Proceedings of the 5th Experimental Chaos Conference: Orlando, Florida June 28-July 1, 1999

210 2. Noisy Synchronization In this paper, we are concerned with noisy synchronization. If the oscillators are weakly enough coupled and if they are also subject to random disturbances, or "noise", then the state of synchronization will not be constant or fixed. Instead, the oscillators may for a time become synchronized, but, due to the disturbances, may drop out of synchrony, then regain it, and so on. The lengths of the time segments, whether in or out of synchrony, are also random. Though originally studied quite early [Stratonovich, 1967], the theory of stochastic synchronization has only recently been applied to biological or medical systems. Examples include studies of the noisy synchrony of the human heartrespiratory system [Schafer, et al, 1998] and in magnetoencephalograms (MEGs) of Parkinsonian patients [Tass, et al, 1998]. The modern theory of stochastic synchronization, especially as applied to two-dimensional systems such as the networks of coupled glial cells studied here, has been recently developed [Neiman, 1994; Shulgin, Neiman and Anishchenko, 1995; Neiman et al., 1998]. Since the synchronization process is itself random, suitable measures, for example a probability density, must be applied. We are interested in quantifying the partly random behavior of A
211 3. T i e Electroreceptor Cells. Oscillating cells, or pacemakers, abound in biology, two examples of which are the caudal photoreceptor cell of the crayfish [Wilkens, 1988; Pei, Wilkens and Moss, 1996] and electroreceptor cells, especially of the paddlefish [Wilkens, et al. 1997]. Figure 1 shows the fish together with a blow-up of a portion of the rostrum showing the pores of its electroreceptor cells.

I I I ! I

Fig. 1. The paddlefish, Polydon Spathula (upper panel). Part of the rostrum showing gel-filled pores (dark spots in groups of 5 to 6) which electrically connect the electroreceptor cells, or Ampullae of Lorenzini, to the external environment (lower panel).

Each electroreceptor cell in the paddlefish projects a long axon to the animal's brain. Each cell is driven by a noisy oscillator [Pei, preprint]. The natural frequencies of the oscillators range from about 20 to 85 Hz for various cells. The oscillatory behavior of an individual cell can be revealed by electrophysiological recordings made by attaching an extracellular electrode to one of the axons. The spontaneous neural firing is quite regular near the natural frequency of the oscillator. Thus the recording from the axon is a sequence of neural spikes with only a relatively small scatter in the interspike time intervals determined by the noise inherent in the oscillator. The imposition of an external electric or magnetic field on the pores shown in Fig. 1 can effectively change the mean spiking frequency or cause the spike train to become synchronized with the field [Neiman, et al, 1999]. We have observed that the spike frequency of these cells is relatively "rigid", that is it takes a relatively strong external electric field ( » 10 jiV/cm) to noticeably change the frequency of the oscillator. However, a relatively we-ak field («1 - 10 jiV/cm) from a periodic source can effectively synchronize the neural spike train, so that m spikes occur at nearly the same phases, cp2i for every cycle of the stimulus field. The baseline sensitivity of the electroreceptor cells, measured in behavioral experiments, is » 0.5 jiV/cm. Thus we are led by these observations to suspect that the animal uses a synchronization code to locate and track prey rather than the rate code more often assumed by sensory biologists. Figure 2 shows a set of synchrograms for three stimulus frequencies: low intermediate and high. The neural spike trains are synchronized to some degree with the stimulus for both the low and intermediate frequencies as indicated by the significant lengths of the horizontal traces. By contrast, the high stimulus frequency results in no appreciable synchrony as indicated by the steep angles of the traces. For the intermediate frequency the 5 traces represent 5 neural spikes for one cycle of the stimulus.

212 The histograms of the phases, corresponding to the synchrograms of Fig. 2 (a), are shown in Fig. 3. One can note the prominent degree of synchronization by the relative heights of the peaks for the low and intermediate frequencies and the lack of synchrony at the high stimulus frequency.

4. The Glial Cell Networks We have applied the same analysis techniques to examine the state of synchronization between groups of glial cells in a network cultured from human brain tissue, in this case a patient suffering from severe epilepsy. The Ca2+ activity in the network was imaged by time lapse confocal microscopy using laser illumination and a fluorescent, calcium sensitive dye. Tissues taken from two locations, the hippocampus and the uncus, were cultured and imaged. In both cases, from the movies, a small group of cells, called the controlling center of oscillation, was identified. Other groups in the culture can be examined for synchronization with the controlling center. Figure 4 shows two frames from the two cultures, the two controlling centers and another group of cells

213

shown by the circles and blown-up images. The diffusion of phase, pixel-by-pixel, is shown in the blow-ups increasing with time (frame number) upwards. Each trace is started at the same absolute time. The blow-ups show that the cells in the hippocampus are not well synchronized with the controlling center. By contrast, those of the uncus are tightly synchronized as shown by the structured groupings of the diffusion traces. In this case, the uncus was near the epileptic focus. Cultures of cells from this region reflect that epilepsy is characterized by coherent synchrony of surrounding tissue. Fig. 4. Glial cell networks from two locations. The blow-ups show the diffusion of phase with time (increasing upward). The phase of the indicated cell group is measured relative to cells in the controlling center.

