)

with the real coefficients p and q. Let us put D = −4p3 − 27q2

(18)

that is called the discriminant of (17). Then: (1) If D < 0, then (17) has one real root and two nonreal complex conjugate roots (2) If D = 0, then all three roots of (17) are real and at least two of them are equal to each other (3) If D > 0, then (17) has three distinct real roots If the eigenvalues of the matrix A are all real, then as we have shown the conditions (5)–(7) hold. Therefore, in this case the characteristic equation (13) takes the form

λ 3 − [1 + (a + b + c − 1) cos2δ ]λ 2 +[(a − 1)(b + c) + (b + c − 1) cos2δ + (bc − 1) cos4δ ]λ −(a − 1)(bc − 1) cos2δ − (bc − c − 1) cos4δ = 0. Using the identity cos 4δ = 2 cos2 2δ − 1 we can rewrite the last equation in the form

λ 3 − [1 + (a + b + c − 1) cos2δ ]λ 2 +{(a − 1)(b + c) − bc + 1 + [b + c − 1 + 2(bc − 1) cos2δ ] cos 2δ }λ +bc − c − 1 − [(a − 1)(bc − 1) + 2(bc − c − 1) cos2δ ] cos 2δ = 0. Hence, taking into account (5)–(7), we get

λ 3 − [1 + 2(b + c) cos2δ ]λ 2 + [(b + c)2 − bc + 1]λ + bc − c − 1 = 0.

(19)

On the Eigenvalues of a Non-Hermitian Hamiltonian

249

Thus, if the eigenvalues of the matrix A defined in (12) are all real, then a, b, c, and δ must satisfy the necessary conditions (5)–(7) and the eigenvalues of A coincide with the roots of (19). Now we define α , β , and γ by (9), (10), and rewrite (19) in the form λ 3 + αλ 2 + β λ + γ = 0. (20) If we put

λ =μ−

α 3

(21)

in (20), then we get the equation

μ 3 + pμ + q = 0

(22)

which does not contain the term with μ 2 , where 1 p = − α2 + β , 3

(23)

2 3 1 α − αβ + γ . (24) 27 3 Note that since a, b, c, and cos 2δ are real, the numbers α , β , and γ defined by (9), (10) and hence the numbers p and q defined by (23), (24) are real. The roots of (20) and (22) are connected by (21). Therefore, the reality of roots of (20) is equivalent to the reality of roots of (22). By Proposition 1, the roots of (22) are all real if and only if its discriminant D is non-negative. On the other hand, for this equation q=

D = −4p3 − 27q2

3

2 1 2 3 1 = −4 − α 2 + β − 27 α − αβ + γ 3 27 3 = α 2 β 2 − 4β 3 − 27γ 2 − 4α 3γ + 18αβ γ . Therefore, the necessity of the condition (8) is also proved. Now we prove the sufficiency of the conditions (5)–(8). Thus assume that the conditions (5)–(8) are satisfied. We have to show that then the eigenvalues of the matrix A are all real. The eigenvalues of A coincide with the roots of (13). Under the conditions (5)–(7), (13) reduces to the (19). Further, in virtue of Proposition 1, the roots of (19) are all real under the condition (8) in which α , β , and γ are defined by (9), (10) The proof is completed. Example 1. The conditions (5)–(7) in general are not sufficient for reality of the eigenvalues of A. Indeed, the numbers 3 a= , 2

1 b= , 2

c = 0,

cos 2δ = −

1 4

250

E. Ergun

satisfy the conditions (5)–(7). Next, for these numbers we have, according to (9), (10), 3 5 α = − , β = , γ = −1, 4 4 and then according to (23), (24), p=

17 , 16

q=−

23 . 32

Therefore, D = −4p3 − 27q2 < 0 and in virtue of Proposition 1 the roots of (22) and hence the eigenvalues of the matrix A are not all real. It follows that, if at least one of the conditions (5)–(8) is not satisfied, then not all eigenvalues of the matrix A are real.

3 4 × 4 Non-Hermitian Hamiltonian Since Δ 2 yn−1 = yn−1 − 2yn + yn+1 and from the second condition in (2) we have y0 − y−1 = e2iδ (y2 − y1 ) so that (taking into account y1 = y−1 ) y0 = y−1 + e2iδ (y2 − y1 ) = (1 − e2iδ )y−1 + e2iδ y2 , problem (1)–(3) can be written as − yn−1 + vn yn − yn+1 = λ ρn yn ,

n ∈ {−M, . . . , −2, −1} ∪ {2, 3, . . ., N},

y0 = (1 − e2iδ )y−1 + e2iδ y2 ,

y1 = y−1 ,

y−M−1 = yN+1 = 0,

(25) (26) (27)

where vn = 2 + q n ,

n ∈ {−M, . . . , −2, −1} ∪ {2, 3, . . ., N}.

In the case M = 2 and N = 3, problem (25)–(27) takes the form −yn−1 + vn yn − yn+1 = λ ρn yn ,

n ∈ {−2, −1, 2, 3},

y0 = (1 − e2iδ )y−1 + e2iδ y2 , y−3 = y4 = 0.

y1 = y−1 ,

(28)

On the Eigenvalues of a Non-Hermitian Hamiltonian

Hence,

251

⎫ −y−3 + v−2y−2 − y−1 = λ ρ−2 y−2 ⎪ ⎪ ⎬ −y−2 + v−1y−1 − y0 = λ ρ−1 y−1 , ⎪ −y1 + v2 y2 − y3 = λ ρ2 y2 ⎪ ⎭ −y2 + v3 y3 − y4 = λ ρ3 y3 y0 = (1 − e2iδ )y−1 + e2iδ y2 ,

(29)

y1 = y−1 ,

(30)

y−3 = y4 = 0.

(31)

Substituting (30) and (31) into (29) and using the explicit expression (4) for ρn , we get ⎫ v−2 e−2iδ y−2 − e−2iδ y−1 = λ y−2 ⎪ ⎪ ⎪ ⎪ −2i δ −2i δ −e y−2 + [1 + (v−1 − 1)e ]y−1 − y2 = λ y−1 ⎬ . (32) ⎪ −e2iδ y−1 + v2 e2iδ y2 − e2iδ y3 = λ y2 ⎪ ⎪ ⎪ ⎭ −e2iδ y2 + v3 e2iδ y3 = λ y3 . Setting ⎡

v−2 = d, de−2iδ

v−1 = a,

−e−2iδ

v2 = b, 0

0

v3 = c, ⎤

⎥ ⎢ −2iδ ⎢ −e 1 + (a − 1)e−2iδ −1 0 ⎥ ⎥, ⎢ A=⎢ ⎥ be2iδ −e2iδ ⎦ −e2iδ ⎣ 0 0 we can write (32) in the form

y−2

⎤

⎥ ⎢ ⎢ y−1 ⎥ ⎥, ⎢ y=⎢ ⎥ ⎣ y2 ⎦

−e2iδ ce2iδ

0

⎡

(33)

y3

Ay = λ y.

Thus, the eigenvalues of problem (1)–(3) in the case M = 2 and N = 3 coincide with the eigenvalues of the matrix A given in (33), where d = v−2 = 2 + q−2, b = v2 = 2 + q2 ,

a = v−1 = 2 + q−1, c = v3 = 2 + q 3 .

4 Reality of the Eigenvalues Now we investigate conditions for reality of the eigenvalues of the matrix A defined in (33). The eigenvalues of the matrix A coincide with the roots of the characteristic equation det(A − λ I4) = 0 that is,

λ 4 + αλ 3 + β λ 2 + γλ + θ = 0,

(34)

252

E. Ergun

where

α = −1 − (a − d − 1)e−2iδ − (b + c)e2iδ , β = de−2iδ + (ad − d − 1)e−4iδ + d(b + c) + (a − 1)(b + c) +(b + c − 1)e2iδ + (bc − 1)e4iδ ,

γ = [−d(a − 1)(bc − 1) + b + c]e−2iδ − d(b + c − 1) −(a + d − 1)(bc − 1)e2iδ − (bc − c − 1)e4iδ ,

θ = (ad − d − 1)(bc − 1) + d(bc − c − 1)e2iδ . If the roots of (34) are all real, then by well-known relations between the roots and the coefficients of a polynomial we get that the coefficients must be real. This gives the necessary conditions (for reality of the eigenvalues of A) (a − b − c − d − 1) sin 2δ = 0, (b + c − d − 1) sin 2δ + (bc − ad + d) sin 4δ = 0, [−d(a − 1)(bc − 1) + b + c + (a + d − 1)(bc − 1)] sin2δ + (bc − c − 1) sin4δ = 0, d(bc − c − 1) sin2δ = 0. Since sin 2δ = 0 for δ ∈ (0, π /2) and sin 4δ = 2 sin 2δ cos 2δ , we get from the last conditions that a − b − c − d − 1 = 0, b + c − d − 1 + 2(bc − ad + d) cos2δ = 0, −d(a − 1)(bc − 1) + b + c + (a + d − 1)(bc − 1) + 2(bc − c − 1) cos2δ = 0, d(bc − c − 1) = 0. Then the coefficients α , β , γ and θ take the form

α = −1 − (a + b + c − d − 1) cos2δ ,

(35)

β = d(b + c) + (a − 1)(b + c) + (b + c + d − 1) cos2δ +(ad + bc − d − 2) cos4δ ,

(36)

γ = −d(b + c − 1) − [d(a − 1)(bc − 1) + (a + d − 1)(bc − 1) − b − c] cos2δ −(bc − c − 1) cos4δ ,

(37)

θ = (ad − d − 1)(bc − 1).

(38)

Thus, we have to investigate conditions for reality of roots of the fourth order polynomial equation (34) with the real coefficients given by formulas (35)–(38).

On the Eigenvalues of a Non-Hermitian Hamiltonian

253

This problem can be reduced to the case of third order polynomial equation (investigated in [6]) as follows. First, if we make in (34) the change of variable

λ =μ−

α , 4

then we get an equation of the form

μ 4 + pμ 2 + qμ + r = 0

(39)

with the real coefficients p, q, and r, which does not contain the term with μ 3 . Next, using an auxiliary parameter t we can transform the left-hand side of this equation as follows: 2  p p2 μ 4 + pμ 2 + qμ + r = μ 2 + + t + qμ + r − − t 2 − 2t μ 2 − pt 2 4

    2 p p2 2 2 2 = μ + + t − 2t μ − qμ + t + pt − r + . (40) 2 4 Choose now t so that the polynomial in μ standing in the brackets is a complete square. For this it must have a single double root, that is, the equality

p2 2 q − 4 · 2t t + pt − r + =0 4 2

(41)

must be held. The equality (41) is a cubic equation in t with real coefficients. Let t0 be a real root of this equation. For t = t0 the polynomial standing in the brackets in (40) has double root q/(4t0 ) and therefore, (39) takes the form 

μ2 +



2 p q 2 + t0 − 2t0 μ − =0 2 4t0

which is splited into the two equations  μ − 2t0 μ +



2

μ2 +

 2t0 μ +



q p + t0 + √ 2 2 2t0 q p + t0 − √ 2 2 2t0

= 0,

(42)

= 0.

(43)



Therefore, the roots of (39) coincide with the roots of quadratic equations (42), (43). Acknowledgments This work was supported by Grant 109T032 from the Scientific and Technological Research Council of Turkey (TUBITAK).

254

E. Ergun

References 1. Bender CM (2007) Making sense of non-Hermitian Hamiltonians. Rep Prog Phys 70:947–1018 2. Bender CM, Boettcher S (1998) Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys Rev Lett 80:5243–5246 3. Davies EB (2002) Non-selfadjoint differential operators. Bull Lond Math Soc 34:513–532 4. Dolph CI (1961) Recent developments in some non-selfadjoint problems of mathematical physics. Bull Am Math Soc 67:1–69 5. Ergun E (2009) On the reality of the spectrum of a non-Hermitian discrete Hamiltonian. Rep Math Phys 63:75–93 6. Ergun E, Bairamov E (2011) On the eigenvalues of a 3 by 3 non-Hermitian Hamiltonian. J Math Chem 49:609–617 7. Guseinov GSh (2009) Inverse spectral problems for tridiagonal N by N complex Hamiltonians. SIGMA 5(Paper 018):28 8. Kurosh A (1975) Higher algebra. Mir Publisher, Moscow 9. Mostafazadeh A (2005) Pseudo-Hermitian description of PT -symmetric systems defined on a complex contour. J Phys A Math Gen 38:3213–3234 10. Mostafazadeh A (2010) Pseudo-Hermitian representation of quantum mechanics. Int J Geom Methods Mod Phys 7:1191–1306

Part III

Nonlinear Physics

Perturbation Methods for Solitons and Their Behavior as Particles L.A. Ostrovsky

Abstract This paper is a partial summary of two plenary presentations by the author at two related conferences “Nonlinear Science and Complexity” and “New Trends in Nanotechnology and Nonlinear Dynamical Systems,” held in July 2010 in Ankara, Turkey. It outlines both well established and some recent achievements in asymptotic perturbation theory of solitary waves (solitons) and its applications to internal gravity waves in the ocean.

1 Introduction The majority of presentations at this meeting dealt with lumped dynamical systems described by ordinary differential equations. However, nonlinear waves can behave in an even more complex way. In this brief review we outline dynamics of the localized nonlinear waves, solitary waves or solitons (here we do not distinguish between these two terms) considered as compact, particle-like entities. When a soliton is not localized but represents a transition between two constant states, it is typically identified as a kink. The particle-like behavior of solitons which can escape collisions unchanged is known since 1960s when Martin Kruskal and Norman Zabusky discovered this property in numerical experiments. Since that soliton interaction was intensely studied in the framework of the exact “inverse scattering theory,” a real achievement in mathematical physics. For this topic see the book by Ablowitz and Segur [1]. However important the new exact theory would be in mathematics and mathematical physics, its practical applications are rather limited, mainly by the so-called fully integrable equations having an infinite number of integrals. As a result the interaction of N solitons which are primarily well separated, ends up as the same

L.A. Ostrovsky () Zel Technologies and University of Colorado, Boulder, Colorado, USA e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 14, © Springer Science+Business Media, LLC 2012

257

258

L.A. Ostrovsky

separated N solitons with their positions inverted in space. If an equation is nonintegrable, such interaction typically adds some radiation. Moreover, most factors of physical reality such as, for example, dissipation, make the nonlinear wave equations non-integrable. Thus, several approximate schemes were suggested since 1970s. Here we briefly outline the direct perturbation scheme suggested in Gorshkov and Ostrovsky [2].

2 Direct Perturbation Method for Solitons Consider a general system of N nonlinear equations: M(u, ut , ux , T, X) = ε R.

(1)

Here u = {u1 , u2 , . . ., uN } is a vector of unknown variables, M is a set of nonlinear functions, R describes perturbations; T = ε t and X = ε x are “slow” time and coordinate on which the equation parameters can depend, and ε << 1 is a small parameter. It is supposed that at ε = 0, the set (1) has a solution in the form of a family of stationary waves, u(r, t) = U(ζ , A), where ζ = x − Vt, V is a constant wave velocity, and A = {A1 , . . . , AN } is a vector of constant parameters. This solution satisfies the unperturbed set of ODEs,   M U, −VUζ ,Uζ = 0.

(2)

Here, we suppose that the basic waves are solitary, so that U(ζ → ±∞) = U± (A). For a perturbed system (1), when ε = 0, we represent the solution by a series J

u (x,t) = U (ζ , T, X) + ∑ ε n u(n) (ζ , T, X) ,

(3)

n=1

where J is an integer determining accuracy of the solution (the order of approxima tion) and ζ = x − V (T, X)dt. Substituting this into (1), we obtain, in each order of ε , a linear inhomogeneous equation P(u(n) ) = H (n) , P =

∂ M(0) d ∂ M(0) + . ∂ Uζ d ζ ∂U

(4)

Here M (0) is taken from the zero approximation (2). The “forcing” H (n) depends on previous-order functions; in particular, H (1) = R(0) −

∂ M(0) ∂ M(0) UT − UX . ∂ Ut ∂ Ux

(5)

Perturbation Methods for Solitons and Their Behavior as Particles

259

General solution to (4) has the form ⎡ u(n) = Y ⎣C(n) +

π

⎤ Y ∗ H (n) d ζ ⎦ ,

(6)

−π

where C(n) (T, X) are integration “constants” independent of the “fast” variable ζ . Here Y and Y ∗ are the matrices of solutions of the homogeneous equation P = 0, and its conjugate counterpart, respectively. It is important that, as long as U is known, m + 1 vectors (columns) of the matrix Yˆ can be immediately written as variations of the basic system over ζ and each Ai : Y1 = Uζ ,

Yi = UAi ,

i = 2, 3, . . . m + 1

(7)

The general solution (6) remains bounded on the entire axis x only if the following compatibility (orthogonality) conditions are met for the matrices Yi (where i = 2, 3, . . . l) tending to zero at infinity: ∞

Yi∗ H (n) d ζ  = 0, i = 1, 2, . . . l.

(8)

−∞

If there are matrices Yi having nonzero limits at infinity, the following algebraic conditions must be added: lim Yi∗ H (n) = 0, i = l + 1, l + 2, . . .m + 1.

θ →±∞

(9)

Together with a proper selection of the integration parameters C, Eqs. (8) and (9) provide the solutions for variation of these parameters and the velocity V as functions of slow variables T and X. The former set of equations describes variation of the velocity and other parameters of a solitary wave. The second set matches the solitary wave with possible non-localized field component, a radiation, which can arise in the next approximation.

3 Interaction of Well-Separated Solitons Gorshkov and Ostrovsky [2] have applied this method to interaction of several solitons which are separated by a finite but large (as compared to their characteristic lengths). In this case, solitons interact by their small “tails” (Fig. 1) Approximate description of weakly interacting solitons with close amplitudes is based on representation of any i-th solution in the form similar to (3): ui = U(ζ − Si ) + ∑ U(ζ − S j ) + ∑ ε n u(n) (ζ − Si, T, X). j=i

n

(10)

260

L.A. Ostrovsky

Fig. 1 Solitons interacting by their “tails.” Si is the coordinates of soliton centers

u

S i −1

1.0

u

S i +1

Si

1

x

3 2

0.5

0.0 0

10

20

30

40

50

x

Fig. 2 Interaction of close-amplitude solitons. At stage 1, the larger soliton (on the left) approaches the smaller one up to the formation of a symmetric structure 2; then the smaller soliton builds up and moves ahead as the soliton 3; the resulting pair of solitons “exchanges” amplitudes (From [3])

Here the second term in the right-hand part describes the fields (“tails”) of other solitons in the vicinity of the i-th soliton, and the last term corresponds to perturbations occurring due to interaction. Applying a procedure similar to that described above, in the first approximation we obtain a system of equations for the motion for solitons: ··

M Si = ∑ f (Si − S j ),

(11)

i= j

where f is defined by soliton asymptotics far from its center (often exponential but also algebraic). These equations are analogous to Newtonian equations for colliding particles having masses close to M. This process can be described by a Lagrangian Leff =

M S˙i2 + ∑ Φ(Si − S j ), 2 ∑ i i= j

(12)

with the pair potential Φ expressed in terms of soliton asymptotics. Here the soliton mass can be defined as M = ∂ P/∂ V , where P is the total field momentum. Such a representation of interacting solitons as classical particles is, in general, valid only to the first approximation. In the higher orders in ε , for non-integrable equations a radiation is possible in the form of a small and widespread perturbations. The sign of the potential Φ can correspond either to repulsion (in most considered cases) or attraction of particles, providing different scenarios of interaction. One of them, “exchange” interaction, is illustrated in Fig. 2.

Perturbation Methods for Solitons and Their Behavior as Particles

261

Fig. 3 Frequency spectrum (upper panel) and two realizations of “soliton turbulence” in a bounded and harmonically forced nonlinear electromagnetic line [4]

Periodic lattices of solitons can support waves propagating along the lattice; for the repulsing solitons such a process is described by the Toda equation [3]. In nonintegrable equations, solitons with oscillating asymptotics are also possible; in this case, complex solitonic structures can exist, up to the formation of a chaotic set (“turbulence”) of solitons (Fig. 3).

4 Interaction of Kinks The scheme presented above needs a modification for the important class of solitary waves having a limiting amplitude. Such solitons were considered in Gorshkov et al. [5] for the so-called extended KdV, or Gardner, equation: ut + (u − u2)ux + uxxx = 0.

(13)

262

L.A. Ostrovsky

Fig. 4 Shapes of Gardner solitons (14) for different parameters ε = 1 − k. Solid lines: exact solution. Dashed lines: approximation according to (15). The difference is noticeable only for the relatively small solitons with k < 0.5. For very small ε , the soliton slopes reduce to the fronts, the kink and anti-kink, represented in (15) (From [5])

Solitons in this equation have the form u=



    k k k k x − k2t + Δ − tanh x − k 2t − Δ , tanh 2 2 2 2

(14)

where Δ = 1/(k tanh(k)) and the parameter k lies in the range (0, 1). Profiles of such solitons for different values of k are shown in Fig. 4. At small k, this solution coincides with the KdV soliton. However, at small ε = 1 − k it can be represented as a sum of well separated transitions, kinks:





 x−t +L x−t −L 1 u(x,t) ≈ u+ + u− − 1 = tanh − tanh , (15) 2 2 2 where L is half of the distance between kink centers. Note that as seen from Fig. 4, such an approximation is good even for a moderate-amplitude soliton, with 0 < ε < 0.5. Within these limits a N-soliton structure is close to a superposition of 2N kinks, and the solution in the vicinity of a selected, i-th kink, can be sought in the form 2N

ui (x,t) = u0i (ξ − Si (T, X)) + ∑ ε n u(n) (ξ − Si , T, X), i n=1

(16)

Perturbation Methods for Solitons and Their Behavior as Particles

263

where u(0) represents a kink of the corresponding polarity, ξ = x − t, and Si (T, X) is a slowly varying coordinate of the kink center. Substituting this into (13) and using the same perturbation procedure as above, we obtain an inhomogeneous linear equation for a perturbation in each order of ε . Here we give only a few results. First, if one has a group (lattice) of kinks modulated in space and time, their coordinates satisfy the equation

∂ Si ∂ Si +3 = 0, ∂T ∂X

i = 1, 2, . . . 2N.

(17)

Hence, a group of kinks propagates with the “group velocity” which is three times the velocity of a single stationary kink. Second, the interacting kinks satisfy 2N equations corresponding to (11)   d 2 Si −(Si+2 −Si ) −(Si −Si+2 ) , Z = X − 3T. = 4 e − e dZ 2

(18)

These equations describe two independent systems, one for even-numbered kinks and another for odd-numbered kinks which can be interpreted as frontal and trailing edges of solitons. These systems (Toda lattices) are connected by the condition of belonging of a kink and anti-kink pair to one and the same soliton. Solitons with the width increasing with their amplitude are typical, in particular, for strongly nonlinear internal waves in the ocean. In a simple model of a twolayer fluid, there exist a maximal displacement of the interface between the layers when a soliton becomes flat top: its length infinitely increases, with two transitionskinks at the edges. Thus, in Gorshkov et al. [5] the general scheme was applied to the description of a sequence of internal solitons observed off the North-West coast of the USA [6]. This sequence was roughly described by the Gardner equation (13) for the normalized displacement of the boundary (pycnocline) between two well-expressed layers with different densities. The internal wave at an initial point of measurements was imitated by the sequence of kinks as explained above, and then the propagation was calculated for up to another observation point 20 km away from the initial one. The result is reproduced in Fig. 5. It shows that the calculated final positions of solitons agree well with the observation, but their widths are not predicted well. The cause of the discrepancy is that the wave is too strong to be described by the weakly nonlinear equation (13), and a strongly nonlinear models should be applied.

5 Strongly Nonlinear Solitons and Kinks Regarding the internal waves in the ocean, an increasing number of field observations shows that the coastal internal waves can be strongly nonlinear, so that characteristic amplitudes of vertical displacement can be comparable with and even

264

L.A. Ostrovsky

Fig. 5 Experimentally, observed displacement of pycnocline at a 20 km distance from the initial point (dashed line) and its shape calculated from (13) (solid line) (From [4])

exceed the background values of vertical wave scale (e.g., the pycnocline depth) [6]. Some attempts to construct strongly nonlinear, long-wave equations similar to the known weakly nonlinear models mentioned above, have been undertaken. In spite of more strong restrictions of applicability than that for the latter, in many cases the long-wave, weakly dispersive models still provide a reasonably good approximation for strong waves, including solitons. In many such models, similarly to the Gardner equation, with the increase of soliton amplitude, its profile, after narrowing at small amplitudes, then broadens up to a limiting tabletop shape corresponding to a pair of fronts (kinks) tending to indefinitely separate from each other when approaching the maximal (limiting) amplitude. Thus it is reasonable to try to apply the above perturbation technique to the strongly nonlinear models. It was done in Gorshkov et al. [7] for the Miyata-Choi-Camassa (MCC) equations obtained as a strongly nonlinear extension of the classic Boussinesq equations for weakly nonlinear, weakly dispersive waves in a two-layer system [8, 9]:

η 1,2t + (η 1,2 u1,2 )x = 0, 1 ρ1,2 (u1,2t + u1,2u1,2x + gξx ) = −px + 3η 1,2

 3 η1,2



∂ ∂ + u1,2 ∂t ∂x

2  ξ , (19) x

Perturbation Methods for Solitons and Their Behavior as Particles

265

c/cm

1.1

1

0.9

0.8

0.7

0.6

e 0.5 0

0.2

0.4

0.6

0.8

1

Fig. 6 Velocity of a compound soliton in (19) (solid line) and its two-kink approximation (dashed line) at h1 /h2 = 0.1, ρ1 /ρ2 = 0.9996, ξm /h1 = −4.495 for different values of ε = |(a − ξm )/ξm |. Solid lines: exact solution, dashed lines: two-kink approximation (From [5])

where η1,2 = h1,2 ∓ ξ , h1,2 are undisturbed thicknesses of the upper and lower layers, respectively; ξ (x,t) is vertical displacement of the interface between the layers; u1,2 are horizontal fluid velocities (averaged over the vertical coordinate), ρ1,2 are fluid densities, p is pressure at the interface; and g is gravity acceleration. This system also has a family of solitary solutions with a limiting soliton amplitude (Fig. 6). It is seen that the two-kink approximation is again true even for a moderateamplitude solitons, within the limits 1 < ε < 0.5. The corresponding perturbation scheme is, in general, similar to that explained above for the Gardner equation. An important (albeit not at the first glance) difference is that whereas a Gardner kink has the same exponential asymptotic at both limits of ξ , in the MCC case the powers of these asymptotics are different, so that upon transition of smaller soliton from one side of a kink to another, its parameters change. An example of interaction between an almost limiting soliton with a smaller one is shown in Fig. 7. Thus, the perturbation theory is equally well applicable to integrable (such as Gardner) and non-integrable (such as MCC) equations.

266

L.A. Ostrovsky

Fig. 7 Interaction of two solitons one of which is close to a limiting one. Profiles of compound solitons at different time moments

6 Conclusions There exist other perturbation schemes differing from that briefly outlined here. One of them is based on the closeness of the basic equation to the integrable equation so that the perturbed inverse scattering problem can be formulated [10]. However, this is only applicable to the latter class, whereas our theory, as mentioned, is not restricted by any integrability conditions. A brief review of the existing schemes is given in Ostrovsky and Gorshkov [11]. The above perturbation scheme has been applied to soliton interactions in a variety of physical applications. As mentioned, one of them refers to internal gravity waves in the ocean, arguably the most ubiquitous case of solitons in nature. At the same time, complex dynamics of solitons treated as classical particles was studied for electromagnetic pulses and in other applications [3]. We expect to see a further development of the method and its applications in near future.

References 1. Ablowitz M, Segur H (2000) Solitons and the inverse scattering transform. SIAM studies in applied mathematics, vol. 4. SIAM, Philadelphia 2. Gorshkov KA, Ostrovsky LA (1981) Interactions of solitons in nonintegrable systems: direct perturbation method and applications. Phys D 3:428–438 3. Gorshkov KA, Ostrovsky LA, Yu A, Stepanyants YA (2010) Dynamics of soliton chains: from simple to complex and chaotic motions. In: Luo A, Afraimovich V (eds) Long-range interactions, stochasticity, and fractional dynamics. Higher Education Press, Beijing and Springer, Berlin 4. Gorshkov KA, Ostrovsky LA, Papko VV (1977) Soliton turbulence in the system with weak dispersion. Sov Phys Dokl 22:378–380 5. Gorshkov KA, Ostrovsky LA, Soustova IA, Irisov VG (2004) Perturbation theory for kinks and application for multisoliton interactions in hydrodynamics. Phys Rev E 69:1–9

Perturbation Methods for Solitons and Their Behavior as Particles

267

6. Stanton T, Ostrovsky L (1998) Observation of highly nonlinear internal solitons over the continental shelf. Geophys Res Lett 25:2695–2698 7. Gorshkov KA, Ostrovsky LA, Soustova IA (2010) Dynamics of strongly nonlinear kinks and solitons in a two-layer fluid. Stud Appl Math. doi:10.1111/j.1467-9590.2010.00497 8. Miyata M (1988) Long internal waves of large amplitude. In: Hirakata K, Mauro H (eds) Nonlinear water waves. Springer, Berlin, Heidelberg, New York 9. Choi W, Camassa R (1999) Fully nonlinear internal waves in a two-fluid system. J Fluid Mech 396:1–36 10. Karpman VI, Maslov EM (1977) Perturbation theory for solitons. J Exp Theor Phys 73:537–541 11. Ostrovsky LA, Gorshkov KA (2000) Perturbation theories for nonlinear waves. In: Christiansen PL, Sørensen MP, Scott AC (eds) Nonlinear science at the dawn of the 21st Century. Springer, Berlin, Heidelberg, pp 47–65

Complex Holomorphic Flows Constantin Udris¸te and Romeo Bercia

Abstract We analyze some properties of complex holomorphic flows and their related flows, obtaining original results. The problems solved include new representations of holomorphic flows, the hyperbolicity and stability of equilibrium points, the behavior near zeros and poles, potential or hamiltonian systems, etc. The tools are not only those of dynamical systems theory and complex functions, but also of differential geometry. The problems tackled are directly or indirectly concerned with new trends in mathematical literature dedicated to dynamics of complex holomorphic functions. One key to the new research results has been the interest for properties of special functions and their evolutions which appear in Applied Sciences.

1 Introduction The dynamics of complex functions has undergone a remarkable resurgence of interest in recent years. This theory underlines the rich dynamical behavior of the elementary maps and their applications ranging from physics and chemistry to ecology and economics. This paper is complementary to the papers [2, 7], and closely related to some aspects developed in [3] and [4]. It involves holomorphic and anti-holomorphic complex functions (if f is an holomorphic complex function, then the complex conjugate function f¯ is called anti-holomorphic).

C. Udris¸te () University Politehnica of Bucharest, Faculty of Applied Sciences, 313 Splaiul Independentei, 060042, Bucharest, Romania e-mail: [email protected] R. Bercia University Politehnica of Bucharest, Faculty of Applied Sciences, 313 Splaiul Independentei, 060042, Bucharest, Romania e-mail: r [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 15, © Springer Science+Business Media, LLC 2012

269

270

C. Udris¸te and R. Bercia

Our work is dedicated to complex dynamical systems as complex differential equations. At this juncture, we should note that sometimes a complex dynamical system is considered as: (1) a discrete dynamical system given by a difference equation (see [8]); (2) an iterative process of functions (see [5] and [6]), obtaining Julia sets and Mandelbrot sets. Section 1 contains historical data. Section 2 gives new representations of complex holomorphic flows and characterizes their geometries. Then the hyperbolicity and stability of the equilibrium points is related to the complex derivative. Section 3 describes the complex gradient flows. Section 4 presents some generic complex flows and, Sect. 5 describes some approximations. The behavior of flows around poles is analyzed in Sect. 6. Section 7 reflects the behavior of flows in the presence of zeros and poles. Section 8 analyzes the P´olya–Latta systems (associated to conjugate complex functions) in relation to potential and/or Hamiltonian systems. Section 9 refers to basic tools in geometric dynamics generated by a complex flow and by the Euclidean metric.

2 Dynamical Systems in Complex Variable Let F = (u, v) : R2 → R2 be a C∞ function. It determines a planar continuous-time dynamical system x(t) ˙ = F(x(t)), x = (x1 , x2 ) ∈ R2 ,t ∈ I ⊂ R, where F(x) = (u(x), v(x)). Explicitly 

x˙1 = u(x1 , x2 ), x˙2 = v(x1 , x2 ),

(x1 , x2 ) ∈ R2 ,

(1)

where u and v are C∞ functions. By introducing a complex variable z = x1 + ix2 and its conjugate z = x1 − ix2 , the system (1) can be written as a system of complex ODEs 

z˙(t) = f (z(t), z(t)), z˙(t) = f (z(t), z(t)),

z ∈ C,

(2)

where f (z, z) is a C∞ function. Suppose that u and v satisfies the Cauchy–Riemann PDEs

∂u ∂v = , ∂ x1 ∂ x2

∂u ∂v =− 1. ∂ x2 ∂x

(3)

Then the dynamical system (2) can be written as a single complex (separable variables) ODE

Complex Holomorphic Flows

271

z˙(t) = f (z(t)),

z ∈ C,

(4)

where f (z) = u(x) + iv(x) is a holomorphic function. Let x0 = (x10 , x20 ) be an equilibrium point of the dynamical system (1). Then the function f (z,z) vanishes at (z0 , z0 ) and the holomorphic function f (z) = u(x)+ iv(x) vanishes at z0 = x10 + ix20 . Remark 1. A complex flow of type (4) consists in conformal transformations if and only if the associated function f is holomorphic or antiholomorphic (see [1]). The fluid flow around airfoils is modelled using conformal mappings. The examples that follow use a variety of elementary functions. Multi-valued functions and special functions will be used in furthermore papers. Theorem 1. If f is a holomorphic function, then the dynamical system (1) or (4) is of the form:   1 0 (i) x˙ = J1 ∇Φ , where Φ is a harmonic function and J1 = . 0 −1   01 (ii) x˙ = J2 ∇Ψ , where Ψ is a harmonic function and J2 = . 10 Proof. (i) A solution of the PDF system (3) is u = ∂∂ xΦ1 , v = − ∂∂ xΦ2 , with Φ harmonic function. It follows Φ (x) =



C

u(x)dx1 − v(x)dx2 .

(ii) Another solution of the system (3) is u = function. It follows Ψ (x) =



C

∂Ψ , ∂ x2 2

∂Ψ , ∂ x1

v=

with Ψ harmonic

v(x)dx1 + u(x)dx .

Remark 2.  (i) The dynamical system x˙ = J1 ∇Φ is a Lorentzian gradient system.   1 0 is a Lorentzian manifold, J1−1 = J1 and J1−1 ∇Φ Indeed R2 , J1 = 0 −1 is a Lorentzian gradient. (ii) The system x˙ = J2 ∇Ψ is a Lorentzian gradient system. Indeed    dynamical 0 1 is a Lorentzian manifold, J2−1 = J2 and J2−1 ∇Ψ is a R2 , J2 = 10 Lorentzian gradient. Theorem 2. If γ is a closed orbit of the flow (4) that bound a set S ⊂ C, then div(u, v) cannot have a fixed sign in S. Proof. The dynamical system (4) is equivalent to the Pfaff system udx1 + vdx2 = (u2 + v2 )dt,

udx2 − vdx1 = 0.

Applying Green theorem, we have 

div(u, v) dx dx = 1

S

2

  S

∂u ∂v + 2 1 ∂x ∂x

 dx1 dx2 =

 γ

udx2 − vdx1 = 0.

272

C. Udris¸te and R. Bercia

Fig. 1 Orthogonal curves u(x1 , x2 ) = c1 and v(x1 , x2 ) = c2 for f = u + iv = sin(z2 )

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

−1

−0.5

0

0.5

1

Remark 3. (i) The function Φ (x1 , −x2 ) is strictly increasing along any nonstationary orbit of the flow x˙ = J1 ∇Φ . Indeed d Φ (x1 , −x2 ) = dt (ii) If γ is a curve in the flow (4), then

  γ

∂Φ ∂ x1



2 +

∂Φ ∂ x2

2 > 0.

f (z)dz > 0.

Each holomorphic function whose derivative does not vanish gives an orthogonal coordinate systems by means of the constant level curves u(x1 , x2 ) = c1 and v(x1 , x2 ) = c2 (Fig. 1). Theorem 3. If f = u + iv is a holomorphic function, then the curves of implicit Cartesian equations u(x1 , x2 ) = c1 and v(x1 , x2 ) = c2 are orthogonal. Proof. The gradient lines of the function u (respectively v) have the equations v(x1 , x2 ) = c2 (respectively u(x1 , x2 ) = c1 ). On the other hand, (∇u, ∇v) = 0, via the Cauchy–Riemann conditions. Remark 4. This theorem is used for geometrical description of the two-dimensional fluid flow, electric flow or heat flow. 

If f is a holomorphic function, then the primitive C

function.

f (z)dz is a holomorphic

Corollary 1.If f is a holomorphic function, thenthe curves of implicit Cartesian   equations ℜ



C

f (z)dz = c1 and ℑ



C

f (z)dz = c2 are orthogonal.

Complex Holomorphic Flows

273

Denote by A the Jacobian matrix of the function F(x) = (u(x), v(x)) evaluated at the equilibrium point x0 . The hyperbolicity and the stability of an isolated equilibrium point x0 , for the dynamical system (4), depend on the eigenvalues λ1,2 = ∂∂xu1 (x0 ) ∓ i ∂∂xu2 (x0 ) of A, which means that λ1 = λ2 = f  (z0 ). Moreover, det(A) = | f  (z0 )| ≥ 0.

3 Complex Gradient Flows Let f (z) = u(x1 , x2 ) + iv(x1 , x2 ) be a holomorphic function and f  = derivative. Let us analyze the complex gradient flow z˙(t) = f  (z(t)).

∂u ∂ x1

− i ∂∂xu2 its (5)

The complex gradient flow (5) is equivalent to one of the dynamical systems:   1 0 (i) x˙ = J1 ∇u, where J1 = and u is a harmonic function; hence, this system 0 −1 corresponds to case (i)from  Theorem 1. 01 (ii) x˙ = J2 ∇v, where J2 = and v is a harmonic function; hence, this system 10 corresponds to case (ii) from Theorem 1. Suppose that z0 = x10 + ix20 is an equilibrium point such that f  (z0 ) = 0. The Jacobian matrix B evaluated at the equilibrium point x0 = (x10 , x20 ) is ⎛ ⎜ B(x0 ) = ⎝

∂ 2u 2 (x0 ) ∂ x1

⎞

∂ 2u (x ) ∂ x1 ∂ x2 0 ⎟

− ∂ x∂1 ∂ux2 (x0 ) − ∂ 2u2 (x0 ) 2

2

⎠.

(6)

∂x

Via the Cauchy PDEs, it follows that  det(B(x0 )) =

2  2 2 ∂ u (x0 ) + (x0 ) ≥ 0. ∂ x1 ∂ x2 ∂ x1 2

∂ 2u

If f  (z0 ) = 0, then det(B(x0 )) = | f  (z0 )|2 implies det(B(x0 )) > 0. Also, the eigenvalues λ1 = λ2 = f  (z0 ) = 0 of the matrix B are complex conjugate. Remark 5. (i) Every complex flow z˙ = g(z) is a complex gradient flow because, if g is holomorphic in the whole plane, then there is a holomorphic function f such that g = f  .

274

C. Udris¸te and R. Bercia

(ii) The evolution of f (z) along the gradient flow z˙(t) = f  (z(t)) is described by d  2 dt f (z) = ( f (z)) or d u(x1 , x2 ) = dt



∂u ∂ x1



2 −

∂u ∂ x2

2 ,

d ∂u ∂u v(x1 , x2 ) = −2 1 2 . dt ∂x ∂x

(iii) If u(x1 , x2 ) is an harmonic function, then the complex function     ∂u 2 ∂u 2 ∂u ∂u − − 2i 1 2 h(z) = ∂ x1 ∂ x2 ∂x ∂x is holomorphic. Open problem. Study the flow z˙ = h(z). Remark 6. The determinant of the Jacobian matrix B(x) and the Gauss curvature K(u(x)) of the surface u = u(x) are of opposite signs. Open problem. Let f be a holomorphic function, f  its derivative and f n = f ◦ f n−1 . Analyze and compare the flows

n z˙ = f  (z) ,

z˙ = ( f n ) (z),

z˙ = ( f  )n (z),

n ∈ N, n ≥ 2.

Remark 7. The function u(x1 , −x2 ) is strictly increasing along any nonstationary orbit of the complex gradient flow (5). Indeed d u(x1 , −x2 ) = | f  (z)|2 > 0. dt If f is a holomorphic function, then f  = f  and hence f  (z) = f  (z). Theorem 4. Let f = u + iv be a holomorphic function. (i) The constant level curves v(x1 , x2 ) = c2 represents the paths followed by a particle in the flow given by f¯ (z). The function u(x1 , x2 ) is increasing along the paths in this flow. (ii) The constant level curves u(x1 , x2 ) = c1 represents the paths followed by a particle in the flow given by i f¯ (z). Proof. (i) Since

∂ v dx1 ∂ v dx2 d v(x1 (t), x2 (t)) = 1 + 2 = 0, dt ∂ x dt ∂ x dt the function v(x1 , x2 ) is a first integral. On the other hand,

∂ u dx1 ∂ u dx2 d u(x1 (t), x2 (t)) = 1 + 2 = | f  (z(t))|2 > 0. dt ∂ x dt ∂ x dt (ii) Similarly, the function u(x1 , x2 ) = c1 is a first integral of the second system.