Cells cultured from h i p p o c a m p u s show iso synchronization behavior

Conrolling center

Cells cultured from It is also possible to uncus tissue show find a global value for the n oi sy sv n c h ron iz at i om diffusion coefficient, Dej/, averaged over time and the space of the whole network. This value characterizes the state of health of the tissue sample from which the network was cultured. Moreover, the effects of various neurotransmitters and neuromodulators on the state of synchrony of the cells of the network can be quantified with our analysis technique. When correlated with EEG recordings obtained in the operating room prior to surgery, this analysis package forms a powerful new tool for studies of normal and abnormal cellular signaling and the effects of neuroactive chemicals.

5. Discussion, Summary and Conclusions We have shown that stochastic synchronization can be used to study the encoding of weak electric and magnetic fields in the paddlefish electroreceptor system. We find that for signals whose strengths are in the range that the animal customarily encounters in the wild, synchronization coding offers a plausible alternative to the more usual rate coding. Significant synchronization was achieved by the noisy electroreceptors only at the lower range of frequencies studied. These correspond to the range of

214 sensitivity previously determined for the paddlefish receptor system and the frequencies of the signatures of its usual planktonic prey [Wilkens, et al, 1997]. The high frequency tested was well beyond this range and showed no statistically significant synchronization. We have also shown that both the global and local synchronization properties of cultured glial cell networks can be quantified using these techniques. Cell cultures of tissues near the epileptic focus (in this case, the uncus) show significant synchrony. Thus these techniques may be developed into useful medical tools.

References Andronov, A., Witt, A. and Khaykin, S. Theory of Oscillations (Pergamon Press, Oxford, 1966) Huygens, C , Horoloqium Oscilatorium (Parisiis, Paris, France, 1673) Glass, L. and Mackey, M.C. From Clocks to Chaos: The Rhythms of Life (Princeton University Press, Princeton, N. J. 1988)

Neiman, A., Synchronizationlike Phenomena in Coupled Stochastic Bistable Systems. Phys. Rev. £49:3484-3487 (1994). Neiman, A. Silchenko A., Anishchenko V., and Schimansky-Geier L. Stochastic Resonance: Noise-Enhanced Phase Coherence, Phys. Rev. E. 58: 7118-7125 (1998). Neiman, A., Pei, X., Russell, D.,Wojtenek, W., Wilkens, L., Moss, F., Braun, H.A., Huber, M., Voigt, K. Synchronization of the Noisy Electrosensitive Cells in the Paddlefish. Phys. Rev. Lett. 82:660-663 (1999). Pei, X., Wilkens, L., and Moss, F. Light enhances hydrodynamic signaling in the multimodal caudal photoreceptor interneurons of the crayfish. J. Neurophysiol. 76:3002-3011(1996). Pei, X. Determining the site of the oscillator in the paddlefish electrosensitive system, preprint. Schafer, C , Rosenblum, M., Kurths J., and Abel, H. Heartbeat synchronized with ventilation. Nature 392:239-240 (1998). Shulgin, B., Neiman, A., and Anishchenko V., Mean Switching Frequency Locking in Stochastic Bistable Systems Driven by a Periodic Force. Phys. Rev. Lett. 75:41574160(1995) Stratonovich, R. Topics in the Theory of Random Noise (Gordon and Breach, New York, 1967), Vol. 2. Tass, P., Rosenblum, M., Weule, J., Kurths J.,. Pikovsky, A., Volkmann, J., Schnitzler A. and Freund, H. Detection oin.m phase locking from noisy data: Application to magnetoencephalography. Phys. Rev. Lett. 81:3291-3294 (1998) Wilkens, L. The crayfish caudal photoreceptor: advances and questions after the first half century. Comp. Biochem. Physiol. 91:61-68 (1988) Wilkens, L., Russell, D., Pei X.,and Gurgens, C. The paddlefish rostrum functions as an electrosensory antenna in plankton feeding. Proc. Roy. Soc. Lond. B 264:1723-1729 (1997).