Complex Holomorphic Flows

275

4 Generic Equilibria of Holomorphic Flows One can perform a coordinate shift, placing the equilibrium point at the origin. Therefore, we may assume without loss of generality that z0 = 0 is the equilibrium point of the system (4). As example, we shall analyze the phase portrait of the polynomial dynamical system z˙ = azn , (7) discussing after the parameters a ∈ C and n ∈ N∗ .

4.1 Case n = 1, z0 = 0 Zero of Order 1 In this case, the dynamical system (7) is a linear system with λ1 = λ2 = a. This means that we have only three possibilities: (i) z0 is node iff a = α , α ∈ R∗ ; the orbits are straight lines z(t) = Ceα t ; (ii) z0 = 0 is focus iff a = α + iβ , α ∈ R∗ , β ∈ R∗ ; the orbits are spirals z(t) = Ceα t (cos β t + i sin β t); (iii) z0 = 0 is centre iff a = iβ , β ∈ R∗ ; the orbits are circles z(t) = C(cos β t + i sin β t). Note that there are no saddle points for this type of systems (Fig. 2).

4.2 Case n≥2, z0 = 0 Zero of Order Greater than 2 This time the dynamical system is nonlinear. Note that z0 = 0 is a nonhyperbolic equilibrium point of system (7) because λ1 = λ2 = f  (0) = 0. C The general solution of the equation z˙ = az2 is z(t) = 1−aCt . This yields homoclinic orbits for aC ∈ / R. There are also two straight semi-lines orbits, arg z = − arga and arg z = − arga ± π . See Fig. 3. For n ≥ 3, the general solution is z (t) = n−1√ C , which yields hon−1 1−(n−1)aC

t

/ R. There are also 2n − 2 straight semi-lines orbits moclinic orbits for aCn−1 ∈ −arg a arg z = kπn−1 , n = 1, 2n − 2. See Figs. 4 and 5.

Fig. 2 Phase portraits of the system z˙ = az for (i) a = 1, (ii) a = 1 + i, (iii) a = i

2

2

2

1

1

1

0

0

0

−1

−1

−1

−2 −2

0

2

−2 −2

0

2

−2 −2

0

2

276

C. Udris¸te and R. Bercia

Fig. 3 Phase portrait of the system z˙ = (1 + i)z2

3

2

1

0

−1

−2

−3 −3

Fig. 4 Phase portrait of the system z˙ = (1 + i)z3

−2

−1

0

1

2

3

−2

−1

0

1

2

3

3

2

1

0

−1

−2

−3 −3

If a = 1, then the flow (1) associated to (7) is defined by pairs of harmonic polynomials un (x) = ℜ(zn ), vn (x) = ℑ(zn ). Theorem 5. The isolated equilibrium point z0 = 0 of the flow z˙ = zn is unstable. Proof. The Lyapunov function

φ : R2 → R, φ (x) = un+1 (x) = ℜ(zn+1 )

Complex Holomorphic Flows Fig. 5 Phase portrait of the system z˙ = (1 + i)z6

277 3

2

1

0

−1

−2

−3 −3

−2

−1

0

1

2

3

determines the open set {x = (x1 , x2 )|φ (x) > 0, ||x|| < 1}. Consider the domain V + = {x = (x1 , x2 )|φ (x) > 0, ||x|| < 1, x1 > 0, x2 > 0}, whose boundary ∂ V + contains the origin. Obviously φ (x) = 0 on ∂ V + . From d d 2n + dt φ (x) = (n + 1)ℜ(z ), it follows that we have dt φ (x) > 0, for any x ∈ V .

5 Approximations Equivalence As is well known by Grobman–Hartman theorem, near a hyperbolic equilibrium point x0 , the dynamical system (1) is locally topological equivalent to its linearization ξ˙ = Aξ . It follows that, near a simple zero z0 of the function f , the system (4) is locally topological equivalent to w˙ = f  (z0 )w if f  (z0 ) = iβ . Example 1. The system z˙ = a sin z, near z0 = π , is locally topological equivalent to w˙ = −aw. Compare the phase portrait in Fig. 6 with those in Fig. 2. The linearization fails when z0 is zero of order n ≥ 2 for f (z). If f is a holomorphic function, at least in the neighborhood of z0 , then its Taylor expansion is

278

C. Udris¸te and R. Bercia

Fig. 6 Phase portraits of the system z˙ = a sin z near z0 = π for (i) a = 1, (ii) a = 1 + i, (iii) a = i

2

2

2

1

1

1

0

0

0

−1

−1

−1

−2

Fig. 7 Phase portrait of z˙ = (1 + i)z2 sin z near z0 = 0

2

4

−2

2

4

−2

2

4

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −2 −1.5 −1 −0.5

f (z) =

0

0.5

1

1.5

f (n) (z0 ) (z − z0 )n + O(|z − z0|n+1 ). n!

2

2.5

(8)

In this case, to the dynamical system (4), we associate the generic system w˙ =

f (n) (z0 ) n w . n!

(9)

We claim that, in this case, the dynamical system (4) is locally topological equivalent to the dynamical system (9). Example 2. The dynamical system z˙ = (1 + i)z2 sin z, near z0 = 0, is locally topological equivalent to w˙ = (1 + i)w3 . Compare the phase portrait in Fig. 7 with those in Fig. 4. Theorem 6. Let f , g : Ω ⊂ C → C be two holomorphic functions. If the dynamical systems z˙ = f (z) and z˙ = g(z) have the same orbits, then there exists α ∈ R\{0} such that f (z) = α g(z). Proof. Let f (z) = u1 (x1 , x2 ) + iv1 (x1 , x2 ) and g(z) = u2 (x1 , x2 ) + iv2 (x1 , x2 ). The symmetric ODE system

Complex Holomorphic Flows

279

dx1 dx2 dx1 dx2 = = dt, = = dt u1 (x1 , x2 ) v1 (x1 , x2 ) u2 (x1 , x2 ) v2 (x1 , x2 ) have the same orbits iff there exists ρ (x1 , x2 ) ∈ R such that (u1 , v1 ) = (u2 , v2 )ρ . Using the Cauchy–Riemann PDEs

∂ u1 ∂ v1 ∂ u1 ∂ v1 ∂ u2 ∂ v2 ∂ u2 ∂ v2 = 2, 2 =− 1; = 2, 2 =− 1, 1 1 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x it follows that

∂ρ ∂ρ = 0, 2 = 0. ∂ x1 ∂x Remark 8. (i) This theorem was formulated in [2], but our proof is original. (ii) The flows z˙ = f (z) and z˙ = 1 have the same orbits. f (z)

6 Behavior of Flows in the Presence of Poles We analyze the behavior of the dynamical system (4) in the presence of an isolated singular point z0 , which is pole of order n ≥ 1 (meromorphic function). One can perform a coordinate shift, placing the pole at the origin. Therefore, we may assume, without loss of generality, that z0 = 0.

6.1 Generic System We consider f (z) = az−n , where a ∈ C∗  and n ∈ N∗ . The general solution of the n+1 −n dynamical system z˙ = az is z(t) = Cn+1 + (n + 1)at, C ∈ C∗ . This yields n−1 −1 unbounded orbits for C a ∈ / R. There are also 2n + 2 straight semi-lines orbits arg z = (n + 1)−1 (kπ + arga), k = 1, 2n + 2. Half of them define the attractive manifold of z0 = 0, while the others define the repulsive one. There are 2n + 2 invariant manifolds defined by argz ∈ (kπ , (k + 1)π ). Example 3. Phase portraits for simple and third order poles, see Figs. 8 and 9.

6.2 Equivalence If z0 is pole of order n for the holomorphic function f , then it follows that f (z) has a Laurent expansion

280 Fig. 8 Phase portrait of the system z˙ = (1 + i)z−1

C. Udris¸te and R. Bercia 3 2 1 0 −1 −2 −3

Fig. 9 Phase portrait of the system z˙ = (1 + i)z−3

−3

−2

−1

0

1

2

3

3

2

1

0

−1

−2

−3 −3

f (z) =

−2

∞

∑

−1

ck (z − z0 )k ,

0

1

2

3

(10)

k=−n

where ck ∈ C, c−n = 0. In this case, to the dynamical system (4), we associate the generic dynamical system w˙ = c−n w−n . (11) We claim that, in this case, the dynamical system (4) is locally topological equivalent to the dynamical system (11).

Complex Holomorphic Flows Fig. 10 Phase portrait of the system z˙ = (1 + i)z−2 sin z

281 3 2 1 0 −1 −2 −3

Fig. 11 Phase portrait of the system z˙ = (1 + i)z−4 sin z

−3

−2

−1

0

1

2

3

3

2

1

0

−1

−2

−3 −3

−2

−1

0

1

2

3

Example 4. The system z˙ = (1 + i)z−2 sin z near z0 = 0 is locally topological equivalent to w˙ = (1 + i)w−1 . Compare the phase portraits in Figs. 10 and 8. Example 5. The system z˙ = (1 + i)z−4 sin z near z0 = 0 is locally topological equivalent to w˙ = (1 + i)w−3 . Compare the phase portraits in Figs. 11 and 9.

282

C. Udris¸te and R. Bercia

7 Behavior of Flows in the Presence of Zeros and Poles Example 6. The homographic function f (z) = az+b cz+d , a = 0, c = 0, has a zero at z0 = −1 −1 −ba and a pole at z1 = −dc . Particularly, the phase portrait of the dynamical system associated to f (z) = z+1 z−1 (see Fig. 12) shows the local equivalence, near z0 = −1, with a generic zero of order 1, respectively, near z1 = 1, with a generic pole of order 1. Example 7. The Jukovski function f (z) = z + z−1 has two zeros at z0 = ±i and a pole at z1 = 0. The phase portrait of the dynamical system associated to f (z) (see Fig. 13) shows the local equivalence, near z0 = ±i, with a generic zero of order 1, respectively, near z1 = 0, with a generic pole of order 1. 3 2 1 0 −1 −2

Fig. 12 Phase portrait of the system z˙ = z+1 z−1

−3 −4

−3

−2

−1

0

1

2

0

1

2

3

3

2

1

0

−1

−2

Fig. 13 Phase portrait of the system z˙ = z + z−1

−3

−3

−2

−1

Complex Holomorphic Flows

283

8 Complex Flows Vs Gradient and Hamiltonian Systems We shall analyse the P´olya–Latta dynamical system z˙ = f (z),

z ∈ C,

(12)

where f is a holomorphic function having a zero at z0 = x10 + ix20 . Note that f and f are simultaneously holomorphic iff f is the constant function. By introducing z = x1 + ix2 and separating the real and imaginary parts, the P´olya–Latta dynamical system can be written as  1 x˙ = u(x1 , x2 ), (13) (x1 , x2 ) ∈ R2 , x˙2 = −v(x1 , x2 ), where u and v satisfies the Cauchy–Riemann equations. Let x0 = (x10 , x20 ) be an equilibrium point of the dynamical system (13) and denote by C the Jacobian matrix evaluated at the equilibrium point x0 . Then det(C) = −| f  (z0 )|2 ≤ 0 and the eigenvalues of C are λ1,2 = ±| f  (z0 )|. Theorem 7. The function f (z) = u(x) + iv(x) is holomorphic if and only if one of the following statements is true: 1. The P´olya–Latta dynamical system is simultaneously a potential and a Hamiltonian system. 2. The P´olya–Latta dynamical system is a Hamiltonian system with harmonic Hamilton function. 3. The P´olya–Latta dynamical system is a potential system with harmonic potential. Proof. The function f (z) = u(x) + iv(x) is holomorphic if and only if it satisfies the Cauchy–Riemann conditions (3). 1. If the function f is holomorphic, then it follows that the P´olya–Latta dynamical system is: (i) A Hamiltonian system with respect to the Hamiltonian h(x) =

 x x0

v(x)dx1 + u(x)dx2 ;

(14)

(ii) A potential system with respect to the potential

φ (x) =

 x x0

u(x)dx1 − v(x)dx2 .

Conversely suppose there exists (i) a Hamiltonian h such that u =

it follows ∂∂xu1 = ∂∂xv2 ; (ii) a potential φ such that u = ∂∂xφ1 , −v follows ∂∂xu2 = − ∂∂xv1 . Consequently, the function f is holomorphic. ∂h ; ∂ x1

(15) ∂h , v= ∂ x2 ∂φ = ∂ x2 ; it

284

C. Udris¸te and R. Bercia

2. If f is a holomorphic function, it follows that the P´olya–Latta dynamical system is a Hamiltonian system with respect to the Hamiltonian h(x). But h(x) is a harmonic function. Conversely, suppose there exists a harmonic function such that u = ∂∂xh2 and v = ∂∂xh1 . Then ∂∂xu1 = ∂∂xv2 , ∂∂xu2 + ∂∂xv1 = Δ h = 0. Hence, f is holomorphic. 3. Similar. The previous formulas yield

Φ (x) = ℜ H(x) = ℑ

 

z

z0 z

z0

 f (ξ )dξ

= ℜ (F(z)) , 

f (ξ )dξ

= ℑ (F(z)) ,

(16)

as potential, respectively Hamiltonian, for the dynamical system (12), where F is a primitive of f . Corollary 2. The P´olya–Latta dynamical system (12) has: 1. No closed orbits, excepting the equilibrium points. 2. No asymptotically stable equilibrium points or asymptotically stable limit cycles. Proof. Since the P´olya–Latta dynamical system (12) is a potential system (respectively Hamiltonian system), the first statement is true, see [9], p. 148, p. 156 (respectively the second statement is true, see [9], p. 144). Theorem 8. If γ is a closed orbit of the flow (12) that bound a set S ⊂ C, then div(u, −v) cannot have a fixed sign in S. Proof. The dynamical system (12) is equivalent to the Faff system udx1 + vdx2 = (u2 + v2 )dt,

udx2 + vdx1 = 0.

Applying Green theorem, we have 

div(u, v)dx dx = 1

S

 

2

S

∂u ∂v + ∂ x1 ∂ x2



Remark 9. If γ is a curve in the flow (12), then

dx1 dx2 =  γ

 γ

udx2 − vdx1 = 0.

f (z)dz > 0.

Example 8. The system z˙ = sin z has H(x) = −ℑ(cos z). The integral curves are constant level sets of the Hamiltonian H. Compare the phase portrait of the system in Fig. 14 with the surface plot of H in Fig. 15 and the surface plot in Fig. 16.

Complex Holomorphic Flows Fig. 14 Phase portrait of z˙ = sin z

285 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.5

0

0.5

1

Fig. 15 Surface plot of the Hamiltonian H(x) = −ℑ(cos z)

Open problem. Study the dynamical systems associated to the functions i f (z), i f (z), iz f  (z). Theorem 9. If f is a holomorphic function, then: (1) The flow z˙ = f (z) is area preserving. (2) The flow z˙ = i f (z) is area preserving.

286

C. Udris¸te and R. Bercia

Fig. 16 Surface plot of the potential Φ (x) = −ℜ(cos z) Fig. 17 Phase portrait of z˙ = z2

1

0.8 0.6 0.4 0.2 0

−0.2 −0.4 −0.6 −0.8 −1 −1

−0.5

Proof. Consequence of the relation div(u, −v) = div(v, u) = ∂∂xv1 + ∂∂xu2 = 0.

0 ∂u ∂ x1

−

0.5 ∂v ∂ x2

1

= 0, respectively of

Example 9. The system z˙ = z2 has H(x) = 13 ℑ(z3 ). Compare the phase portrait of the system in Fig. 17 with the surface plot of H in Fig. 18 and the surface plot in Fig. 19.

Complex Holomorphic Flows

287

Fig. 18 Surface plot of the Hamiltonian H(x) = 13 ℑ(z3 )

Fig. 19 Surface plot of the potential Φ (x) = 13 ℜ(z3 )

9 Flow Induced by a Given Curve If we use the polar coordinates (r, ϕ ), then z = reiϕ . Suppose f (z) is holomorphic and one-to-one on the unit open disk |z| < 1. One also assumes f (0) = 0.

288

C. Udris¸te and R. Bercia

The image of the circle z = reiϕ , r = const < 1, ϕ ∈ [0, 2π ] by the function f is the curve Γ : w = f (reiϕ ), r = const < 1, ϕ ∈ [0, 2π ]. The geometry of the curve Γ is characterized by the following geometrical vector fields (see [1]): the velocity z f  (z)  vector field ddw ϕ = iz f (z), the unit normal vector field N = − |z f  (z)| , the acceleration vector field

d2w dϕ 2

= −z( f  (z) + z f  (z)). The curvature of Γ is

κ= The curve Γ is convex iff

  z f  (z) 1 . ℜ 1 + |z f  (z)| f  (z)   z f  (z) ℜ 1+  ≥ 0. f (z)

The family of circles z = reiϕ , r = const is transformed into the family of curves Γ . The flow defined by the family Γ is dw = i f −1 (w) f  ( f −1 (w)). dϕ The trajectories in this flow are images of the trajectories in the flow ddzϕ = iz, i.e., the images via the conformal map f of the family of circles z = reiϕ , r = const.

10 Geometric Dynamics The basic idea of geometric dynamics (see [9]) is to imbed a flow into a geodesic motion under a gyroscopic field of forces. To formulate this theory on the complex plane, we need a holomorphic function f = u + iv and its associated semi-modulus G = 12 | f (z)|2 = 12 (u2 + v2 ). (i) To the flow z˙ = f (z), we can associate the least square Lagrangian

1 1 1 (x˙ − u)2 + (x˙2 − v)2 . L = |˙z − f (z)|2 = 2 2 The Euler-Lagrange ODEs

∂ v dx2 ∂ G d 2 x2 ∂ v dx1 ∂ G d 2 x1 + + 2 = −2 , = 2 dt 2 ∂ x1 dt ∂ x1 dt 2 ∂ x1 dt ∂x describe a geodesic motion in a gyroscopic field of forces (geometric dynamics) that contains the initial flow. Coming back to the complex variable z, we can write

Complex Holomorphic Flows

289

∂v ∂G ∂G z˙ + 1 + i 2 . ∂ x1 ∂x ∂x (ii) Similarly, the anti-holomorphic flow z˙ = f¯(z) suggests the least square Lagrangian z¨ = 2i

1 1 1 (x˙ − u)2 + (x˙2 + v)2 . L1 = |˙z − f¯(z)|2 = 2 2 The Euler–Lagrange ODEs d 2 x1 ∂ G d 2 x2 ∂G = , = 2 dt 2 ∂ x1 dt 2 ∂x describe a geodesic motion in a conservative field of forces (geometric dynamics) that contains the initial flow. Also, we remark that the Lagrangian L1 and the Lagrangian 1 L2 = δi j x˙i x˙ j + G 2 gives the same Euler–Lagrange ODE. Coming back to the complex variable z, we can write ∂G ∂G z¨ = 1 + i 2 . ∂x ∂x

References 1. Blair D (2000) Inversion theory and conformal mapping. American Mathematical Society, Providence 2. Broughan KA (2003) Holomorphic flows on simply connected regions have no limit cycles. Meccanica 38:699–709 3. Broughan KA (2005) The holomorphic flow of Riemann’s function ξ (z). Nonlinearity 18: 1269–1294 4. Broughan KA, Barnett AR (2003) The holomorphic flow of the Riemann zeta function. Math Comput 73:987–1004 5. Devaney RL (1989) Chaotic dynamical systems. Addison-Wesley, Redwood City 6. Milnor J (1990) Dynamics in one complex variable. http://arxiv.org/abs/math/9201272. Cited 20 Apr 1990 7. Newton T, Lofaro T (1996) On using flows to visualize functions of a complex variable. Math Mag 69:28–34 8. Sedaghat H (2003) Nonlinear difference equations. Springer, Dordrecht/Boston 9. Udris¸te C (2000) Geometric dynamics. Kluwer Academic, Dordrecht

Unsteady MHD Flow Past a Stretching Sheet Due to a Heat Source/Sink A.K. Banerjee, A. Vanav Kumar, and V. Kumaran

Abstract This paper deals with the unsteady heat transfer effects due to a sudden introduction of heat source/sink on a steady viscous boundary layer MHD flow and heat transfer over a linearly stretching sheet subjected to a constant temperature. Governing boundary layer equations have been solved by an implicit finite difference method. Numerical results show that the steady state is reached quickly for a heat sink or for a large Prandtl number. The time to reach steady state increases under magnetic field. Upto a critical value of the strength of heat source, steady solution exists.