REENTRANT WAVES INDUCED BY LOCAL BISTABILITIES IN A CARDIAC MODEL S. Bahar Box 90305, Department of Physics, Duke University, Durham NC 27708, USA [email protected]

Abstract Rate-dependent bistability and hysteresis have recently been observed to be highly prevalent in periodically stimulated bullfrog ventricular muscle. Similar bistabilities have been found in in vivo sheep atria at interstimulus intervals for which spatiotemporally complex behaviors, possibly atrial flutter and fibrillation, are observed. Might bistability play a role in the onset of spatiotemporal disorganization in the whole heart? We investigate the role of local testability in a coupled map/cellular automaton model of cardiac dynamics. This two-dimensional model is based on a simple mapping which gives good qualitative agreement with many of the local features of cardiac dynamics. Under some conditions, local regions of bistability are found to result in phase singularities and rotors in the spatially extended model presented here.

Introduction and Background Different patterns of dynamical response can be elicited under different conditions in cardiac tissue. For example, when stimuli are delivered at a slow rate, one action potential response is elicited for every stimulus (denoted as a 1:1 state). As the stimulus rate (basic cycle length, or BCL) is increased, the action potential duration (APD) begins to oscillate, entering a state called "alternans", in which every stimulus elicits a response but the responses are of alternating duration (2:2 state) [1,2,3], As the stimulus rate is increased further, the tissue becomes unable to recover fast enough to respond to every applied stimulus, resulting in a pattern of "skipped beats", where every other stimulus elicits a response (2:1 pattern). Under other conditions (e.g., different range of applied current amplitude), more complicated behaviors arise [4,5,6]. As pointed out by Guevara et al. [2], rate-dependent bistability and hysteresis between 1:1 (or 2:2) and 2:1 states can occur in simple models of cardiac dynamics as the interstimulus interval (here called BCL, or basic cycle length) is swept up and down. Bistability between the 1:1 branch and the 2:1 branch of APD response was first observed experimentally by Mines in 1913 in a bullfrog cardiac preparation [7], and later by Guevara et al. [4] in an aggregate of spontaneously beating chick embryonic ventricular myocytes. Recently it has been shown that bistability is a highly prevalent local behavior in bullfrog ventricular myocardium, occurring in 74% of preparations studied [8]. Bistability was subsequently observed in in vivo sheep atrium for BCL values over which arrhythmias and spatiotemporal disorganization (possibly flutter or fibrillation) also occurred [9]. Can local bistabilities play a role in the onset of spatial disorganization in excitable media? We investigate this question using a simple cardiac model. 215

216 The Model Experimentally it has been found [3] that APDn is dependent on the previous diastolic interval DIn-i, i.e., APD n =f(DI n .i) (1) where the diastolic interval is defined as DI„., =N*BCL-APD„.i. (2) Here, N is the smallest integer such that N*BCL>APD„.i +0 (3) where 9 is the minimal diastolic interval the tissue can sustain. The experimentally measured values of APD„ and DI n .i can be fit with an exponential function APD n =A 0 -Ae• DIn - l/,

(4)

which, after substituting (1), constitutes a mapping which takes APD n.i into APD„ [2,3]. When the slope of the function f is of magnitude less than one, there is a single steady state solution, corresponding to the 1:1 state. When the magnitude of the slope of f exceeds unity, the 1:1 solution becomes unstable and a stable 2:2 solution is born through a forward bifurcation [10]. When BCL-APD n falls below 9, the tissue can only recover fast enough to respond to every other stimulus, and the system moves to the 2:1 branch (N=2). While only a simple approximation to cardiac dynamics, this model does preserve many of the observed features of experimental preparations (transitions from 1:1 to 2:2, bistability between N=l and N=2 branches). The coupled map/cellular automaton model considered here consists of a 170 by 170 grid of elements, with a map of the form (2)-(4) at each element, with parameters Ao=290 msec, A=203 msec, T = 1 8 7 msec and 9=-20 msec. The window of bistability occurs for BCL values 118 msec to 185 msec. (Compare the physically observed range of BCL values, 90 msec to 160 msec, over which bistability is observed in vivo in sheep atrium [9].) Each element in the grid can exist in one of three states, defined as follows: State 0. In this "off" or "excitable " state, the element is considered to have recovered from its previous action potential and to be susceptible to stimulation by its neighbors. In this state the element cannot excite its neighbors. State 1. An "on", state, where the element is considered capable of exciting its neighbors, but is not susceptible to stimulation by neighboring elements. State 2. A "refractory" state, where the element cannot excite its neighbors or be excited by them. The dwell time of each element in a particular state is determined by the states of the neighboring elements and by the mapping model. We define a variable APD,/ at each