1 Introduction Steady heat transfer analysis in the presence of heat source has been receiving wide attention among the researchers due to its applications in polymer extrusion process, metallurgical process, drawing of artificial fibres, etc. Recently, Liu [1] studied the heat and mass transfer in a MHD flow past a stretching sheet including the chemically reactive species of order one and internal heat generation or absorption. Xu [2] studied the free convective heat transfer characteristics in an electrically conducting fluid near an isothermal sheet with internal heat generation or absorption. Viscous dissipation effect along with heat transfer in MHD viscoelastic fluid flow over a stretching sheet was studied by Abel and Mahesha [3]. Their study also includes the effect of variable thermal conductivity, non-uniform heat source, and

A.K. Banerjee () · V. Kumaran Department of Mathematics, National Institute of Technology, Tiruchirappalli - 620015, India e-mail: [email protected]; [email protected] A.V. Kumar Department of Mathematics, Vellore Institute of Technology, Chennai - 600048, India e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 16, © Springer Science+Business Media, LLC 2012

291

292

A.K. Banerjee et al.

radiation. Khan [4] studied the effect of heat transfer on a viscoelastic fluid flow over a stretching sheet with heat source/sink, suction/blowing, and radiation. Pal and Talukdar [5] studied the unsteady MHD heat and mass transfer along with heat source past a vertical permeable plate using a perturbation analysis, where the unsteadiness is caused by the time dependent surface temperature and concentration. Flow and heat transfer over an unsteady stretching sheet was studied by Ishak et al. [6]. Liu and Andersson [7] have also studied the heat and flow transfer over an unsteady stretching sheet. Flow and heat transfer with viscous dissipation of a non-Newtonian fluid over an unsteady stretching sheet was considered by Chen [8]. Radiation effect on the flow and heat transfer over an unsteady stretching sheet was studied using fifth order Runge–Kutta–Fehlberg integration scheme by El-Aziz [9]. Chebyshev finite difference method was used by Tsai et al. [10] to study the flow and heat transfer over an unsteady stretching surface with a nonuniform heat source. Mukhopadhyay [11] studied the heat and flow transfer along with radiation effect over an unsteady stretching sheet. It must be noted that the unsteadiness in [5–11] are due to either the time-dependent stretching rate or the time dependent temperature at the sheet. In the present paper, the transient changes in the temperature field is due to the sudden introduction of the heat source/sink on the steady MHD flow and heat transfer past a linearly stretching isothermal sheet. The effects of heat source/sink, magnetic field and Prandtl number on the temperature are analysed.

2 Formulation 2.1 Initial State (t  ≤ 0) Consider a steady two dimensional laminar boundary layer flow and heat transfer of an incompressible electrically conducting Newtonian fluid past a linearly stretching isothermal sheet under a transverse magnetic field of strength B0 . The sheet issues from a thin slit at x = 0, y = 0, where x -axis is along the horizontal direction of the flow, y -axis is normal (vertically upwards) to the flow; u and v are the horizontal and vertical components of velocity along x and y -directions respectively. The stretching speed is proportional to the distance from the origin along the x -direction, with a constant stretching rate β (> 0). The sheet is assumed to be at a constant temperature Tw , far away the constant ambient fluid temperature is T∞ and T0 is the temperature of the fluid. Under these assumptions, the governing steady state  boundary layer equations for the time t ≤ 0 are given by Char [12] as,

∂ u ∂ v + = 0, ∂ x ∂ y

(1)

Unsteady MHD Flow Past a Stretching Sheet Due to a Heat Source/Sink

u

293

 ∂ 2 u σ B20  ∂ u  ∂u + v = ν − u, ∂ x ∂ y ρ ∂ y 2

(2)

∂ T0 ∂T ∂ 2 T0 + v 0 = α ,  ∂x ∂y ∂ y 2

(3)

u

subjected to the boundary conditions, y = 0 : u = β x , v = 0, T0 = Tw , for x ≥ 0, y → ∞ : u → 0, T0 → T∞ , for x ≥ 0,

(4)

where ρ is the density, μ is the viscosity, ν is the kinematic viscosity, σ is the electric conductivity and α is the thermal diffusivity of the fluid. Defining the dimensionless variables and parameters,   (x, y, u, v, T0 ) =



x

β  ,y ν



T  − T∞ v β u ,  ,  , 0 ν νβ νβ Tw − T∞

 ,M =

σ B20 ν , Pr = ρβ α

(5)

the equations governing the initial state take the dimensionless form,

∂u ∂v + = 0, ∂x ∂y ∂u ∂ u ∂ 2u +v = u − Mu, ∂x ∂ y ∂ y2 u

∂ T0 ∂ T0 1 ∂ 2 T0 , +v = ∂x ∂y Pr ∂ y2

(6) (7) (8)

subjected to the boundary conditions, y = 0 : u = x, v = 0, T0 = 1, for x ≥ 0, y → ∞ : u → 0, T0 → 0, for x ≥ 0.

(9)

The (6)–(9) admit a closed form solution, which is given by, u = xe− T0 (y) = e

√ 1+My

− √ Pr y 1+M

 √ √ , v = e− 1+My − 1 1 + M,





√ F a, a + 1, −ae− 1+My F a, a + 1, −a ,

(10) (11)

where√ a = Pr/(1 + M) and F(a0 , a1 , z) is the Kummer’s function [12]. Substituting η = y 1 + M in (11), one can get the solution obtained by Char [12], and Kumari and Nath [13].

294

A.K. Banerjee et al.

2.2 Transient Heat Transfer (t  >0) Assuming that the flow and temperature field when t  ≤ 0 are given by (10) and (11), a heat source/sink of constant strength Q is introduced at time t  = 0 in the fluid flow and maintained for t  > 0. When t  > 0, the equation governing the transient heat transfer is given by,   ∂ 2T  ∂T ∂T  ∂T + u + v = α + Q (T  − T∞ ), ∂t ∂ x ∂ y ∂ y 2

(12)

subjected to the boundary conditions, t > 0 :

T  = Tw at y = 0, T  → T∞ as y → ∞.

(13)

Introducing the following dimensionless quantities (along with (5)), t = β t  , T (x, y,t) =

T  − T∞ Q , Q = , Tw − T∞ β

(14)

the dimensionless transient temperature field due to the sudden introduction of heat source/sink is given by,

∂T ∂T ∂T 1 ∂ 2T + QT, (t > 0) +u +v = ∂t ∂x ∂y Pr ∂ y2 t>0:

T = 1 at y = 0, T → 0 as y → ∞,

(15) (16)

(17) t ≤ 0 : T = T0 (y), for x ≥ 0, y ≥ 0, where (u, v) is governed by (10) for t > 0 also and T0 (y) is given by (11). At steady state (when t → ∞), (15) and (16) admit a closed form solution, which is given by,



√ √ K+a K+a −( K+a ) 1+My − 1+My 2 F F T =e , K +1, −ae , K + 1, −a , (18) 2 2  where K = a2 − 4aQ, Q ≤ Qc with Qc = Pr/4(1 + M). The dimensionless form of the local Nusselt number is given by,



1 ∂T ∂T = −x . (19) Nux = −x T ∂ y y=0 ∂ y y=0 The dimensionless average Nusselt number averaged over 0 ≤ x ≤ 1 is given by

1 ∂T Nu = − dx. (20) ∂ y y=0 0 Computations reveal that, T is independent of x for all time (including t > 0). Hence,

Nux ∂ T

=− Nu = . (21) x ∂ y y=0

Unsteady MHD Flow Past a Stretching Sheet Due to a Heat Source/Sink

a

295

b 0.2 1

0.6

0.1

1 - Pr = 7.0, Q = 1.75/2 2 - Pr = 0.71, Q = 0.1775/2

0.4

TQ≠0-TQ=0

TQ≠0-TQ=0

1

M=0 0.2 2

0 3

-0.2

3 - Pr = 7.0, Q = -1 4 - Pr = 0.71, Q = -1

4

-0.4 0

2

4

6

0 -0.1 3

M=2

-0.2 -0.3

8

10

1 - Pr = 7.0, Q = 1.75/6 2 - Pr = 0.71, Q = 0.1775/6

2

-0.4

3 - Pr = 7.0, Q = -1 4 - Pr = 0.71, Q = -1

4

0

2

4

6

y

y

M= 0

M= 2

8

10

Fig. 1 Profiles of the steady state excess temperature due to heat source/sink. (a) M = 0, (b) M = 2

3 Numerical Solution The numerical results for the unsteady state dimensionless temperature distribution is computed by solving (15)–(17) using the implicit finite difference method of Crank–Nicholson type [14, 15]. For the present problem when the Prandtl number Pr = 0.71, the mesh size was taken as Δ x = 0.002, Δ y = 0.0125 and Δ t = 0.01. The domain of computation was taken as 0 ≤ x ≤ 1 and 0 ≤ y ≤ 35. Whereas for Pr = 7.0, the domain of computation was taken as 0 ≤ x ≤ 1 and 0 ≤ y ≤ 12. The



n+1 n convergence criteria was set as Ti, j − Ti, j ≤ 1 × 10−5, where i, j denote the mesh point along the x-axis, y-axis respectively and n denotes the number of iterations with respect to time.

4 Results and Discussion The effects of heat source/sink have been studied for various values of the parameters namely, the heat source (Q > 0)/heat sink (Q < 0) parameter, the magnetic parameter M and the Prandtl number Pr. The Fig. 1 describe the profiles of steady state (t → ∞) excess temperature due to the heat source/sink. From the Fig. 1a,b it is observed that the thermal boundary layer is thin for Pr = 7.0 whereas it is thick for Pr = 0.71. The temperature raises or drops if the magnitude of the heat source/sink increases respectively. These effects are more pronounced under the magnetic field. From the Table 1, it is seen that the values of the computed steady local Nusselt number at x = 1 are in good agreement with the values of the exact steady local Nusselt number obtained from (11).The Figs. 2a–5b describe the evolution of the excess (drop in) temperature profiles with respect to time. Increase in temperature and decrease in temperature is seen to occur for positive Q (heat source) and negative Q (heat sink) respectively. Tables 2 and 3 give the time to reach steady state for Pr = 0.71 and Pr = 7.0 for various values of Q. It is observed

296

A.K. Banerjee et al.

Table 1 Values of the steady local Nux at x = 1 Pr 0.71

7

M

0

0

2

2

Q −1 Qc /2 −1 Qc /2 −1 Qc /2 −1 Qc /2 Exact Nux −0.9826 −0.3657 −0.9521 −0.2850 −3.2271 1.1703 −3.1714 −1.0058 −1.0136 Computed Nux −0.9822 −0.3699 −0.9517 −0.2919 −3.2257 1.1625 −3.17

a

b 0.08

0.8

Pr = 0.71, M = 0.0, Q = 0.1775/2

0.7

0.06

T⏐t≠0 - T⏐t=0

T⏐t≠0 - T⏐t=0

0.07

t = 0.2, 0.4, 2.0, 4.0, t*

0.05 0.04 0.03 0.02 0.01 0

Pr = 7.0, M = 0.0, Q = 1.75/2

0.6 0.5 0.4

t = 0.2, 0.4, 2.0, 4.0, t*

0.3 0.2 0.1

0

1

2

3

4

5

6

7

0

8

0

0.5

1

y

1.5

2

y

Fig. 2 Profiles of the transient excess temperature due to heat source for M = 0. (a) Pr = 0.71, (b) Pr = 7.0

a

b

0

0 -0.05

T⏐t≠0 - T⏐t=0

T⏐t≠0 - T⏐t=0

-0.05 -0.1 -0.15

t = 0.2, 0.4, 2.0, t*

-0.2 -0.25

-0.1 t = 0.2, 0.4, 2.0, t*

-0.15 -0.2 -0.25

Pr = 0.71, M = 0.0, Q = -1.0 -0.3

0

1

2

3

4

y

5

6

7

Pr = 7.0, M = 0.0, Q = -1.0 8

-0.3

0

1

2

3

4

5

y

Fig. 3 Profiles of the transient temperature drop for M = 0, Q = −1 (heat sink). (a) Pr = 0.71, (b) Pr = 7.0

that for an increase in strength of the heat sink, the steady time decreases. The value Qc represents the critical value given by Qc = Pr/(4(1 + M)). The computations converges for Q ≤ Qc only. The steady state time increases with an increase in Q, M and decreases with an increase in Pr. Steady state is reached very quickly for Pr = 7.0 when compared to Pr = 0.71. For Q > Qc the computation does not terminate justifying the fact that steady solution exists only for Q ≤ Qc . The excess average Nusselt number increases with M whereas it decreases with Q, see Fig. 6.

Unsteady MHD Flow Past a Stretching Sheet Due to a Heat Source/Sink

297

b

a 0.06

0.16

Pr = 0.71, M = 2.0, Q = 0.1775/6 0.04 0.03 0.02 0.01 0

Pr = 7.0, M = 2.0, Q = 1.75/6

0.14

t = 0.2, 0.4, 2.0, 4.0, t*

T⏐t≠0 - T⏐t=0

T⏐t≠0 - T⏐t=0

0.05

0.12

t = 0.2, 0.4, 2.0, 4.0, t*

0.1 0.08 0.06 0.04 0.02

0

1

2

3

4

5

6

7

0

8

0

0.5

1

y

1.5

2

y

Fig. 4 Profiles of the transient excess temperature due to heat source for M = 2. (a) Pr = 0.71, (b) Pr = 7.0

a

b

0

-0.1 -0.15 -0.2 -0.25

t = 0.2, 0.4, 2.0, 4.0, t*

-0.3

-0.1

t = 0.2, 0.4, 2.0, t*

-0.15 -0.2 -0.25

-0.35 -0.4

0 -0.05

T ⏐t≠0 - T ⏐t=0

T⏐t≠0 - T⏐t=0

-0.05

Pr = 7.0, M = 2.0, Q = -1.0

Pr = 0.71, M = 2.0, Q = -1.0 0

1

2

3

4

5

6

7

8

-0.3

0

1

2

y

3

4

5

y

Fig. 5 Profiles of the transient temperature drop for M = 2 and Q = −1 (heat sink). (a) Pr = 0.71, (b) Pr = 7.0 Table 2 The values of t = t ∗ , the time to reach steady state for Pr = 0.71 Q M

0 (Qc = 0.1775) 1 (Qc = 0.08875) 2 (Qc = 0.05917)

−1.0 3.71 4.30 4.62

−0.5 5.02 6.25 6.96

0 6.48 8.52 9.36

0.04 5.86 11.79 20.62

Qc /4 6.00 7.08 7.43

Qc /2 10.34 13.04 14.74

Qc 40.57 40.43 42.53

The important observations made from this study are: A critical value Qc of strength of heat source exists beyond which no steady solution exists. Reaching steady state is delayed under the magnetic field. The steady state time decreases with an increase in Pr. Steady state is reached quickly for a heat sink whereas it is delayed for a heat source of same strength. The local Nusselt number decreases with an increase in M, Q and for a decrease in Pr.

298

A.K. Banerjee et al. Table 3 The values of t = t ∗ , the time to reach steady state for Pr = 7.0 Q M

b

1

Pr = 0.71 Q = -1.0, M = 2.0

0.6

Q = -1.0, M = 0.0

—

0.4 0.2 0

Q = 0.1775/6, M = 2.0

-0.2

Q = 0.1775/2, M = 0.0

0 3.14 3.61 4.01

0.04 3.16 3.62 4.01

Qc /4 5.84 5.01 4.95

Qc /2 16.93 9.26 8.51

3

Pr = 7.0

2

Q≠0- Nu Q=0

—

Q≠0 -Nu Q=0

0.8

— Nu

−0.5 3.78 3.15 3.47

— Nu

a

−1.0 2.40 2.65 2.85

0 (Qc = 1.75) 1 (Qc = 0.875) 2 (Qc = 0.5833)

Q = -1.0, M = 2

1

Q = -1.0, M = 0

0 Q = 1.75/6, M = 2

-1 -2 -3

-0.4

Q = 1.75/2, M = 0

0

2

4

6

8

t

10

12

14

-4

0

5

10

15

20

t

Fig. 6 Profiles of the excess average Nusselt number (averaged over 0 ≤ x ≤ 1). (a) Pr = 0.71, (b) Pr = 7.0

References 1. Liu IC (2005) A note on heat and mass transfer for a hydromagnetic flow over a stretching sheet. Int Commun Heat Mass Transf 32:1075–1084 2. Xu H (2005) An explicit analytic solution for convective heat transfer in an electrically conducting fluid at a stretching surface with uniform free stream. Int J Eng Sci 43:859–874 3. Abel MS, Mahesha N (2008) Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation. Appl Math Model 32:1965–1983 4. Khan SK (2006) Heat transfer in a viscoelastic fluid flow over a stretching surface with heat source/sink, suction/blowing and radiation. Int J Heat Mass Transf 49:628–639 5. Pal D, Talukdar B (2010) Perturbation analysis of unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction. Commun Nonlinear Sci Numer Simul 15:1813–1830 6. Ishak A, Nazar R, Pop I (2009) Heat transfer over an unsteady stretching permeable surface with prescribed wall temperature. Nonlinear Anal Real World Appl 10:2909–2913 7. Liu IC, Andersson HI (2008) Heat transfer in a liquid film on an unsteady stretching sheet. Int J Therm Sci 47:766–772 8. Chen C-H (2006) Effect of viscous dissipation on heat transfer in a non-Newtonian liquid film over an unsteady stretching sheet. J Non-Newton Fluid Mech 135:128–135 9. El-Aziz MA (2009) Radiation effect on the flow and heat transfer over an unsteady stretching sheet. Int Commun Heat Mass Transf 36:521–524 10. Tsai R, Huang KH, Huang JS (2008) Flow and heat transfer over an unsteady stretching surface with non-uniform heat source. Int Commun Heat Mass Transf 35:1340–1343 11. Mukhopadhyay S (2009) Effect of thermal radiation on unsteady mixed convection flow and heat transfer over a porous stretching surface in porous medium. Int J Heat Mass Transf 52:3261–3265

Unsteady MHD Flow Past a Stretching Sheet Due to a Heat Source/Sink

299

12. Char M-I (1994) Heat transfer in a hydromagnetic flow over a stretching sheet. Warme-und ¨ Stoffubertragung ¨ 29:495–500 13. Kumari M, Nath G (2009) Analytical solution of unsteady three-dimensional MHD boundary layer flow and heat transfer due to impulsively stretched plane surface. Commun Nonlinear Sci Numer Simul 14:3339–3350 14. Muthucumaraswamy R, Ganesan P (2000) Flow past an impulsively started vertical plate with constant heat flux and mass transfer. Comput Methods Appl Mech Eng 187:79–90 15. Kumaran V, Vanav Kumar A, Pop I (2010) Transition of MHD boundary layer flow past a stretchng sheet. Commun Nonlinear Sci Numer Simul 15:300–311

Effect of Chemical Kinetics on Permeability of a Porous Rock Scaling by Concentration of Active Fluid Tapati Dutta, Supti Sadhukhan, and Sujata Tarafdar

Abstract Pores and fractures in rocks are continuously being reshaped through different chemical and physical processes. Fluids filling the pore space and carrying different chemical species are responsible for these changes. In the present work, we study the effort of chemical kinetics on the reshaping of pore structure and thereby on permeability. A simulation study is carried on a two-dimentional random porous structure. The particles permeate with a constant Peclet number, and their diffusion is represented through a random walk. Changing the probability of interaction varies the strength of the chemical reaction between the fluid and the rock. This study is done for different concentrations for the active material in the fluid. A scaling law is found to exist between the changes in permeability with reaction rate.

1 Introduction Changes in the rock structure and therefore in their transport properties are often brought about through the flow of reactive fluids in the pore spaces of such rocks. Competition between fluid flow and chemical kinetics cause either dissolution or disposition along the pore surfaces of rocks. Where local kinetics is fast relative to reactant transport (Damkohlar number Da1), fracture dissolution is strongly

T. Dutta () Physics Department, St. Xavier’s College, 30, Mother Teresa Sarani, Kolkata 700032, India e-mail: tapati [email protected] S. Sadhukhan Physics Department, Jogesh Chandra Choudhury College, Kolkata 700033, India e-mail: [email protected] S. Tarafdar Condensed Matter Physics Research Centre, Physics Department, Jadavpur University, Kolkata 700032, India e-mail: sujata [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 17, © Springer Science+Business Media, LLC 2012

301

302

T. Dutta et al.

influenced by the Peclet number (Pe). Pe gives the relative magnitude of advective and diffusive transport of reactants. The local kinetics is determined by the reactivity of the fluid with the pore wall. The process of channel formation is a contributing factor to digenesis in sedimentary rocks. The growth of channels through dissolution has been studied through experiments by Detwiler et al. [3] and simulations by Szymczak et al. [15] in quasi 2-dimensional systems. In the present work, we report a computer simulation in two dimension to simulate the flow through a porous medium for fluids having different reactivity for a particular choice of Pe number. We first simulate fluid flow through an irregular channel and calculate permeability from Darcy’s law. A numerical finite difference solution of the Navier–Stokes equation is employed for this. The ratio of fluid flux to the applied constant pressure gradient gives initial permeability. Channel reshaping is simulated next for a constant Peclet number and different reactivity of the fluid to erode the rock wall. The fluid flow reactive species in the fluid is simulated through a random walk. A random walker traversing the channel represents a corrosive particle carried by the fluid which has a probability of eroding a point on the channel wall. Through this formalism, we can insert the desired ratio of drift to diffusion through “hand” by imposing a suitable basis on the walker. The reaction between the fluid and the channel wall is introduced through a nonzero probability. The walker either “reacts” and disappears or exits the channel. The erosion continues until the channel has been exposed to certain “dose” of corrosive fluid, i.e., a certain number of walkers. The channel geometry has changed and so too the permeability. We then recalculate the fluid flow flux and the new permeability. The next section describes briefly the flow simulation and the procedure to calculate the permeability of the channel with definite boundaries that have been reshaped by interacting walkers. The following section describes the random walk algorithm that reshapes the boundary wall. Finally, our results for permeability variation for different reactive fluids are presented.