217

element, where i and j are the element's coordinates in the grid, and t is the number of total timesteps in the simulation (each update of the map is a timestep of 1 msec). We define the phase Pi/ as the time since the last activation at coordinates (i,j). At each update of the lattice, Pi/ at coordinates where an activation does not occur is advanced by 1 msec, and the APD/ is unchanged. When the element (i,j) is excited by a neighbor, P / is reset to 1 and APD/ is calculated using the previous APD/ according to the map (2)(4), with BCL=Py'. The state of an element at time t is determined from P/and APD;/ as follows: Pi/^aKAPDij'+e)^ statel ai(APD/+0)< Pi/^APDij'+e)^ state2 Pi/StAPDij'+S)-* stateO Thus during some early fraction ai of the action potential duration the element is in its excited state (state 1). After this period the element enters a refractory state (state 2), which lasts (l-criXAPDjj'+e) msec. After a total time (APDy'+e) msec since its activation, the element returns to an excitable state (state 0). Coupling between the elements of the map is defined as follows: in each timestep t an element (i,j) can be excited by its three nearest neighbors in the longitudinal directions, its two nearest neighbors in the transverse directions, and its closest nearest neighbors diagonally. This results in "propagation" consistent with the distribution of conduction velocities in ventricular myocardium, where excitation propagates three times as fast along the longitudinal direction as along the transverse direction [11]. Each element is be considered to approximate a group of 3 X 3 cells, and the entire array an 51 mm by 10 mm piece of "tissue". In the simulations illustrated here we have stretched the aspect ratio in order to elongate the vertical direction. Simple Periodic Stimulation To simulate periodic stimulation of the "tissue", we deliver a stimulus to one element of the grid at intervals BCL. In Figure 1, we show a sequence of snapshots of the coupled map model as it is periodically stimulated with a BCL of 150 msec. In Figure 2 we show results from a run identical to that shown in Fig. 1, except that APDij°=182 msec. Here, the first stimulus propagates as before (2a, b). However, the second stimulus elicits no response (c), since the APD on the 2:1 branch gives APD+6 >BCL, and thus the element (25,25) is still in a refractory state (state 2) when the second stimulus is given. By the time the third stimulus is delivered, however, the tissue has returned to its excitable state (state 0), and is now ready to respond, as shown in (e,f). The patterns shown in Figs. 1 and 2 are bistable — they coexist for the same set of parameter values, and the system falls into one pattern or another depending on initial conditions.

218

a

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t=153msec

d

t=240 msec

Fig. 1 Snapshots of a simulation with <Ti=0.60. All elements (ij) are initially set with APDij°=118.5 msec, the steady state value on the N=l branch at BCL=150 msec. Periodic stimulation at BCL=150 msec is delivered from element (25,25). In Frame la, a small region of elements centered about (25,25) has become excited (state 1, black region), while the rest of the tissue is unexcited (state 0, white region). In frame lb, at t=90 msec, the excited region has spread to the boundaries of the "tissue", and the center region is now in a refractory state (state 2, grey region). When another stimulus is delivered at t=150 msec, the tissue is ready to respond (frames c and d).

Reentrant Waves, Phase Singularities and Rotors: A Small Region of 2:1 Suppose that a small region of the map, though in phase with its neighbors, has fallen onto the initial conditions for another branch. Specifically, suppose we have the entire tissue initially in a 1:1 state, except for a small region in a 2:1 state. In order for the 2:1 region to remain active until the surrounding 1:1 wavefront has passed, we must require Oi(APD2+0)> APD,+e. If this condition is met, the active 2:1 elements will "reenter" the recovered region in the wake of the passing front, and restimulate the just-recovered 1:1 elements.

219

t=3 msec

b

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t=303 msec

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t=390 msec

Fig. 2 Snapshots from a simulation with conditions identical to Fig. 1, except that the initial conditions APD jj° are set to 182 msec, the steady state value of APD on the N=2 branch at BCL=150 msec. Since APD+9 >BCL, the tissue at the pacing site (25,25) is still refrartory at t=150 msec, as shown in Fig 2c. No response is elicited by a stimulus delivered at this point (marked with an x). By t=240, the tissue has recovered, shown in Fig. 2d. At t=2BCL=300 msec, the tissue is ready to respond again to an apphed stimulus. This illustrates a 2:1 pattern, in contrast to the 1:1 pattern shown in Fig 1.

220

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Fig. 3. Phase resetting plots for a simulation in which the "1:1" elements are set with APDij°=78.85 msec, a value in the basin of attraction of the 1:1 fixed point (APD=103.82 msec) at BCL=120 msec. A strip of elements (j=75 to 78, i=20 to 80) is set with APDij°=158.63 msec, the 2:1 fixed point at this BCL. Periodic stimulation is delivered from element [25][25] at BCL=120 msec. Panels show the evolution of P;/ and APDi/ at (a) the stimulation site, (b) an element in the reentrant 2:1 region and (c) and an element initially on the "1:1" (or "transient 2:2") branch. By t=200 msec, all elements are being periodically stimulated by the rotor at intervals Tf. Coordinates [i][j] are shown in the top right corner of each panel.

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Fig. 4 A phase singularity in the simulation illustrated in Fig. 3, at t=235 msec. Because of the size and location of the strip of 2:1 elements with respect to the stimulation site, reexcitation of the recovered tissue by the 2:1 elements generates a clearly visible pair of "twin rotors". The locations of the rotor cores do not appear to drift during the duration of the simulation. Color code: black: 0 45. Once this "reentry" has occurred, what happens next? The crucial thing is that the restimulation of the recovered 1:1 elements creates a phase singularity (Figure 4). If the 1:1 elements are all placed squarely on the 1:1 fixed point, the singularity is drowned out by the periodic stimuli delivered from the pacing site. However, we find that if the 1:1 elements are in a state of transiently alternating about the 1:1 steady state (i.e., in the basin of attraction but not at the fixed point), then the singularity acts as a self-sustaining rotor with period Tt=f(0)+6, drowning out the regular stimuli and subsuming the entire extent of the tissue. The long-term behavior of the tissue is illustrated in Figure 3. This shows three "phase-resetting" plots, which illustrate the phase at a given element as a function of time t (thin diagonal lines) and the APD at that point as a function oft (thick horizontal lines). We show phase-resetting plots for three different elements: the stimulation site (Fig. 3a), an element in the initial 2:1 region (Fig. 3b), and an element in the initial "1:1" (or "transient 2:2") region (Fig. 3c). By t=200 msec, all three points have fallen into the wake of the rotor, and the phase is periodically reset at intervals Tf= f(0)+6. Figure 4 shows a pair of rotors in this simulation at t=235 msec. Similar results have been found for a small region of 1:1 elements embedded in a 2:1 field (not shown). If the 1:1 elements are transiently alternating about the fixed point, under certain restrictions on cji, a stable rotor can be formed. It remains to be determined whether there is an analytical reason which limits the formation of a sustained rotor to cases where the initial conditions are alternating about, rather than settled at, a fixed

222 point. It may be that this limitation is unique to certain choices of parameters in the map (2)-(4), or to certain ranges of Oi. Conclusions We have shown that under some conditions local bistabilities can lead to phase singularities and rotor formation. However, the model presented here is, certainly, only a caricature of cardiac dynamics. First, the map (2)-(4) is itself an imperfect description of cardiac dynamics [8], and a spatial extension of an imperfect local map is bound to give an incomplete picture. Also, the values required of ai may be unrealistically large. The model presented here does not take into account current attenuation as the signal is transmitted further from the stimulus site, nor the dependence of propagation velocity on wave front curvature (e.g., a convex front propagates faster than a concave one). The model does not account for diffusive coupling between elements. The rotation rate of the rotor found here Tf=f(0)+0 ~ 48 msec is considerably faster than that observed in experimental preparation [12], A realistic picture of inhomogeneities such that some regions of tissue would be in a 1:1 state and others in a 2:1 state must also be developed. Certainly more realistic models must be investigated before a clear hypothesis can be formed about the role of bistability in rotor formation. A next step might be to explore the role of local bistabilities in a spatially extended version of the Luo-Rudy [13] model, for example, using the method of Barkley [14]. Despite these shortcomings, however, this model does reproduce some of the most basic features of propagation in cardiac tissue, and therefore may be considered a "ground-zero" testbed for a study of the spatial effects of bistability. References 1. D. R. Chialvo and J. Jalife, Nature 330 (1987) 749. 2. M Guevara, G. Ward, A. Shrier and L. Glass, Computers in Cardiology, IEEE Comp. Soc.,(1984) 167. 3.J. B. Nolasco, R. W. Dahlen, J. Appl. Physiol. 25 (1968) 191. 4. M. Guevara, A. Shrier and L. Glass, in Zipes DP, Jalife J (eds): Cardiac Electrophysiology, From Cell to Bedside. Philadelphia, PA, WB Saunders Co., 1990, ch. 23. 5. D. R. Chialvo, D. C. Michaels and J. Jalife, Circ. Res. 66 (1990) 525. 6. D. R. Chialvo and J. Jalife, in Zipes DP, Jalife J (eds): Cardiac Electrophysiology, From Cell to Bedside. Philadelphia, PA, WB Saunders Co., 1990, ch. 24. 7. G. R. Mines, J Physiol. (Lond) 46 (1913) 349. 8. G. M Hall, S. Bahar and D. J. Gauthier, Phys. Rev. Lett. 82 (1999) 2995 9. R. A. Oliver, W. Krassowska, G. M Hall, S. Bahar, P. D. Wolf and D. J. Gauthier, submitted abstract. 10. P. Berge, Y. Pomeau and C. Vidal, Order Within Chaos: Towards a Deterministic Approach to Turbulence. New York, NY, J. Wiley and Sons, 1984. 11. M. L. Pressler, P. N. Munster and X. Huang, in Zipes DP, Jalife J (eds): Cardiac Electrophysiology, From Cell to Bedside, 2nd ed. Philadelphia, PA, WB Saunders Co., 1995, ch. 16 12. J. M. Davidenko, A. V. Pertsov, R. Salomonsz, W. Baxter and J. Jalife, Nature 355 (1992):349. 13. C. H. Luo and Y. Rudy, Circ. Res. 68 (1991): 1501. 14. D. Barkley. Physica 49D (1991):61.