2 Fluid Flow in Connected Channels We have generated the initial “two-dimensional-rock” by using the Relaxed Bidisperse Ballistic Deposition Model (RBBDM) by Dutta and Tarafdar [5] and Sadhukhan et al. (2007a). In BBDM, by Dutta and Tarafdar [4], elongated 2 × 1 grains are deposited on a square lattice to get a porous structure with a connected solid phase. The RBBDM allows unstable overhangs generated by the grains to topple over. It is well known that natural sand are elongated and angular [8]. So the aspect ratio 2 is realistic. The Hoshen and Kopelman algorithm by [7] is used to identify the sample spanning channel and the Navier–Stokes equation is solved numerically to find the flow pattern by Sadhukhan et al. (2007b).

Effect of Chemical Kinetics on Permeability of a Porous Rock...

303

The pressure and velocity fields are solved by the procedure described by Sarkar et al. [13] with some necessary departures appropriate to our problem. The equation of motion of the fluid is given by the Navier–Stokes equation

ρ [∂ V/∂ t + (V.∇)V] + ∇P − μ ∇2 V = f e ,

(1)

where V, P, and f e represent the velocity, pressure, and external force per unit volume, respectively. ρ and μ are the density and the dynamic viscosity of the fluid respectively. Neglecting the inertial term and assuming no external forces acting on the fluid (1) reduces to

∂ V/∂ t = −1/ρ ∇P + η ∇2 V,

(2)

where η = μ /ρ is the kinematical viscosity. For an incompressible fluid the equation of continuity becomes ∇·V = 0 (3) Taking divergence of (2) and substitution of (3) in it, time discretization is done to obtain ∇2 Pn+1 = (ρ /Δ t)(∇. V n ) (4) Time discretization of (2) yields V n+1 − η∂ t∇2 V n+1 = V n − (∂ t/ρ )∇Pn+1

(5)

The initial condition is taken as n = 0 and V n = V 0 = 0. The updated pressure Pn+1 are the steady-state solutions for pressure and velocities to compute the updated velocity V n+1 . The new velocity V n+1 is then used as V n and (4) and (5) are iterated till the steady state is reached. Numerically, this condition is satisfied when |V n+1 − V n | < ε , where ε is a very small quantity. The pore channel is divided into square grid cells. P is the pressure at the center of the grid-cell, and the velocity components are defined suitably on the cell walls obeying the boundary conditions. To calculate the pressure and velocity components numerically, sequential updating of the discretized versions of (4) and (5) is done. The detailed steps are discussed in next chapter. The boundary conditions required for solving the equations are as follows – velocity components vanish at pore-rock boundary and terms outside the flow region are treated using imaginary image-grids by Aziz and Settari [1]. The rigid interface between pore and matrix exhibit a no-slip boundary condition. Let v be the y component of the velocity beyond a vertical rigid wall in the imaginary grid. For no-slip condition at the boundary, v = −v1, , where v1 is the y component of the velocity in the pore grid. Also, since ∇,V = 0, in the fluid cell, it follows that for ∇.V  = 0, to vanish in the imaginary cell u = u1 . The left-hand figure in Fig. 1 summarizes the boundary condition across the vertical wall. Here, u1 is the x component of the velocity in the fluid cell. Analogous boundary conditions for velocity are applied at a horizontal wall (right hand figure in Fig. 1.). To summarize, the velocity boundary conditions at no-flow boundaries – the normal velocity component remains the same, while the tangential velocity reverses.

304

T. Dutta et al.

Fig. 1 The grid cells for pressure P and velocity components u and v are shown here. The unprimed coordinates represent a vacant site in the real pore space, the heavy black line is the pore–solid interface and the primed quantities represent the pressure and velocity introduced in the image cell to preserve proper boundary conditions described in the text. The figure on the left is for the vertical boundary and the right-hand side figure is for the horizontal

In order to apply the finite difference method for numerical solution described below, it is necessary for the minimum width of the pore channel to be at least three units in terms of the smallest unit of resolution. This is because, with noslip boundary conditions, the two sites (or cells), adjacent to the walls of the pore have zero velocity, so to have a finite flow; there must be a third cell between these two, having nonzero velocity. Increasing this number from three, as much as practicable will generate the true parabolic velocity profile. However, in view of limited computer time and memory, we have at present restricted this minimum number to four. This means that, each site (either solid or vacant), we have generated by RBBDM, described above, has to be replaced by a 4 × 4 square grid, increasing the linear size of our sample 4 times. Each of these smaller squares is henceforth our new unit. This ensures that fluid may flow through even the narrowest channel in our system. The permeability at steady state is determined from Darcy’s law averaging over 250 percolating configurations. It has been shown that in BBDM by Dutta and Tarafdar [4]. the sample attains a constant porosity only after a sufficient number of grains (depending on the system size) have been deposited to overcome substrate effects. Here, a 128 × 128 size sample was generated, from which a 32 × 32 size sample was selected after the porosity had stabilized. The 32 × 32 size sample was magnified to 128 × 128 and simulation was carried out on it. This magnification was necessary to make the channels wide enough so that a realistic velocity profile with no-slip could be implemented at the walls. The average values of initial permeability (κ ) and the porosity (φ ) thus obtained were 0.002165 × 10−4 m2 and 0.152, respectively.

Effect of Chemical Kinetics on Permeability of a Porous Rock...

305

3 Channel Reshaping A biased random walker mimics the reactive chemical species in the fluid flowing through the rock pores. The reactive particles have a random diffusive motion and a drift along the pressure gradient. The Peclet number decides whether drift or diffusion dominates. The combined effect is simulated by imposing a bias (π ) on the motion of the walker related to the Peclet number Pe by Pe = UL/D,

(6)

where U, L, and D represent the drift velocity, a suitable length scale, and diffusivity, respectively. The distance traveled by a random walker can be obtained from Reif [9]. For motion along the Y -direction, we have after N steps (or equivalently, time intervals), < m > = N (p − q) , (7) and

Δ < m2 > = 4Npq

(8)

where p(q = 1 − p) is the probability of going in the +Y ( −Y ) direction. For biased motion U = < m > /N

(9)

and the diffusion coefficient D is given by D = Δ < m2 > /N = 4pq

(10)

We let a walker move with a probability py = π in the Y -direction, probability 0 in the −Y direction and px = (1 − π)/2 in both the +X and −X directions, we have therefore Pe = py /(4p2x ) (11) Drift is due to motion along Y and diffusion along X directions. L is taken as 1, the lattice spacing. A constant pressure difference of unity is maintained across the faces of the sample in the Y -direction while a periodic boundary condition is applied along the X-direction. A random number decides where the walker moves. The reactivity of the fluid with the wall is introduced through a parameter Pr . The probability of a solid state being converted to a void is introduced through a second random number for all values less than Pr , which is a measure of reaction kinetics. On dissolution of the rock particle, the walker walks into the new pore and ends its journey. A new walker is then released from the top surface and the entire process is repeated. In [own], we have studied the effect of Pe on the porosity and permeability of the system for a particular fluid. The number of walkers W represents the “dose” of the reactive

306

T. Dutta et al.

chemical in the sample. Physically, W is the product of the concentration C of the reactive species and the time T of their flow W = CT

(12)

W is varied in steps of 100 up to a maximum of 500 walkers for each Pr . This may correspond to an increase in C for a constant T or vice versa. π is taken as 0.5 corresponding to a Pe value of 2. The assumption of no flow in the −Y direction makes the simulation less time-consuming. If back flow was allowed, a large number of particles would simply flow out of the sample by retracing their paths. px and py are each taken to be 0.25.

4 Results In Sadhukhan et al. [12], we had studied the effect of different Pe on the restructuring of pore channel. For any given Pe, the maximum enhancement of permeability increased with the number of walkers. In this work, we have studied the effect of the reactivity of the fluid on the permeability of the sample for a constant value of Pe = 2. Figures 1 and 2 show the restructuring of the pore channel for a typical configuration. The initial pore space is shown in Fig. 1, the final configuration for Pe = 2 is shown in Fig. 2. The flow direction in the figure is from top to bottom, but the system is considered to be horizontal, the effect of gravity is absent.

Fig. 2 Restructuring of the channel geometry. The variation in velocity shown in the color code is in cm sec−1 . (a) shows the initial pore space and velocity distribution for fluid flow. (b) shows the final pore space for W = 10, 000 and Pe = 2.0

Effect of Chemical Kinetics on Permeability of a Porous Rock...

307

Fig. 3 Variation of porosity vs. reactivity for different W

Figure 3 shows the variation of porosity ϕ with fluids of different reactivity Pr for different doses W of the reactive species. The porosity for a particular Pr increases almost proportionately with W . From Fig. 3, it is evident that every curve shows a definite change in the slope at around Pr = 0.25. Beyond this value of Pr , ϕ remains almost independent of Pr and is a function of W alone. Figure 4 is a study of the change in relative permeability (κ /κ0 ) with Pr , with W as a parameter. Figure 5 is a study of the same variables on a logarithmic scale with relative permeability now scaled by the concentration of the reactive species W . The data shows a good collapse when the relative permeability is scaled as (κ /κ0 )/W 1.95 = APr0.37 ,

(13)

where A is some system constant. The scaling factor is 1.95. From (13), it is evident that the relative permeability has a power law dependence on the reactivity Pr , the exponent being 0.37. Figure 6 is a study of the variation of relative permeability with porosity for different reaction rates. The data points for the different Pr ’s almost coincide and are best fitted by a cubic law. This matches the results reported by Detwiler et al. [3].

308

Fig. 4 Relative permeability vs. reactivity

Fig. 5 Scaled fit of relative permeability vs. reactivity

T. Dutta et al.

Effect of Chemical Kinetics on Permeability of a Porous Rock...

309

Fig. 6 Variation of relative permeability vs. porosity

5 Discussions The focus of this study is the effect of the reactivity of a fluid in channel reshaping through dissolution and its effect on permeability. Szymczak and Ladd [14], Sadhukhan et al. [12], and Detwiler et al. [3] have done simulation studies and experiments where effect of Peclet number on channel reshaping has been studied. According to Hartmann et al. [6], leaching is promoted by rapid fluid flow while a slower flux enhances cementation. This effect on the diagenesis of sedimentary rocks has been studied in Sadhukhan et al. (2007a). In this study, we study only dissolution due to chemical reaction between fluid and rock. Bekri et al. [2] have done a comprehensive study of dissolution and deposition in fractures, varying Peclet and Peclet–Damkohler numbers. They simulated reactive flow in three-dimensional in a single fracture with a rough surface. However, the role of chemical kinetics on channel reshaping has not been studied extensively. The salient features that emerge from this study is that the permeability changes with the reactivity of the fluid with the rock. This happens due to the formation of new channels and the widening of the existing channels. The rate of change of porosity with the reactivity however becomes almost negligible beyond a certain value Pr = 0.25, for a constant Pe.

310

T. Dutta et al.

The relative (κ /κ0 ), when suitably scaled by the dose of the fluid, shows a power law dependence on the reactivity. The exponent is 0.37. It needs to be seen whether there is a dependence of this exponent on the Peclet number before any claim to the universal nature of such curves may be made. The permeability increases with porosity faster than linear as supported by experiments of Detwiler et al. [3]. We have not studied the variation of the Damkohler number in this problem, which needs to be done. A more realistic three-dimensional simulation incorporating cementation and dissolution is hoped to be reported in the near future. Acknowledgments The computation has been carried out using IBM-P690 at the Mobile Computing Centre, Jadavpur University.

References 1. Aziz K, Settari A (1979) Petroleum reservoir simulation. Applied Science Publisers Ltd., London 2. Bekri S, Thovert JF, Adler PM (1997) Eng Geo 48:283 and references therein 3. Detwiler RL, Glass RJ, Bourcier WL (2003) Simulation of fracture dissolution. Geophys Res Lett 30:1648 4. Dutta T, Tarafdar S (2003) Fractal pore structure of sedimentary rocks: simulation by ballistic deposition. J Geophys Res 108:2062 5. Dutta T, Tarafdar S Simulation of sedimentary rocks by ballistic deposition: presented at the 9th world multiconference on systemics, cybernetics and informatics, Orlando, 10–13 July 2005 6. Hartmann DJ, Beaumont EA, Coalson E (2000) Predicting sandstone reservoir system quality and example of petrophysical evaluation search and discovery article 40005, e-book 7. Hoshen J, Kopelman R (1976) Percolation and cluster distribution. 1. Cluster multiple labeling technique and critical concentration algorithm. Phys Rev B 14:3438 8. Pettijohn FJ (1984) Sedimentary rocks. Harper and Row Publishers Inc., New York 9. Reif F (1965) Fundamentals of statistical and thermal physics. McGraw-Hill, Singapore 10. Sadhukhan S, Dutta T, Tarafdar S (2007a) Simulation of diagenesis and permeability variation in two dimensional rock surface. Model Simul Mater Sci Eng 15:773 11. Sadhukhan S, Dutta T, Tarafdar S (2007b) Simulation of diagenesis and permeability variation in two-dimensional rock structure. Geophys J Int 169:1366 12. Sadhukhan S, Mal D, Dutta T, Tarafdar S (2008) Permeability variation with fracture dissolution: role of diffusion versus drift. Physica A 387:4541–4546 13. Sarkar S, Toksoz MN, Burn DR (2004) Fluid flow modelling in fractures: presented at the MIT earth resources laboratory industry consortium meeting, MIT Earth Resources Laboratory 14. Szymczak P, Ladd AJC (2004) Microscopic simulation of fracture dissolution. Geophys Res Lett 31:L23606 15. Szymczak P, Ladd AJC (2006) A network model of channel competition in fracture dissolution. Geophys Res Lett 33:L05401

Exciton–Phonon Dynamics with Long-Range Interaction Nick Laskin

Abstract Exciton–phonon dynamics on a 1D lattice with long-range exciton– exciton interaction have been introduced and elaborated. Long-range interaction leads to a nonlocal integral term in the motion equation of the exciton subsystem if we go from discrete to continuous space. In some particular cases for power-law interaction, the integral term can be expressed through a fractional order spatial derivative. A system of two coupled equations has been obtained, one is a fractional differential equation for the exciton subsystem, the other is a standard differential equation for the phonon subsystem. These two equations present a new fundamental framework to study nonlinear dynamics with long-range interaction. New approaches to model the impact of long-range interaction on nonlinear dynamics are: fractional generalization of Zakharov system, Hilbert–Zakharov system, Hilbert– Ginzburg–Landau equation and nonlinear Hilbert–Schr¨odinger equation. Nonlinear fractional Schr¨odinger equation and fractional Ginzburg–Landau equation are also part of this framework.

1 Introduction Dynamic lattice models are widely used to study a broad set of physical phenomena and systems. In the early 1970s, a novel mechanism for the localization and transport of vibrational energy in certain types of molecular chains was proposed by A.S. Davydov [1]. He pioneered the concept of solitary excitons or the Davydov soliton [2]. His theoretical model to study solitary excitons is based on exciton– phonon lattice dynamics with nearest-neighbor exciton–exciton interaction, the so-called Davydov model. Our primary focus is the analytical developments of quantum 1D exciton–phonon dynamics with power-law long-range exciton–exciton N. Laskin () TopQuark Inc., Toronto, ON, M6P 2P2, Canada e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 18, © Springer Science+Business Media, LLC 2012

311

312

N. Laskin

interaction Jn,m = J/|n − m|−s, (s > 0) for excitons located at lattice sites n and m. In addition to the well-known interactions with integer values of s, some complex media can be described by fractional values of s (see, for example, references in [14]). Using the ideas first developed in [8], we elaborate the Davydov model for an exciton–phonon system with a fractional power-law exciton–exciton interaction. It has been shown that 1D lattice exciton–phonon dynamics in the long-wave limit can be effectively presented by the general system of two coupled equations for exciton and phonon dynamic variables. The dynamic equation describing the exciton subsystem is the fractional differential equation, which is a manifestation of non-locality of interaction, originating from the long-range interaction term. These two dynamic equations can be considered as a new general approach to study nonlinear quantum dynamics with long-range interaction. Particular physical cases include: nonlinear fractional Schr¨odinger equation, fractional Ginzburg–Landau equation, fractional generalization of Zakharov system, Hilbert–Zakharov system, Hilbert–Ginzburg–Landau equation, and nonlinear Hilbert–Schr¨odinger equation. The Chapter is organized as follows. In Sect. 2, we generalize Davydov’s Hamiltonian for the case of long-range power-law exciton–exciton interaction. The system of two coupled discrete equations of motion for exciton and phonon subsystems has been found using the Davydov anzatz. Transformation to the system of two continuous equations of motion has been performed in the long wave limit. Section 3 focuses on new nonlinear fractional differential equations resulting from our general approach to study the 1D exciton–phonon system with long-range interaction. In conclusion, we outline our new developments.

2 Lattice Exciton–Phonon Hamiltonian with Long-Range Interaction 2.1 Davydov’s Hamiltonian To model 1D quantum lattice dynamics with long-range exciton–exciton interaction we follow [3] and consider a linear, rigid arrangement of sites with one molecule at each lattice site. Davydov’s Hamiltonian reads H = Hex + H ph + Hint .

(1)

Here Hex is the Hamiltonian operator of the exciton system, which describes dynamics of intramolecular excitations or simply excitons, Hph is phonon Hamiltonian operator, which describes molecular displacements or, in other words, the lattice vibrations, and Hint is the exciton–phonon operator, which describes the interaction of an exciton with lattice vibrations. The exciton Hamiltonian is Hex = ε

∞

∑

n=−∞

b+ n bn −

∞

∑

n,m=−∞

Jn.m b+ n bm ,

(2)

Exciton–Phonon Dynamics with Long-Range Interaction

313

where b+ n is creation and bn is annihilation operators of an exciton on the n site. + + Operators b+ n and bn satisfy the relations [bn , bm ] = δn,m , [bn , bm ] = 0, [bn , bm ] = 0. Parameter ε is exciton energy on the site, Jn,.m is the exciton transfer matrix, which describes exciton–exciton interaction between sites n and m. To extend Davydov’s model and go beyond the nearest-neighbor interaction, we introduce the power-law interaction between excitons on sites n and m Jn,.m = Jn−m =

J , n = m, |n − m|s

(3)

where J is the interaction constant, parameter s covers different physical models; the nearest-neighbour approximation (s = ∞), the dipole–dipole interaction (s = 3), and the Coulomb potential (s = 1). Our main interest will be in fractional values of s that can appear for more sophisticated interaction potentials attributed to complex media. The phonon Hamiltonian Hph is 1 ∞ H ph = ∑ 2 n=−∞



 pˆ 2n 2 + w(uˆn+1 − uˆ n ) , m

(4)

where w is the elasticity constant of the 1D lattice, uˆn is the displacement operator from the equilibrium position of site n, pˆ n is the momentum operator of site n, and m is molecular mass. Finally, the exciton–phonon Hamiltonian Hint is Hint = χ

∞

∑

(uˆn+1 − uˆn )b+ n bn ,

(5)

n=−∞

with coupling constant χ . Furthermore, aiming to obtain a system of dynamic equations for the exciton–photon system under consideration, we introduce Davydov’s ansatz.

2.2 Davydov’s Anzatz and Motion Equations To study system (1), we introduce quantum state vector |φ (t) > following [1, 3, 11] |φ (t) >= |Ψ(t) > |Φ(t) >,

(6)

where quantum vectors |Ψ(t) > and |Φ(t) > are defined by |Ψ(t) >= ∑ ψn (t)b+ n |0 >ex , n

(7)

314

N. Laskin

  i |Φ(t) >= exp − ∑ (ξn (t) pˆn − ηn (t)uˆn |0 > ph , h¯

and

(8)

here, h¯ is the Planck’s constant, |0 >ex and |0 >ph are vacuum states of exciton and phonon subsystems, and ξn (t) is the diagonal matrix element of the displacement operator uˆn in the basis defined by (6), while ηn (t) is diagonal matrix element of the momentum operator pˆn in the same basis,

ξn (t) = < φ (t)|uˆn |φ (t) >,

ηn (t) = < φ (t)| pˆ n |φ (t)>.

State vector |φ (t) > satisfies the normalization condition < φ (t)|φ (t) > = ∑ |ψn (t)|2 = N, n

with |ψn (t)|2 being the probability to find exciton on the nth site and N is the total number of excitons. Therefore, the study of dynamics of an exciton–photon system (1) can be performed in terms of the functions ψn (t), ξn (t) and ηn (t). In other words, Davydov’s ansatz defined by (6)–(8) allows us to go from the quantum Hamiltonian operator introduced by (1) to the Hamiltonian function developed below. In the basis of vectors |φ (t) >, Hamiltonians Hex , Hph , and Hint become functions of dynamic variables ψn (t), ψn∗ (t), ξn (t) and ηn (t) < ϕ (t)|Hex |ϕ (t) > = Hex (ψn , ψn∗ ) =ε

∞

∑

ψn∗ (t) ψn (t) −

n=−∞

and

∞

∑

Jn−mψn∗ (t) ψm (t),

(9)

n,m=−∞

 2  1 ∞ ηn 2 < ϕ (t)|H ph |ϕ (t) > = H ph (ξn , ηn ) = ∑ m + w(ξn+1 − ξn) , (10) 2 n=−∞

and < ϕ (t)|Hint |ϕ (t) > = Hint (ψn , ψn∗ ; ξn , ηn ) = χ

∞

∑

(ξn+1 − ξn )ψn∗ (t) ψn (t),(11)

n=−∞

From (9)–(11), we obtain the system of dynamic equations in discrete space for ψn (t), ξn (t) and ηn (t), i¯h

∂ ψn (t) = Λ ψn (t) − ∑ Jn−m ψm (t) + χ (ξn+1(t) − ξn (t))ψn (t), ∂t m(n=m) ∂ ξn (t) ηn (t) = , ∂t m

(12)

(13)

Exciton–Phonon Dynamics with Long-Range Interaction

315

and

∂ ηn (t) = w(ξn+1 (t) − 2ξn (t) + ξn−1 (t)) + χ (|ψn+1 (t)|2 − |ψn (t)|2 ), ∂t

(14)

where constant Λ is

    ∂ ξn (t) 2 1 ∞ 2 Λ=ε+ + w(ξn+1 (t) − ξn (t)) . ∑ m ∂t 2 n=−∞

Substituting ηn (t) from (13) into (14) yields m

∂ 2 ξn (t) = w(ξn+1 (t) − 2ξn (t) + ξn−1(t)) + χ (|ψn+1(t)|2 − |ψn (t)|2 ). (15) ∂ t2

Our focus now is the system of two coupled discrete dynamic equations (12) and (15).

2.3 From Lattice to Continuum To go from the discrete to continuum version of (12) and (15), let us introduce

ϕ (k,t) =

∞

∑

e−ikn ψn (t),

∞

v(k,t) =

n=−∞

∑

e−ikn ξn (t),

n=−∞

where ψn (t) is related to ϕ (k,t) as 1 ψn (t) = 2π

π

dkeikn ϕ (k,t),

−π

and ξn (t) is related to v(k,t) as 1 ξn (t) = 2π

π

dkeikn v(k,t), −π

and k can be considered as a wave number. In the long-wave limit, when the wavelength exceeds the intersite scale a (let’s put for simplicity a = 1), we may consider ϕ (k,t) as a kth Fourier component of continuous function ψ (x,t), ψn (t) −→ ψ (x,t) k→0

and v(k,t) as a kth Fourier component of function ξ (x,t), ξn (t) −→ ξ (x,t). That is, k→0

functions ψ (x,t) and ϕ (k,t) are related to each other by the Fourier transform

ψ (x,t) =

1 2π

∞ −∞

dkeikx ϕ (k,t),

ϕ (k,t) =

∞ −∞

dxeikx ψ (x,t),

316

N. Laskin

and similarly for ξ (x,t) and v(k,t), 1 ξ (x,t) = 2π

∞ ikx

dke v(k,t),

v(k,t) =

−∞

∞

dxeikx ξ (x,t).

−∞

Therefore, we conclude that in the long wave limit, (12) and (15) become continuous equations of motion

i¯h

∂ ψ (x,t) = λ ψ (x,t) − ∂t

∞

dy∂x K(x − y)∂x ψ (x,t) + χ

−∞

∂ ξ (x,t) ψ (x,t), ∂x

(16)

and m

∂ 2 ξ (x,t) ∂ 2 ξ (x,t) ∂ |ψ (x,t)|2 , =w + 2χ 2 2 ∂t ∂x ∂x

(17)

where kernel K(x) in (16) has been introduced as

K(x) =

1 π

∞

dkeikx −∞

G(k) , k2

with function G(k) defined by G(k) = J(0) − J(k),

J(k) =

∞

∑ e−ikn Jn, n=−∞ n=0

here, Jn is given by (3), and finally, λ = Λ − J(0). Thus, we obtained a new system of coupled dynamic equations (16) and (17) to model 1D exciton–phonon dynamics with long-range exciton–exciton interaction (3). Field ψ (x,t) describes the exciton subsystem and field ξ (x,t) describes the phonon subsystem. Equation (16) is the integro-differential equation while (17) is the differential one. The integral term in (16), which is a manifestation of non-locality of interaction, comes from the long-range interaction term in Hamiltonian (2).

Exciton–Phonon Dynamics with Long-Range Interaction

317

3 Exciton–Phonon Dynamics To transform system (16) and (17) into the system of coupled differential equations, we use the properties of function G(k) at limit k → 0, which can be obtained from the polylogarithm asymptotics [8] G(k) ∼ Ds |k|s−1 ,

2 ≤ s < 3,

(18)

G(k) ∼ −Jk2 ln k,

s = 3,

(19)

G(k) ∼

J ζ (s − 2) 2 k , 2

s > 3,

(20)

where Γ(s) is Γ-function, ζ (s) is the Riemann zeta function and coefficient Ds is defined by Ds =

πJ . Γ(s) sin(π (s − 1)/2)

(21)

It is seen from (18) that the fractional power of k occurs for interactions with 2 < s < 3 only. In the coordinate space, fractional power of |k| gives us the fractional Riesz derivative [9, 10], and we come to a fractional differential equation i¯h

∂ ψ (x,t) ∂ ξ (x,t) = λ ψ (x,t) − Ds ∂xs−1 ψ (x,t) + χ ψ (x,t), ∂t ∂x

2 < s < 3, (22)

where, ∂xs−1 is the Riesz fractional derivative of order s − 1

∂xs−1 ψ (x,t) =

1 − 2π

∞

dkeikx |k|s−1 ϕ (k,t).

−∞

Thus, our main result is the new system of coupled equations (17) and (22) to study exciton-phonon dynamics with long-range interaction on a 1D lattice. The system of equations (17) and (22) is in fact a new general framework to model nonlinear dynamic phenomena with long-range interaction. Now let us introduce and briefly discuss new theoretical approaches originating from the framework. They are: fractional generalization of the Zakharov system, the nonlinear fractional Schr¨odinger equation, the fractional Ginzburg–Landau equation, the Hilbert–Zakharov system, the nonlinear Hilbert–Schr¨odinger equation, and the fractional Hilbert–Ginzburg–Landau equation.

318

N. Laskin

3.1 Fractional Generalization of Zakharov System Introducing a new variable σ (x,t) = ∂ ξ∂(x,t) turns (22) and (17) into the following x new system of equations for fields ψ (x,t) and σ (x,t), i¯h

∂ ψ (x,t) = λ ψ (x,t) − Ds ∂xs−1 ψ (x,t) + χσ (x,t)ψ (x,t), ∂t

and



∂2 ∂2 − v2 2 2 ∂t ∂x



σ (x,t) =

2 < s < 3, (23)

2χ ∂ 2 |ψ (x,t)|2 , m ∂ x2

(24)

where v = w/m is the velocity of sound. The system of two coupled equations (23) and (24) can be considered as a fractional generalization of the Zakharov system introduced in 1972 to study the Langmuir waves propagation in an ionized plasma [13].

3.2 Nonlinear Fractional Schr¨odinger Equation Assuming the existence of a stationary solution ∂ ξ (x,t)/∂ t = 0 in the system of (17) and (22) results in the following fractional differential equation for wave function ψ (x,t), i¯h

∂ ψ (x,t) 2χ = λ ψ (x,t) − Ds∂xs−1 ψ (x,t) − |ψ (x,t)|2 ψ (x,t), ∂t w

2 < s < 3, (25)

which can be rewritten in the form of a nonlinear fractional Schr¨odinger equation, i¯h

∂ φ (x,t) 2χ = −Ds ∂xs−1 φ (x,t) − |φ (x,t)|2 φ (x,t), ∂t w

(26)

where 2 < s < 3 and the wave function φ (x,t) is related to the wave function ψ (x,t) by

φ (x,t) = exp{iλ t/¯h}ψ (x,t).

(27)

It follows from (20) that for s > 3, (26) turns into the nonlinear Schr¨odinger equation i¯h

2χ ∂ φ (x,t) J ζ (s − 2) 2 ∂x φ (x,t) − =− |φ (x,t)|2 φ (x,t). ∂t 2 w

where ∂x2 = ∂ 2 /∂ x2 . Finally, note that the linear fractional Schr¨odinger equation in 1D and 3D has been developed at first in [4–7]. Three quantum mechanical problems were studied in these papers; a quantum particle in an infinite potential well, fractional quantum

Exciton–Phonon Dynamics with Long-Range Interaction

319

oscillator, and fractional Bohr atom. The energy spectra for these three fractional quantum mechanical problems were found using the linear fractional Schr¨odinger equation.

3.3 Fractional Ginzburg–Landau Equation In the case of propagating waves, we can search for the solution of system (17) and (22) in the form of travelling waves, ψ (x,t) = ψ (x − vt), and ξ (x,t) = ξ (x − vt), where v is velocity of the wave. From (17) to (22), let us go to (23) and (24) and substitute ψ (x,t) = ψ (ζ ), and σ (x,t) = σ (ζ ), where ζ = x − vt. It is easy to see that the solution of (24) is

σ (x,t) =

2χ |ψ (ζ )|2 . m(v2 − v2 )

(28)

Then (23) results in nonlinear equation i¯hv

∂ ψ (ζ ) = λ ψ (ζ ) − Ds ∂ζs−1 ψ (ζ ) + γ |ψ (ζ )|2 ψ (ζ ), ∂ζ

2 < s < 3,

(29)

where γ is the nonlinearity parameter

γ=

2χ . m(v2 − v2 )

(30)

Introducing wave function φ (ζ ) related to wave function ψ (ζ ) by

φ (ζ ) = exp{iλ ζ /¯hv}ψ (ζ ), results in fractional Ginzburg–Landau equation i¯hv

∂ φ (ζ ) = −Ds ∂ζs−1 φ (ζ ) + γ |φ (ζ )|2 φ (ζ ), ∂ζ

2 < s < 3,

which was initially proposed in [12].

3.4 Hilbert-Zakharov System It follows from (18) that in the case when s = 2, the function G(k) at limit k → 0 takes the form G(k) ∼ π J|k|, s = 2.

320

N. Laskin

Hence, (23) becomes i¯h

∂ ψ (x,t) = λ ψ (x,t) − π JH {∂x ψ (x,t)} + χσ (x,t)ψ (x,t), ∂t

s = 2, (31)

Here, H is the Hilbert integral transform defined by H {ϕ (x,t)} = P

∞

−∞

dy

ϕ (y,t) , y−x

where P stands for the Cauchy principal value of the integral. We will call the system of equations (31) and (24) as the Hilbert–Zakharov system.

3.5 Nonlinear Hilbert–Schr¨odinger Equation In the case when s = 2 and ∂ ξ (x,t)/∂ t = 0, the system of equations (16) and (17) results in the following nonlinear quantum mechanical equation for wave function ψ (x,t), i¯h

∂ ψ (x,t) 2χ = λ ψ (x,t) − π JH {∂x ψ (x,t)} + |ψ (x,t)|2 ψ (x,t). ∂t m

(32)

Introducing wave function φ (x,t) related to wave function ψ (x,t) by means of (27), brings nonlinear Hilbert–Schr¨odinger equation i¯h

2χ ∂ φ (x,t) = −π JH {∂x φ (x,t)} − |φ (x,t)|2 φ (x,t). ∂t ω

(33)

This equation was first developed in [8].

3.6 Hilbert–Ginzburg–Landau Equation In the case when s = 2, let’s search for the solution of the system of equations (31) and (24) in the form of travelling waves, ψ (x,t) = ψ (x− vt), and ξ (x,t) = ξ (x− vt), where v is velocity of the wave. The solution of (24) has the form of (28). Thus, (31) results in i¯hv

∂ ψ (ζ ) = λ ψ (ζ ) − π JH {∂x ψ (ζ )} + γ |ψ (ζ )|2 ψ (ζ ), ∂ζ

(34)

where γ is the nonlinearity parameter introduced by (30) and ζ = x − vt. We will call (34) the Hilbert–Ginzburg–Landau equation.

Exciton–Phonon Dynamics with Long-Range Interaction

321

4 Conclusion We have introduced and developed exciton–phonon dynamics with long-range exciton–exciton interaction. It has been shown that the long–range power-law interaction leads, in general, to a nonlocal integral term in the motion equation of an exciton subsystem if we go from a 1D lattice to continuous space. In some particular cases for power-law interaction with non-integer power s, the nonlocal integral term can be expressed through a spatial derivative of fractional order. We have obtained the system of two coupled equations, where one is fractional differential equation for exciton subsystem and the other one is a standard differential equation for phonon subsystem. It has been found that this system of two coupled equations can be further simplified to present new fundamental approaches for studying a wide range of nonlinear physical phenomena with long-range interaction. These approaches are: nonlinear fractional Schr¨odinger equation, fractional Ginzburg–Landau equation, fractional generalization of Zakharov system, Hilbert–Zakharov system, Hilbert– Ginzburg–Landau equation, and nonlinear Hilbert–Schr¨odinger equation. The results presented will further advance applications of fractional calculus to study waves and chaos phenomena in media with long-range interaction. Implementation of new theoretical approaches will initiate developing numerical algorithms to simulate impact of long-range interaction on nonlinear dynamics.

References 1. Davydov AS (1973) The theory of contraction of proteins under their excitation. J Theor Biol 38:559–569. doi:10.1016/0022-5193(73)90256-7 2. Davydov AS, Kislukha NI (1973) Solitary Excitons in One-Dimensional Molecular Chains. Physica Status Solidi (B) 59:465–470. doi:10.1002/pssb.2220590212 3. Davydov AS (1991) Solitons in molecular systems, 2nd edn. Reidel, Dordrecht 4. Laskin N (2000a) Fractional quantum mechanics and L´evy path integrals. Phys Lett 268A: 298–304. doi:10.1016/S0375-9601(00)00201-2 5. Laskin N (2000b) Fractional quantum mechanics. Phys Rev 62E:3135–3145. doi:10.1103/ PhysRevE.62.3135 6. Laskin N (2000c) Fractals and quantum mechanics. Chaos 10:780–790. doi:10.1063/ 1.1050284 7. Laskin N (2002) Fractional Schr¨odinger equation. Phys Rev 66E:056108 (7 pages). doi:10. 1103/PhysRevE.66.056108 8. Laskin N, Zaslavsky GM (2006) Nonlinear fractional dynamics of lattice with long-range interaction. Physica 368A:38–54. doi:10.1016/j.physa.2006.02.027 9. Saichev AI, Zaslavsky GM (1997) Fractional kinetic equations: solutions and applications. Chaos 7:753–764. doi:10.1063/1.166272 10. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives, theory and applications. Gordon and Breach, Amsterdam 11. Scott A (1992) Davydov’s soliton. Phys Rep 217:1–67. doi:10.1016/0370-1573(92)90093-F 12. Weitzner H, Zaslavsky GM (2003) Some applications of fractional equations. Commun Nonlinear Sci Numer Simulat 8:273–281. doi:10.1016/S1007-5704(03)00049-2

322

N. Laskin

13. Zakharov VE (1972) Collapse of Langmuir waves. Sov Phys JETP 35:908–914 14. Zaslavsky GM, Stanislavsky AA, Edelman M (2006) Chaotic and pseudochaotic attractors of perturbed fractional oscillator. Chaos 16:013102 (6 pages). doi:10.1063/1.2126806

Time Evolution of the Spectral Data Associated with the Finite Complex Toda Lattice Aydin Huseynov and Gusein Sh. Guseinov

Abstract Spectral data for complex Jacobi matrices are introduced and the time evolution of the spectral data for the Jacobi matrix associated with the solution of the finite complex Toda lattice is computed.

1 Introduction In this chapter, we deal with the finite Toda lattice a˙n = an (bn+1 − bn ),

b˙ n = 2(a2n − a2n−1 ),

n = 0, 1, . . . , N − 1,

(1)

subject to the boundary conditions a−1 = aN−1 = 0

(2)

and the complex-valued initial conditions an (0) = a0n

(0 ≤ n ≤ N − 2),

bn (0) = b0n

(0 ≤ n ≤ N − 1),

(3)

where an = an (t) (0 ≤ n ≤ N − 2), bn = bn (t) (0 ≤ n ≤ N − 1) form the desired solution and a0n , b0n are given complex numbers such that a0n = 0 (n = 0, 1, . . . , N −2).

A. Huseynov Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, AZ1141 Baku, Azerbaijan e-mail: [email protected] G.Sh. Guseinov () Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 19, © Springer Science+Business Media, LLC 2012

323

324

A. Huseynov and G.Sh. Guseinov

If we define the N × N matrices J and A by ⎡

b0 ⎢a ⎢ 0 ⎢0 ⎢ ⎢ . J=⎢ ⎢ .. ⎢ ⎢0 ⎢ ⎣0 0 ⎡

0 ⎢a ⎢ 0 ⎢ ⎢0 ⎢ . A=⎢ ⎢ .. ⎢ ⎢0 ⎢ ⎣0 0

a0 b1 a1 .. .

0 a1 b2 .. .

··· ··· ··· .. .

0 0 0 .. .

0 0 0 .. .

0 0 . . . bN−3 aN−3 0 0 · · · aN−3 bN−2 0 0 · · · 0 aN−2

0 0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ 0 ⎥ ⎥ aN−2 ⎦ bN−1

⎤ −a0 0 · · · 0 0 0 0 −a1 · · · 0 0 0 ⎥ ⎥ ⎥ a1 0 · · · 0 0 0 ⎥ .. .. .. ⎥ .. .. . . ⎥ . . . . ⎥, . . ⎥ 0 0 . . . 0 −aN−3 0 ⎥ ⎥ 0 0 · · · aN−3 0 −aN−2 ⎦ 0 0 · · · 0 aN−2 0

(4)

(5)

then the system (1) with the boundary conditions (2) is equivalent to the Lax equation d J = [J, A] = JA − AJ. (6) dt The Lax representation (6) plays important role in the investigation of nonlinear initial-boundary value problem (1)–(3). There is a huge number of papers devoted to the investigation of the Toda lattices and their various generalizations, from which we indicate here only [1–6]. However, the finite Toda lattice subject to the complex-valued initial conditions was not considered in the literature to the best of our knowledge. For some techniques of investigation of nonlinear discrete equations (difference equations), see [7–9].

2 Spectral Data for Complex Jacobi Matrices An N × N complex Jacobi matrix is a matrix of the form ⎡

b0 ⎢a ⎢ 0 ⎢0 ⎢ ⎢ . J=⎢ ⎢ .. ⎢ ⎢0 ⎢ ⎣0 0

a0 b1 a1 .. .

0 a1 b2 .. .

··· ··· ··· .. .

0 0 0 .. .

0 0 0 .. .