223 COUPLED OSCILLATORS SYSTEM IN THE TRUE SLIME MOLD A. TAKAMATSU, T. FUJII* and I. ENDO Biochemical Systems Laboratory, The Institute of Physical and Chemical Research (The RIKEN Institute), 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, JAPAN E-mail: [email protected] ABSTRACT The Plasmodium of true slime mold, Physarum polycephalum, which shows various oscillatory phenomena, can be regarded as a coupled nonlinear oscillators system. The partial bodies of the Plasmodium are interconnected by microscale tubes, whose dimension can be related to the coupling strength between the plasmodial oscillators. Investigation on the collective behavior of the oscillators under the condition that the configuration of the tube structure can be manipulated gives significant information on the characteristics of the Plasmodium from the viewpoint of nonlinear dynamics. In this study, we propose a living coupled oscillators system. Using a microfabricated structure, we patterned the geometry and the dimensions of the microscale tube structure of the Plasmodium. As the first step, the Plasmodium was grown in the microstructure for coupled two oscillators system that has two wells (oscillator part) and a microchannel (coupling part). We investigated the oscillation bahavior by monitoring the thickness oscillation of Plasmodium in the strucutre with various width (W) and length (L) of microchannel. We found that there are various types of oscillation bahavior, such as anti-phase and in-phase oscillations depending on the channel dimension W and L. The present method is suitable for further studies of the network of the Plasmodium as a collective nonlinear oscillators system.

1.

Introduction Nonlinear dynamics becomes important for the studies of collective cells, for

example, cardiac cells, neurons and other oscillatory or excitable cells1'2. In these biological systems, the connections between the elements are not highly designed individually. Nevertheless, the collectives of such elements self-organize to show various biological functions, such as information processing in nervous systems and cooperative beating in cardiac systems. In order to understand the mechanism of biological selforganization, it is, therefore, important to investigate the behavior of the systems under the condition that the connections of the elements can be flexibly controlled. In this study, we use the true slime mold, Physarum polycepharum

as a living coupled oscillators

system, in which we can control the connections among the oscillators , systematically. The Plasmodium of Physarum polycephalum

is a huge amoeboid multinucleated

unicellular organism. The Plasmodium is an aggregate of endoplasm without any highly ' Present address: Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi Minato-ku Tokyo, 106-8558, Japan.

224 differentiated structure like nervous systems. Multiple Plasmodia can be fused into a single one and a single Plasmodium can be easily divided into multiple ones. Despite its simple structure, the Plasmodium demonstrates sophisticated biological functions. Namely, a large cluster of the Plasmodium is able to behave as a single entity based on the information from its surroundings. The information acquired by a local partial body drives the cooperative behavior by transmitting the information throughout the whole cluster. The information is considered to be processed through the interaction among the partial bodies. This leads to, for example, the attracting/escaping behavior of the Plasmodium to/from attractants/repellents3. These functions, information transmission and processing, are said to be realized through the intrinsic oscillatory phenomena in the plasmodium, such as thickness oscillation4 which synchronizes with the shuttle streaming of endoplasm5, and the oscillation in the concentrations of ATP6 and Ca2+ 7. These phenomena are supposed to be generated by mechanochemical reactions caused by complicated interactions among intracellular chemicals, proteins, organelles, etc8. From the viewpoint of nonlinear dynamics, the plasmodium can, therefore, be modeled as a collective system that consists of nonlinear oscillators interconnected with certain coupling strength , ' ,0 ' n . In these models, an oscillator can be defined for each partial body in the plasmodium and the coupling strength among the oscillators would correspond to the amount of the shuttle streaming of endoplasm inside the tube structures which interconnect the partial bodies. It is, therefore, expected that by controlling the growth pattern and the dimension of the tube structure, i.e., the geometry and the coupling strength of the oscillator network, we could observe various types of collective behaviors which could help us to understand the characteristics of the plasmodium as a nonlinear dynamical system. In this paper, we describe the advanced patterning method and the results of observation of the Plasmodium in the microstructure for coupled oscillators system, as the first step. Materials and Methods'2,13 The plasmodium prefers wet medium such as agar plate, on which the plasmodium is usually cultured, to dry one. It is possible to control the geometry of the plasmodium by using the agar plate covered with the dry media having variously designed openings. As the dry media, we adopted an ultra-thick negative photoresist resin (NANO™ SU-8 50, Microlithography Chemical Corp.), which enables us to fabricate microstructures with about 10jo.m precision that is the same order as the tube diameter of the plasmodium in the natural situation. The microstructures are thick sheets (~100(xm) having an opening consists of two circular wells connected by a channel ( Fig.l). The diameter of the wells are determined as 2mm so that we can consider the plasmodium in a well as an oscillator. 2.

225 The size is comparable to half of the spatial wave length of oscillation in the natural Plasmodium and, within this area, the Plasmodium shows almost synchronized oscillation. The plasmodia locally put at the wells in the microstructure on the agar plate (Fig. 1 (c)) grew along the channel pattern during the culture for 5-15 hours at the condition of 25°C and RH85%. Finally, the plasmodia from both sides of wells contacted at around the center of the channel to be physically connected through the tube structure (Fig. 1 (a), (b)), where the endoplasmic streaming were observed. The plasmodia grown in the microstructure were set under a microscope (SMZ-2T, Nikon). The transmitted light through the Plasmodium was detected by the CCD camera (C2400, Hamamatsu). The image data converted by the flame grabber (LG-3, Scion Corp.) were sequentially stored in the PC (Power Macintosh G3, Apple) every 4 seconds. We have confirmed that the change in the transmitted light intensity is inversely proportional to the change in thickness within the range of thickness oscillation (data not shown). The experiments were performed in the thermostat and humidistat chamber (PR2K, ESPEC) under the condition of 25 ±0.3°C and RH85 ±2.5%. 3.

Results and discussion

We observed the thickness oscillation of various width (W; 50-1000 |0.m) and length (L; two oscillators system (Fig. 1). Various types channel dimension W and L were observed, for etc. as shown in Fig. 2.

the plasmodium in the wells with the 3-15 mm) of the channel on the coupled of oscillation pattern depending on the example, anti-phase, in-phase oscillation

Fig. 2 (a) shows an example of the anti-phase oscillation, in which the alternating wave propagation from one plasmodial oscillator to another was observed. Fig. 3 (a) shows the change in thickness of each oscillator that was obtained by averaging the images over each well part. Fig. 3 (b) shows the phase difference between two oscillators that was determined by the conventional method14. These graphs demonstrate that the two oscillators are mutually entrained and their phases are locked in anti-phase. This phenomenon was observed when L is small (L=4mm) and W is relatively large (W=400500u.m). When W is smaller (W=200-300u.m), the stability of anti-phase oscillation becomes smaller and the oscillators show quasi-periodic oscillation (Fig. 4). When W is much smaller (W=50-100ixm), the oscillators was not entrained and oscillate with their own periods (Fig. 5). In the narrow channel width below 50 u.m, the tube structure was never formed. We have also confirmed that the tube diameter can be controlled by the

226

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microfabricated channel structure13 and the velocity of the endoplasmic streaming observed inside the tube structure increases with the tube diameter15. That indicates the amount of the endoplasm exchanged by the endoplasmic streaming affect on the coupling strength in a reverse manner (as antiferromagnetic like interaction), whose strength can be controlled by the microfabricated channel structure. It is considered that the synchronous and asynchronous phenomena as shown in Fig. 2(a) and Fig. 3-5 were caused by the difference of the coupling strength. Fig. 2(b) and Fig. 6 shows an example of the in-phase oscillation, in which the thickness changes in the both oscillators synchronized in phase. This phenomenon was observed when L is large (L^lOmm). The propagation velocity v of the endoplasmic streaming between oscillators was ~0.2mm/sec under this condition15, which suggests that the oscillators interact each other with time delay -40 sec ((L-2)/v; note that L is the distance from center to center of both wells). A periods of oscillation under this condition is -100 sec so that the delay time is not negligible. We have applied the model of coupled limit cycle oscillators with time delay16 to this system and found that in-phase solutions appear when delay time is considerably large even if the coupling is the antiferromagnetic typels.

• ^ ^ ^ ^ ^ ^ ^ ^ ^ & ">wk& ^

.?mm Fig/7 Multiple coupled oscillators system

229 4.

Conclusion

We developed a new method to control the coupling strength between oscillators in the Physarum Plasmodium by using the microfabricated structure and observed the various oscillation behaviors depending on the dimension of the microfabricated channel structures. With this system, it is also possible to integrate a stimulator device on the same structure and observe the response to stimuli in a systematic way. More rich oscillation behaviors will be observed by using the microstructure designed for the multiple coupled oscillators system as shown in Fig. 7, which could give us further information about the collective nonlinear oscillators systems. 5.