0 0 . . . bN−3 aN−3 0 0 · · · aN−3 bN−2 0 0 · · · 0 aN−2

0 0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ 0 ⎥ ⎥ aN−2 ⎦ bN−1

(7)

Finite Complex Toda Lattice

325

where for each n, an and bn are arbitrary complex numbers such that an is different from zero: an , bn ∈ C, an = 0. (8) If the entries of the matrix J are real, then the eigenvalues of J are real and simple. However, if the matrix J is complex, then its eigenvalues may be nonreal and multiple. Let R(λ ) = (J − λ I)−1 be the resolvent of the matrix J (by I we denote the identity matrix of needed dimension) and e0 be the N-dimensional column vector with the components 1, 0, . . . , 0. The rational function  w(λ ) = − (R(λ )e0 , e0 ) = (λ I − J)−1 e0 , e0

(9)

we call the resolvent function of the matrix J, where (·, ·) stands for the standard inner product in CN . Denote by λ1 , . . . , λ p all the distinct eigenvalues of the matrix J and by m1 , . . . , m p their multiplicities, respectively, as the roots of the characteristic polynomial det (J − λ I), so that 1 ≤ p ≤ N and m1 + . . . + m p = N. We can decompose the rational function w(λ ) into partial fractions to get w(λ ) =

p

mk

βk j

∑ ∑ (λ − λk ) j ,

(10)

k=1 j=1

where βk j are some complex numbers uniquely determined by the matrix J. Definition 1. The collection of the quantities {λ k ,

βk j

( j = 1, . . . , mk ,

k = 1, . . . , p)},

(11)

we call the spectral data of the matrix J. For each k ∈ {1, . . . , p} the (finite) sequence {βk1 , . . . , βkmk } we call the normalizing chain (of the matrix J) associated with the eigenvalue λk . Let us indicate a convenient way for computation of the spectral data of complex Jacobi matrices. For this purpose we should describe the resolvent function w(λ ) of the Jacobi matrix. Given a Jacobi matrix J of the form (7) with the entries (8), N−1 consider the eigenvalue problem Jy = λ y for a column vector y = {yn }n=0 , that is equivalent to the second-order linear difference equation an−1 yn−1 + bn yn + an yn+1 = λ yn , n ∈ {0, 1, . . ., N − 1},

a−1 = aN−1 = 1,

(12)

326

A. Huseynov and G.Sh. Guseinov

for {yn }Nn=−1, with the boundary conditions y−1 = yN = 0. Denote by {Pn (λ )}Nn=−1 and {Qn (λ )}Nn=−1 the solutions of (12) satisfying the initial conditions P−1 (λ ) = 0, P0 (λ ) = 1; (13) Q−1 (λ ) = −1,

Q0 (λ ) = 0.

(14)

For each n ≥ 0, Pn (λ ) is a polynomial of degree n and is called a polynomial of first kind and Qn (λ ) is a polynomial of degree n − 1 and is known as a polynomial of second kind. The equality det (J − λ I) = (−1)N a0 a1 · · · aN−2 PN (λ )

(15)

holds so that the eigenvalues of the matrix J coincide with the zeros of the polynomial PN (λ ). The entries Rnm (λ ) of the matrix R(λ ) = (J − λ I)−1 (resolvent of J) are of the form (see [10])

Rnm (λ ) =

Pn (λ )[Qm (λ ) + M(λ )Pm (λ )], Pm (λ )[Qn (λ ) + M(λ )Pn (λ )],

where M(λ ) = −

0 ≤ n ≤ m ≤ N − 1, 0 ≤ m ≤ n ≤ N − 1,

(16)

QN (λ ) . PN (λ )

Therefore according to (9) and using initial conditions (13), (14), we get w(λ ) = −R00 (λ ) = −M(λ ) =

QN (λ ) . PN (λ )

(17)

Denote by λ1 , . . . , λ p all the distinct roots of the polynomial PN (λ ) (which coincide by (15) with the eigenvalues of the matrix J) and by m1 , . . . , m p their multiplicities, respectively: PN (λ ) = c(λ − λ1)m1 · · · (λ − λ p)m p , where c is a constant. We have 1 ≤ p ≤ N and m1 + . . . + m p = N. Therefore by (17) decomposition (10) can be obtained by rewriting the rational function QN (λ )/PN (λ ) as the sum of partial fractions. We can also get another convenient representation for the resolvent function as follows. If we delete the first row and the first column of the matrix J given in (7), then we get the new matrix

Finite Complex Toda Lattice

327

⎡

(1)

b0 ⎢ (1) ⎢ a0 ⎢ ⎢ 0 ⎢ ⎢ J (1) = ⎢ ... ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0 0 where

(1)

a0 (1) b1 (1) a1 .. . 0 0 0

0 (1) a1 (1) b2 .. .

··· ··· ··· .. .

0 0 0 .. .

0 0 0 .. .

(1)

(1)

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ 0 ⎥ ⎥ (1) ⎥ aN−3 ⎦

0 . . . bN−4 aN−4 (1) (1) 0 · · · aN−4 bN−3 (1) (1) 0 · · · 0 aN−3 bN−2

(1)

n ∈ {0, 1, . . . , N − 3},

(1)

n ∈ {0, 1, . . . , N − 2}.

an = an+1, bn = bn+1,

⎤

0 0 0 .. .

The matrix J (1) is called the first truncated matrix (with respect to the matrix J). We show that det(J (1) − λ I) w(λ ) = − (18) det(J − λ I) holds. To this end, we denote the polynomials of the first and the second kinds, (1) (1) corresponding to the matrix J (1) , by Pn (λ ) and Qn (λ ), respectively. It is easily seen that (1)

Pn (λ ) = a0 Qn+1 (λ ), (1)

Qn (λ ) =

n ∈ {0, 1, . . . , N − 1},

1 {(λ − b0 )Qn+1 (λ ) − Pn+1(λ )}, a0

n ∈ {0, 1, . . . , N − 1}.

(19)

Indeed, both sides of each of these equalities are solutions of the same difference equation (1)

(1)

(1)

an−1 yn−1 + bn yn + an yn+1 = λ yn ,

n ∈ {0, 1, . . . , N − 2},

(1)

aN−2 = 1,

and the sides coincide for n = −1 and n = 0. Therefore the equalities hold by the uniqueness theorem for solutions. Consequently, taking into account (15) for the matrix J (1) instead of J and using (19), we find (1) (1)

(1)

(1)

det(J (1) − λ I) = (−1)N−1 a0 a1 · · · aN−3 PN−1 (λ ) = (−1)N−1 a1 · · · aN−2 a0 QN (λ ). Comparing this with (15), we get QN (λ ) det(J (1) − λ I) =− PN (λ ) det(J − λ I) so that formula (18) follows by (17).

328

A. Huseynov and G.Sh. Guseinov

3 Time Evolution of the Spectral Data Theorem 1. Let {an (t), bn (t)} be a solution of (1), (2) and J = J(t) be the Jacobi matrix defined by this solution according to (4). Then there exists an invertible N ×N matrix-function X(t) such that X −1 (t)J(t)X(t) = J(0)

f or

all

t.

(20)

Proof. Let A(t) be defined according to (5). Then the Lax equation (6) holds. Denote by X = X(t) the matrix solution of the initial value problem ˙ = −A(t)X(t), X(t)

(21)

X(0) = I.

(22)

Such solution X(t) exists and is unique. From the Liouville formula

 t det X(t) = exp − trA(τ )d τ 0

it follows that detX(t) = 0 so that the matrix X(t) is invertible. Using the formula dX −1 dX −1 = −X −1 X , dt dt equation (21), and the Lax equation (6), we have dJ d  −1 dX −1 dX X JX = JX + X −1 X + X −1 J dt dt dt dt dJ = X −1 AXX −1 JX + X −1 X − X −1JAX dt 

dJ = X −1 − [JA − AJ] X = 0. dt Therefore X −1 (t)J(t)X(t) does not depend on t and by initial condition (22) we get (20). The proof is complete. It follows from (20) that for any t the characteristic polynomials of the matrices J(t) and J(0) coincide. Therefore we arrive at the following statement. Corollary 1. The eigenvalues of the matrix J(t), as well as their multiplicities, do not depend on t. Remark 1. From the skew-symmetry AT = −A of the matrix A(t) defined by (5) it follows that for the solution X(t) of (21), (22) we have X −1 (t) = X T (t), where T stands for the matrix transpose. Indeed, by (21),

Finite Complex Toda Lattice

329

d  T dX T dX X X = X + XT = −X T AT X − X T AX dt dt dt = X T AX − X T AX = 0. Therefore X T (t)X(t) does not depend on t and by initial condition (22) we get X T (t)X(t) = X T (0)X(0) = I. Lemma 1. The dynamics of the solution of the Toda equations (1) and (2) with the arbitrary complex initial data (3) corresponds to the following evolution of the resolvent function w(λ ;t) of the matrix J(t) : w(λ ;t) =

e2 λ t 2 w(λ ; 0) − S(t) S(t)

where S(t) =

1 2π i

t

S(u)e2λ (t−u)du,

(23)

0

Γ

w(z; 0)e2zt dz

(24)

in which Γ is any closed contour that encloses all the eigenvalues of the matrix J(0). Proof. Let an = an (t) (0 ≤ n ≤ N − 2), bn = bn (t) (0 ≤ n ≤ N − 1) form a solution of problem (1), (2). We construct from this solution of (1), (2) the Jacobi matrix J = J(t) by (7). Next, take these an and bn together with a−1 = aN−1 = 1 as coefficients in (12) and denote by {Pn (λ ;t)}Nn=−1 and {Qn (λ ;t)}Nn=−1 its solutions satisfying the initial conditions (13) and (14), respectively. Let us introduce the N-dimensional vectors (columns) ⎡

P0 (λ ;t) P1 (λ ;t) .. .

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ P=⎢ ⎥, ⎢ ⎥ ⎣ PN−2 (λ ;t) ⎦

⎡ ⎢ ⎢ ⎢ F =⎢ ⎢ ⎣

PN−1 (λ ;t)

⎡

Q0 (λ ;t) Q1 (λ ;t) .. .

⎢ ⎢ ⎢ Q=⎢ ⎢ ⎣Q

⎤

⎥ ⎥ ⎥ ⎥, ⎥ ⎦ N−2 (λ ;t)

⎡ ⎢ ⎢ ⎢ G=⎢ ⎢ ⎣

QN−1 (λ ;t)

0 0 .. . 0 −PN (λ ;t)

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦

1 0 .. . 0 −QN (λ ;t)

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

Then we have equations JP = λ P + F,

(25)

JQ = λ Q + G.

(26)

330

A. Huseynov and G.Sh. Guseinov

Differentiating (25) with respect to t, we get ˙ ˙ + J P˙ = λ P˙ + F. JP Using (6) in the left-hand side, we obtain ˙ JAP − AJP + J P˙ = λ P˙ + F. Hence, replacing JP by (25), J(P˙ + AP) = λ (P˙ + AP) + F˙ + AF. Therefore,

P˙ + AP = (J − λ I)−1(F˙ + AF).

(27)

Similarly, we get from (26), Q˙ + AQ = (J − λ I)−1 (G˙ + AG). Besides,

⎡

0 0 0 .. .

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥, F˙ + AF = ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎣ aN−2 PN ⎦ −P˙N

⎡

(28)

0 a0 0 .. .

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥. ⎢ ˙ G + AG = ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎣ aN−2 QN ⎦ −Q˙ N

Equating in (27) the first coordinates of the left and right sides and taking into account the matrix elements (16) of (J − λ I)−1, we get P˙0 − a0 P1 = R0,N−2 aN−2 PN − R0,N−1 P˙N or P˙0 − a0 P1 = aN−2 P0 (QN−2 − wPN−2 )PN − P0 (QN−1 − wPN−1 )P˙N . Hence, taking into account P0 = 1,

P1 =

λ − b0 , a0

(29)

we find that P˙N =

λ − b0 QN−2 − wPN−2 + aN−2 PN . QN−1 − wPN−1 QN−1 − wPN−1

(30)

Finite Complex Toda Lattice

331

Similarly, equating in (28) the first coordinates of the left and right sides, we get Q˙ 0 − a0 Q1 = R0,1 a0 + R0,N−2 aN−2 QN − R0,N−1 Q˙ N or Q˙ 0 − a0 Q1 = a0 P0 (Q1 − wP1 ) + aN−2P0 (QN−2 − wPN−2 )QN −P0 (QN−1 − wPN−1 )Q˙ N . Hence, taking into account (29) and Q0 = 0, we find that

Q1 =

1 , a0

QN−2 − wPN−2 2 − (λ − b0 )w Q˙ N = + aN−2 QN . QN−1 − wPN−1 QN−1 − wPN−1

(31)

Now multiply (30) by QN and (31) by PN and subtract then side by side to get

λ − b0 2 − (λ − b0 )w Q˙ N PN − P˙N QN = PN − QN . QN−1 − wPN−1 QN−1 − wPN−1 Dividing by PN2 gives Q˙ N PN − P˙N QN 2 − (λ − b0 )w (λ − b0 )w = − . (QN−1 − wPN−1 )PN (QN−1 − wPN−1 )PN PN2 Hence

w˙ = 2(λ − b0)w − 2

(32)

because (QN−1 − wPN−1 )PN = QN−1 PN − QN PN−1 = −1 (see [10, Lemma 4]). Solving the first-order linear nonhomogeneous differential equation (32), we obtain   t   −2 0t [b0 (u)−λ ]du 2 0u [b0 (τ )−λ ]d τ du + w(λ ; 0) . w(λ ;t) = e −2 e 0

This implies (23) with S(t) = e2

t

0 b0 (u)du

.

To derive formula (24) for S(t), we integrate (23) with respect to λ to get 1 2π i



1 w(λ ;t)d λ = S(t)2π i Γ −



1 S(t)π i

Γ

e2λ t w(λ ; 0)d λ

 t Γ

0

 S(u)e2λ (t−u)du d λ ,

(33)

332

A. Huseynov and G.Sh. Guseinov

where Γ is any closed contour that encloses all the eigenvalues of the matrix J(0). Next, the left-hand side of (33) is equal to p

p

k=1

k=1

∑ resλ =λk w(λ ;t) = ∑ βk1 (t)

by the residue theorem and equals in turn 1 by (18) in virtue of which resλ =∞ w(λ ;t) = −1. The second integral in the right-hand side of (33) is equal to 0 because the integrand is an analytic function of λ . Therefore we get (24) and the lemma is proved. By Corollary 1 the eigenvalues of the matrix J(t) and their multiplicities do not depend on t. However, the normalizing numbers of the matrix J(t) will depend on t. The following theorem describes this dependence. Theorem 2. For the normalizing numbers

βk j (t) ( j = 1, . . . , mk ;

k = 1, . . . , p)

of the matrix J(t) the following evolution holds:

βk j (t) = where S(t) =

e2λkt S(t)

mk

(2t)s− j

∑ βks (0) (s − j)! ,

(34)

s= j

p

mk

k=1

j=1

(2t) j−1

∑ e2λkt ∑ βk j (0) ( j − 1)! .

(35)

Proof. According to (10), we have w(λ ;t) =

mk

p

βk j (t)

∑ ∑ (λ − λk ) j .

(36)

k=1 j=1

We will deduce (34), (35) from (23), (24) of Lemma 1. First we compute the function S(t) defined by (24). We use the known formula f (n) (z) 1 = n! 2π i

γ

f (ζ ) dζ , (ζ − z)n+1

where γ is the boundary of the disk D = {ζ ∈ C : |ζ − z| ≤ δ } with δ > 0 and f (z) is analytic in a domain of C containing D.

Finite Complex Toda Lattice

333

For any k ∈ {1, . . . , p} we set Γk = {z ∈ C : |z − λk | = δ } and choose δ > 0 so small that Γ1 , . . . , Γp are disjoint and all inside Γ . Then we have, by (24) and (36), S(t) =

1 p mk ∑ ∑ βk j (0) 2π i k=1 j=1

Γ

e2zt dz. (z − λk ) j

Further, 1 2π i

Γ

1 e2zt dz = (z − λk ) j 2π i

Γk

(2t) j−1 2λk t e2zt e . dz = j (z − λk ) ( j − 1)!

Therefore for S(t) we have the formula (35). Now we rewrite (23) in the form, using (36),

βk j (t) e2λ t p mk βk j (0) 2 ∑ ∑ (λ − λk ) j = S(t) ∑ ∑ (λ − λk) j − S(t) k=1 j=1 k=1 j=1 p

mk

t

S(u)e2λ (t−u) du.

(37)

0

Take any l ∈ {1, . . ., p} and s ∈ {1, . . . , ml }, multiply (37) by (2π i)−1 (λ − λl )s−1 and integrate both sides over Γl to get 1 2π i

ml

∑ βl j (t)

j=1

Γl

1 1 dλ = j−s+1 (λ − λl ) 2π i S(t)

ml

∑ βl j (0)

j=1

Γl

e2λ t d λ . (λ − λl ) j−s+1

We have used the fact that the integral of the second term of the right side of (37) gave 0 because of the analyticity of that term with respect to λ . Hence

βls (t) =

e2λl t S(t)

ml

(2t) j−s

∑ βl j (0) ( j − s)!

(38)

j=s

because 1 2π i 1 2π i

Γl

Γl

dλ = (λ − λl ) j−s+1 e2λ t d λ = (λ − λl ) j−s+1



1 0

if if 0

j = s, j = s, if

(2t) j−s 2λl t ( j−s)! e

j < s, if

j ≥ s.

Now, (38) means that (34) holds. Note that finding the time evolution of the spectral data forms a crucial step in the solution procedure of nonlinear evolution equations by means of inverse spectral problems. A complete solution of problem (1)–(3) will be presented by the authors elsewhere.

334

A. Huseynov and G.Sh. Guseinov

References 1. Kac M, van Moerbeke P (1975) On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices. Adv Math 16:160–169 2. Moser J (1975) Three integrable Hamiltonian systems connected with isospectral deformations. Adv Math 16:197–220 3. Toda M (1981) Theory of nonlinear lattices. Springer, New York 4. Teschl G (2000) Jacobi operators and completely integrable nonlinear lattices. In: Math. surveys monographs, vol 72. American Mathematical Society, Providence 5. Aptekarev AI, Branquinbo A (2003) Pad´e approximants and complex high order Toda lattices. J Comput Appl Math 155:231–237 6. Berezanskii Yu M, Mokhonko AA (2008) Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices. Funct Anal Appl 42:1–18 7. Huseynov A (2008) On solutions of a nonlinear boundary value problem for difference equations. Trans Natl Acad Sci Azerb Ser Phys-Tech Math Sci 28(4):43–50 8. Bereketoglu H, Huseynov A (2009) Boundary value problems for nonlinear second-order difference equations with impulse. Appl Math Mech (English Ed.) 30:1045–1054 9. Huseynov A (2010) Positive solutions for nonlinear second-order differernce equations with impulse. Dyn Contin Discret Impuls Syst Ser A: Math Anal 17:115–132 10. Guseinov GSh (2009) Inverse spectral problems for tridiagonal N by N complex Hamiltonians. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 5:1–28, Paper 018

Efficient Dynamic Modeling of a Hexa-Type Parallel Manipulator Ant´onio M. Lopes

Abstract Dynamic modeling of parallel manipulators is difficult due to the existing multiple closed-loops and kinematic constraints. Generally speaking, all dynamic modeling approaches use classical mechanics principles, leading to equivalent dynamic equations. Nevertheless, these equations may present different levels of complexity and computational loads. This chapter presents the generalized momentum approach to derive the inverse dynamic model of a Hexa-type fully-parallel manipulator. This approach showed high computational efficiency, measured by the number of arithmetic operations involved in the computation of the manipulator’s inertia and Coriolis and centripetal terms matrices.

1 Introduction A parallel manipulator is a complex multi-body dynamic system comprising a (usually) fixed platform (the base) and a moving platform, linked together by two or more independent, open kinematic chains. Dynamic modeling of parallel manipulators presents an inherent complexity, mainly due to the existing multiple closed-loops and kinematic constraints. Despite intensive study carried out in this specific topic of robotics, mostly conducted in the last two decades, additional research still has to be done in this area. Dynamic models play an important role in parallel manipulators simulation and control. Mainly in the later case, the efficiency of the involved computations is of paramount importance, because manipulator real-time control is usually necessary [1]. The dynamic model is usually derived in Cartesian space coordinates

A.M. Lopes () Unidade de Integrac¸a˜ o de Sistemas e Processos Automatizados, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465, Porto, Portugal e-mail: [email protected] A.C.J. Luo et al. (eds.), Dynamical Systems and Methods, DOI 10.1007/978-1-4614-0454-5 20, © Springer Science+Business Media, LLC 2012

335

336 Fig. 1 Manipulator structure

A.M. Lopes

Arm θ 4

θ5 B5 B y B4 B

B3

Base

θ6

B6

xB

θ1

zB B 1

B2

θ3

θ2

Rod P6

P5 P

P4

yP zP P3

P2

xP

P1 Moving platform

and, mathematically, it can be represented by a set of nonlinear differential equations that may be written in matrix form as I (x) · x¨ + V (x, x˙ ) · x˙ + G (x) = f

(1)

I(x) being the inertia matrix, V(x, x˙ ) the Coriolis and centripetal terms matrix, G(x) a vector of gravitational generalized forces, x the generalized position of the moving platform (or end-effector) and f the controlled generalized force applied on the endeffector. Thus, the statics equation is f = JT (x) · τ

(2)

where τ is the generalized force or torque developed by the actuators and J(x) is the inverse jacobian matrix [2]. Several approaches have been applied to the dynamic analysis of parallel manipulators, the Newton-Euler and the Lagrange methods being the most popular ones [3–5]. These methods use the classical mechanics principles, as is the case for all the approaches found in the literature, namely, the ones based on the principle of virtual work, screw theory, recursive matrix method, and Hamilton’s principle [6–10]. Thus, all approaches are equivalent, leading to equivalent dynamic equations. Nevertheless, these equations can present different levels of complexity and associated computational loads. Minimizing the number of operations involved in the computation of the manipulator dynamic model has been the main goal of recent works [11, 12]. This chapter presents the generalized momentum approach to derive the inverse dynamic model of a Hexa-type parallel kinematic structure [13]. In brief, a Hexa parallel manipulator comprises a (usually) fixed platform (the base) and a moving platform, linked together by six independent, identical, open kinematic chains (Fig. 1). Each chain (leg) comprises an arm and a rod that are connected together

Efficient Dynamic Modeling of a Hexa-Type Parallel Manipulator

337

by a passive universal joint. The upper end of each leg is connected to the base by an active revolute joint, θi , whereas the lower end is connected to the moving platform by a passive spherical joint. Points Bi and Pi (i = 1, . . . , 6) (Fig. 1) are the connecting points to the base and moving platforms, respectively. Compact analytical expressions corresponding to the contributions of each rigid body to the dynamic model are presented in this chapter and it is shown that the used approach results in a computational efficient model as measured by the (small) number of arithmetic operations involved in the computation of both the inertia and the Coriolis and centripetal terms matrices.

2 Hexa Kinematic Structure The Hexa is a six degrees-of-freedom (DOF) fully parallel manipulator. The configuration that is considered in this chapter has its six legs connected to the base and moving platforms at points Bi and Pi , respectively. Figure 2 shows the base platform and points Bi . These points are located at the vertices of a semi-regular hexagon inscribed in a circumference of radius rB . The separation angles between points B1 and B6 , B2 and B3 , and B4 and B5 are denoted by 2φB . Points Pi are located on the moving platform in a similar way. The separation angles between points P1 and P6 , P2 and P3 , and P4 and P5 are denoted by 2φP and they are located on a circumference of radius rP . The coordinates of points Bi and Pi with reference to the base and moving platform coordinate systems, respectively, are T  bi = rB · cos λi rB · senλi 0 (3)   T P pi = rP · cos γi rP · senγi 0 (4) B5 B4

φB

φB B6

120 rB

φB B

xB

φB B1

B3 Fig. 2 Position of the legs to base platform connecting points

φB

φB B2

yB

338

A.M. Lopes

where T  λ = φB 120◦ − φB 120◦ + φB 240◦ − φB 240◦ + φB 360◦ − φB  T γ = φP 120◦ − φP 120◦ + φP 240◦ − φP 240◦ + φP 360◦ − φP

(5) (6)

For kinematic modeling purposes, two frames, {P} and {B}, are attached to the moving and base platforms, respectively. Its origins are the platforms’ centers of mass. The generalized position of frame {P} relative to frame {B} may be represented by the vector: B

T  xP |B|E = xP yP zP ψP ξP ϕP  T = B xTP(pos) | B xTP(o) | B

(7)

E

where B xP(pos)|B = [xP yP zP ]T is the position of the origin of frame {P} relative to frame {B}, and B xP(o)|E = [ψP ξP ϕP ]T defines an Euler angles system representing orientation of frame {P} relative to {B}. The used Euler angles system corresponds to the basic rotations: ψP about zP ; ξP about the rotated axis yP ; and ϕP about the rotated axis xP . The rotation matrix is given by: ⎡ CψPCξP ⎢ B RP = ⎣ SψPCξP −SξP

CψP SξP SϕP − SψPCϕP SψP SξP SϕP + CψPCϕP

CψP SξPCϕP + SψPSϕP

⎤

⎥ SψP SξPCϕP − CψPSϕP ⎦

CξP SϕP

(8)

CξPCϕP

S(·) and C(·) correspond to the sine and cosine functions, respectively. The manipulator inverse jacobian matrix is given by (9) which can be computed using vector algebra [14]. ⎡ ⎢ ⎢ JC = ⎢ ⎣

Pp hT1 1 |B ×h1 k1 ·(l1 ×h1 ) k1 ·(l1 ×h1 )

.. .

.. .

hT6 k6 ·(l6 ×h6 )

6 |B ×h6 k6 ·(l6 ×h6 ) Pp

⎤ ⎥ ⎥ ⎥ ⎦

(9)

k 1 = k2 =

b1 − b6 b1 − b6 

(10)

k 3 = k4 =

b3 − b2 b3 − b2 

(11)

k 5 = k6 =

b5 − b4 b5 − b4 

(12)

Efficient Dynamic Modeling of a Hexa-Type Parallel Manipulator Fig. 3 Schematic representation of a kinematic chain

Base

339

B

ei

x P ( pos ) B

Moving platform

Pi P zP

Bi

P

yP

bl li

yB

zB

B

bi

xB

hi bh

pi

xP

All vectors are obtained analyzing each kinematic chain of the parallel manipulator (Fig. 3). hi = ei − li

(13)

ei = B xP(pos)|B − bi + P pi |B ⎤ ⎡ l cos θi cos αi li = ⎣ l cos θi sin αi ⎦ l sin θi T  α = 0 120◦ 120◦ 240◦ 240◦ 0

(14) (15)

(16)

pi|B = B RP ·P pi are the positions of points Pi , expressed in the base frame, and l and h represent arms and rods lengths, respectively. Using (13), the actuators’ displacements, θi , are obtained solving (17)

P

ai = bi cos θi + ci sin θi ai = h

2

− e2ix − e2iy − e2iz − l 2

(17) (18)

bi = −2eix l cos αi − 2eiy l sin αi

(19)

ci = −2eiz l

(20)

The velocity kinematics is represented by the Euler angles jacobian matrix, JE , or the kinematic jacobian, JC . These jacobians relate the velocities of the active revolute joints to the generalized velocity (linear and angular) of the moving platform:

B ˙θ = JE · B x˙ P | = JE · x˙ P(pos) |B (21) Bx B|E ˙ P(o) |E

340

A.M. Lopes

θ˙ = JC · x˙ P |B = JC · B

Bx ˙ P(pos) |B Bω P |B

(22)

with T  θ˙ = θ˙1 θ˙2 · · · θ˙6

(23)

ωP |B = JA · B x˙ P(o) |E

(24)

B

and

⎡

0 −SψP CξPCψP

⎤

⎢ ⎥ JA = ⎣ 0 CψP CξP SψP ⎦ 1 0 −SξP

(25)

Vectors B x˙ P(pos)|B ≡ B vP|B and B ωP|B represent the linear and angular velocity of the moving platform relative to {B}, and B x˙ P(o)|E represents the Euler angles time derivative.

3 Dynamic Modeling The generalized momentum of a rigid body, qc , may be obtained using the following general expression: qc = Ic · uc

(26)

Vector uc represents the generalized velocity of the body and Ic is its inertia matrix. Vectors qc and uc , and inertia matrix Ic must be expressed in the same referential. Equation (26) may also be written as:

   Ic(tra) 0 v Qc qc = · c = Hc ωc 0 Ic(rot) 

(27)

where Qc is the linear momentum vector due to rigid body translation, and Hc is the angular momentum vector due to body rotation. Ic(tra) is the translational inertia matrix, and Ic(rot) the rotational inertia matrix. vc and ωc are the body linear and angular velocities. The inertial component of the generalized force acting on the body can be computed from the time derivative of (26): fc(ine) = q˙ c = I˙c · uc + Ic · u˙ c with force and momentum expressed in the same frame.

(28)

Efficient Dynamic Modeling of a Hexa-Type Parallel Manipulator

341

3.1 Moving Platform Modeling The linear momentum of the moving platform, written in frame {B}, may be obtained from the following expression: QP |B = mP · B vP |B = IP(tra) · B vP |B

(29)

IP(tra) is the translational inertia matrix of the moving platform, mP being its mass. IP(tra) = diag([ mP mP mP ])

(30)

The angular momentum, also written in frame {B}, is: HP |B = IP(rot) |B · B ωP |B

(31)

IP(rot)|B represents the rotational inertia matrix of the moving platform, expressed in the base frame {B}. The inertia matrix of a rigid body is constant when expressed in a frame that is fixed with relation to that body. Furthermore if the frame axes coincide with the principal directions of inertia of the body, then all inertia products are zero and the inertia matrix is diagonal. Therefore, the rotational inertia matrix of the moving platform, when expressed in frame {P}, may be written as: IP(rot) |P = diag([ IPxx IPyy IPzz ])

(32)

This inertia matrix can be written in frame {B} using the following transformation: IP(rot) |B = B RP · IP(rot) |P · B RTP

(33)

The generalized momentum of the moving platform, expressed in frame {B}, can be obtained from (29) and (31): qP |B =



IP(tra)

0

0

IP(rot) |B

·

Bv

P |B

Bω P |B

(34)

where the moving platform inertia matrix written in the base frame {B} is IP |B =

IP(tra)

0

0

IP(rot) |B

(35)

The combination of (24) and (31) results in: HP |B = IP(rot) |B · JA · B x˙ P(o) |E

(36)

342

A.M. Lopes

Accordingly, (34) may be rewritten as:

qP |B



 B IP(tra) 0 vP |B ℑ 0 = · · B 0 JA x˙ P(o) |E 0 IP(rot) |B

qP |B = IP |B · T · B x˙ P |B|E

(37) (38)

T being a matrix transformation defined by:  T=

ℑ 0 0 JA

 (39)

The time derivative of (38) results in: P

fP(ine) |B = q˙ P |B =

 d  IP |B · T · B x˙ P |B|E + IP |B · T · B x¨ P |B|E dt

(40)

fP(ine)|B is the inertial component of the generalized force acting on {P} due to the moving platform motion, expressed in frame {B}. The corresponding actuators forces, τP(ine) , may be computed from the statics equation: P

τP(ine) = JC−T · P fP(ine) |B where P

fP(ine) |B =



P FT P T P(ine) |B MP(ine) |B

(41) T (42)

Vector P FP(ine)|B represents the force vector acting on the center of mass of the moving platform, and P MP(ine)|B represents the moment vector acting on the moving platform, about its center of mass, expressed in the base frame, {B}. From (40) it can be concluded that two matrices playing the roles of the inertia matrix and the Coriolis and centripetal terms matrix are: IP |B · T  d  IP |B · T dt

(43) (44)

It must be emphasized that these matrices do not have the properties of inertia or Coriolis and centripetal terms matrices and therefore should not, strictly, be named as such. Nevertheless, throughout the chapter the names “inertia matrix” and “Coriolis and centripetal terms matrix” may be used if there is no risk of misunderstanding.

Efficient Dynamic Modeling of a Hexa-Type Parallel Manipulator

343

3.2 Rods Modeling Considering that the center of mass of each rod is located at distance bh from the moving platform to rod connecting point (Fig. 3), then its position relative to frame {B} is: bh B phi |B = B xP(pos) |B + P pi |B − · hi (45) h The linear velocity of the rod center of mass, B p˙ hi |B , relative to {B} and expressed in the same frame, may be computed as: B

   b bh B h x˙ P(pos) |B + B ωP |B × B pi |B + θ˙i ki × li p˙ hi |B = 1 − h h

(46)

which can be rewritten as B

p˙ hi |B = JBi ·

Bv P |B B ω P |B

(47)

The linear momentum of each rod can be represented in frame {B} by (48), where mh is the rod mass. Qhi |B = mh · B p˙ hi |B (48) Introducing jacobian JBi in the previous equation, results Qhi |B = mh · JBi · B x˙ P |B

(49)

The inertial component of the force applied to the rod center of mass due to its translation and expressed in {B} can be obtained from the time derivative of (49): hi

˙ h | = mh · J˙ B · B x˙ P | + mh · JB · B x¨ P | Fhi (ine)(tra) |B = Q i i i B B B

(50)

When (50) is multiplied by JTBi , the inertial component of the generalized force (force and momentum) applied to {P}, due to each rod translation, is obtained in frame {B}: P

fhi (ine)(tra) |B = mh · JTBi · J˙ Bi · B x˙ P |B + mh · JTBi · JBi · B x¨ P |B

(51)

The corresponding actuators forces, τhi (ine)(tra) , may be computed from the following relation: τhi (ine)(tra) = JC−T · P fhi (ine)(tra) |B

(52)

Now, regarding rod rotation, the angular velocity of each rod, B ωhi |B , can be obtained from the linear velocities of two points belonging to it. If these two points are taken

344

A.M. Lopes

as the arm to rod and rod to moving platform connecting points, the following expression results B

ωhi |B × hi = B vP |B + BωP |B × P pi |B − θ˙i ki × li

(53)

As the rod cannot rotate about its own axis, the angular velocity about hˆ i is always zero and vectors hi and B ωhi |B are always perpendicular. This property enables (53) to be rewritten as B

ωhi |B =

   1 B B P ˙ · h × v + ω × p − θ k × l i i i i P |B P |B i |B hTi · hi

and,

B

ω hi | B

(54)

B

v = JDi · B P |B ωP |B

(55)

Thus, the angular momentum of each rod can be represented in frame {B} by: Hhi |B = Ihi (rot) |B · B ωhi |B

(56)

It is convenient to express the inertia matrix of the rotating rod in a frame fixed to the rod itself, {Hi } ≡ {xhi , yhi , zhi }. So, Ihi (rot) |B = B Rhi · Ihi (rot) |H · B RThi

(57)

Ihi (rot) |H = diag([ Ihxx Ihyy Ihzz ])

(58)

i

and i

Ihxx , Ihyy , and Ihzz are the rod moments of inertia, expressed in its own frame, and B R is the orientation matrix of each rod frame, {H }, relative to the base frame, i hi {B}. This matrix can be written as B

Rhi = B ROi · Oi Rhi

(59)

where the fist matrix of the product is a constant for each leg, representing the rotation matrix of intermediate referential {Oi } relative to {B} (Fig. 4): ⎤ ⎡ cos αi 0 sin αi B (60) ROi = ⎣ sin αi 0 − cos αi ⎦ 0 1 0 Matrix Oi Rhi can be computed using the Denavit-Hartenberg (D-H) algorithm for each leg and (13). Introducing jacobian JDi and matrix transformation T in (56) results in: Hhi |B = Ihi (rot) |B · JDi · T · B x˙ P |B|E

(61)

Efficient Dynamic Modeling of a Hexa-Type Parallel Manipulator Fig. 4 Frames {O1 }, {O3 } and {O5 } used as the ground frames for D-H algorithm (similar frames for legs 6, 2 and 4, respectively; frames origins are coincident with points Bi )

345

xO5 B5

zO5

B4 B6 B z O3

B1

xB xO1

B3 xO3

z O1 B2

yB

The inertial component of the momentum applied to the rod, due to its rotation and expressed in {B} can be obtained from the time derivative of (61): hi

˙h | Mhi (kin)(rot) |B = H i B =

 d  Ihi (rot) |B · JDi · T · B x˙ P |B|E + Ihi (rot) |B · JDi · T · B x¨ P |B|E dt

(62)

When (62) is pre-multiplied by JTDi the inertial component of the generalized force applied to {P} due to each rod rotation is obtained in frame {B}: P

fhi (kin)(rot) |B = JTDi · hi Mhi (kin)(rot) |B  d  Ihi (rot) |B · JDi · T B x˙ P |B|E = JTDi · dt +JTDi · Ihi (rot) |B · JDi · T · B x¨ P |B|E

(63)

The corresponding actuators forces, τhi (ine)(rot) , may be computed from the following relation: τhi (ine)(rot) = JC−T · P fhi (ine)(rot) |B

(64)

The inertia matrices and the Coriolis and centripetal terms matrices of the translating and rotating rod may be written as, respectively: mh · JTBi · JBi

(65)

JTDi · Ihi (rot) |B · JDi · T

(66)

mh · JTBi · J˙ Bi

(67)

346

A.M. Lopes

JTDi ·

 d  Ihi (rot) |B · JDi · T dt

(68)

These matrices represent the inertia matrices and the Coriolis and centripetal terms matrices of a virtual moving platform that is equivalent to each translating and rotating rod.

3.3 Arms Modeling Each arm rotates at the corresponding joint axis. Considering the center of mass of each arm is located at a constant distance, bl , from the base platform to arm connecting point (Fig. 3), then the corresponding inertial actuators forces, τli (ine) , may be computed through the following equation:  τli (ine) = Izz θ¨i + ml

bl l

2

θ¨i

(69)

where, Izz is the arm inertia along an axis parallel to the joint axis and about the arm’s center of mass, and ml is the arm mass.

3.4 Gravitational Components Given a general frame {x, y, z}, with z ≡ gˆ , the potential energy of a rigid body is given by: Pc = mc · g · zc

(70)

where mc is the body mass, g is the modulus of the gravitational acceleration and zc the distance, along z, from the frame origin to the body center of mass. The gravitational components of the actuators forces can be easily obtained from the potential energy of the different rigid bodies that compose the system: τP(gra) = J−T E ·

τli (gra) = J−T E ·

τhi (gra) = J−T E ·

∂ PP



Bx P |B|E



∂ B xP |B|E   ∂ Pli B xP |B|E ∂ B xP |B|E   ∂ Phi B xP |B|E ∂ B xP |B|E

(71)

(72)

(73)

Efficient Dynamic Modeling of a Hexa-Type Parallel Manipulator

347

Table 1 Operations involved in the computation of the Coriolis and centripetal terms matrices Lagrange method Generalized momentum Mobile platform Translating rod Rotating rod

Sum. 94 972 2103

Mult. 226 2160 3986

Div. 0 7 12

Sum. 310 197 395

Mult. 590 447 920

Div. 0 14 16

Total operations

18544

37102

19

3862

8792

30

4 Computational Load The computational efficiency of the proposed dynamic model was evaluated. The results were obtained using Maple symbolic computational software package, and are presented in Table 1. The dynamic model obtained using the generalized momentum approach is computationally much more efficient than the one obtained using the Lagrange method. The main advantage is in the computation of the matrices that require the largest relative computational effort: Coriolis and centripetal terms matrices of the moving platform and the six legs’ rods. Regarding the total number of sums, multiplications, and divisions involved with the two models, the ratio is almost oneto-five. On the other hand, comparing the results from both methods, they are nearly identical up to eight significant digits.

5 Conclusions In this chapter an approach based on the manipulator generalized momentum was applied to the dynamic modeling of a Hexa-type parallel manipulator. Analytic expressions for the rigid bodies’ inertia and Coriolis and centripetal terms matrices were obtained, which can be added, as they are expressed in the same frame. Computational efficiency of the proposed model was evaluated by counting the number of scalar operations needed in the calculations (sums, multiplications, and divisions). The results were then compared with the ones obtained by using the Lagrange formulation. The generalized momentum approach resulted in a much more efficient dynamic model. Regarding the total number of sums and multiplications involved in the two models, the ratio is almost five.

References 1. Zhao Y, Gao F (2009) Inverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual work. Robotica 27:259–268 2. Merlet J-P (2006) Parallel robots. Springer, Dordrecht

348

A.M. Lopes

3. Do W, Yang D (1988) Inverse dynamic analysis and simulation of a platform type of robot. J Robotic Syst 5:209–227 4. Ji Z (1994) Dynamics decomposition for Stewart platforms. ASME J Mech Design 116:67–69 5. Khalil W, Ibrahim O (2007) General solution for the dynamic modelling of parallel robots. J Intell Robot Syst 49:19–37 6. Staicu S, Liu X-J, Wang J (2007) Inverse dynamics of the HALF parallel manipulator with revolute actuators. Nonlinear Dynam 50:1–12 7. Tsai L-W (2000) Solving the inverse dynamics of Stewart-Gough manipulator by the principle of virtual work. J Mech Design 122:3–9 8. Gallardo J, Rico J, Frisoli A, Checcacci D, Bergamasco M (2003) Dynamics of parallel manipulators by means of screw theory. Mech Mach Theory 38:1113–1131 9. Staicu S, Zhang D (2008) A novel dynamic modelling approach for parallel mechanisms analysis. Robot Cim-Int Manuf 24:167–172 10. Liu M-J, Li C-X, Li C-N (2000) Dynamics analysis of the Gough-Stewart platform manipulator. IEEE T Robot Autom 16:94–98 11. Abdellatif H, Heimann B (2009) Computational efficient inverse dynamics of 6-DOF fully parallel manipulators by using the Lagrangian formalism. Mech Mach Theory 44:192–207 12. Lopes AM (2009) Dynamic modeling of a Stewart platform using the generalized momentum approach. Commun Nonlinear Sci Numer Simul 14:3389–3401 13. Pierrot F, Uchiyama M, Dauchez P, Fournier A (1992) A new design of a 6-DOF parallel robot. In: Proceedings of the 23rd international symposium on industrial robots, Barcelona, pp 771–776 14. Tartari Filho S, Cabral E (2006) Kinematics and workspace analysis of a parallel architecture robot: the Hexa. In: ABCM symposium series in mechatronics, pp 158–165