Acknowledgement

The authors thank Dr. K. Hosokawa for helping in the microfabrication processes. This study was partly supported by Special Coordination Funds for Promoting Science and Technology of the Science and Technology Agent of the Japanese Government and was partly supported by the RIKEN Special Postdoctoral Research Program. 6.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

A. T. Winfree, The geometry of biological time (Springer-Verlag, N.Y., Heiderberg, Berlin, 1980) D. Hoyer, B. Pompe, H. Herzel, U. Zwiener, IEEE Engineering in Medicine and Biology 17 (6,1998) 17. D. J. C. Knowles, M. J. Carlile, J Gen Microbiol 108 (1978) 17. Z. Baranowski, Acta. Protozoologica 17 (1978) 377. N. Kamiya, Cytologia 15 (1950) 194. Y. Yoshimoto, T. Sakai, N. Kamiya, Protoplasma 109 (1981) 159. K. Natsume, Y. Miyake, M. Yano, H. Shimizu, Protoplasma 166 (1992) 55. T. Ueda, K. Matsumoto, T. Akitaya, Y. Kobatake, Exp. Cell Res. 162 (1986) 486. K. Matsumoto, T. Ueda, Y. Kobatake, J. theor. Biol 131 (1988) 175. K. Takahashi, G. Uchida, Z. Hu, Y. Tsuchiya, J. theor. Biol 184 (1997) 105. A. Takamatsu, K. Takahashi, M. Nagao, Y. Tsuchiya, J. Phys. Soc. Jpn. 66 (1997) 1638. A. Takamatsu, T. Fujii, K. Hosokawa, I. Endo, Proceedings of the 20,h annual international conference of the IEEE-EMBS 20 (1998), 2987 A. Takamatsu, T. Fujii, H. Yokota, K. Hosokawa, T. Higuchi, I. Endo, (1999) submitted. S. Nakata, T. Miyata, N. Ojima, K. Yoshikawa, PhysicaD 115 (1998) 313. A. Takamatsu, T. Fujii, I. Endo, (1999) in preparation. H. G. Schuster, P. Wagner, Prog. Theor. Phys. 81 (1989) 939.

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VII. Synchronization

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EXPERIMENTAL MANIFESTATIONS OF PHASE AND LAG SYNCHRONIZATIONS IN COUPLED CHAOTIC SYSTEMS YING-CHENG LAI Departments

of Mathematics and Electrical Engineering, Center for Systems Science and Engineering Research, Arizona State University, Tempe, AZ 85287-1804 E-mail: [email protected]

VICTOR ANDRADE, RUSLAN DAVIDCHACK, AND SAEED TAHERION Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045

C h a o t i c phase a n d lag synchronizations are subtle d y n a m ical p h e n o m e n a in which t h e phases of two m u t u a l l y coupled, nonidentical chaotic oscillators b e c o m e correlated. We investigate t o w h a t extent phase a n d lag synchronizations can be observed in l a b o r a t o r y experiments. Specifically, for phase synchronization, we s t u d y t h e effect of noise. It is found t h a t additive white noise can induce phase slips in integer multiples of 27r's in p a r a m e t e r regimes where p h a s e synchronization is observed in t h e absence of noise. T h e average t i m e d u r a t i o n of t h e t e m p o r a l phase synchronization scales with t h e noise a m p l i t u d e in a way t h a t can be described as super-persistent transient. We give two indep e n d e n t heuristic derivations t h a t yield t h e same n u m e r ically observed scaling law. For lag synchronization, we perform l a b o r a t o r y experiments using chaotic electronic circuits. O u r m e a s u r e m e n t s indicate t h a t due t o t h e influence of noise, lag synchronization occurs only i n t e r m i t t e n t l y in t i m e . Numerical confirmation and a heuristic e x p l a n a t i o n t o t h e observed i n t e r m i t t e n t behavior are given. 1

Introduction

Synchronization is a fundamental phenomenon in nature. It has been recognized since 1983 that synchronization can occur in chaotic systems 1<2<3'i<5-6. It was, however, first suggested by Pecora and Carroll 3 that synchronous chaos could be utilized for nonlinear digital communication 7 ' 8 ' 9 . The issue of security of such chaotic communication scheme was also explored 1 0 ' n . Since the work of Pecora and Carroll, synchronization in chaotic systems has received a tremendous amount of attention and it still remains to be one of the most active research areas in chaotic dynamics. Recently, a more delicate class of chaotic synchronization phenomena has been discovered and analyzed 12 - 13 ' 14 . These are the phase and lag synchronizations of coupled chaotic systems. In particular, consider two slightly different chaotic oscillators described by: dx.i/dt = Fi(xi),

and

dyLifdt = F 2 (x 2 ),

where both xi(i) and x2(£) are d-dimensional vectors and Fi RJ F 2 . Typically, the trajectories x ^ i ) and x 2 (t) correspond to some rotational motion in the phase space so that the corresponding phase variables <j>i(t) and (j>i(t) can be defined 13 . If there is no interaction between the two oscillators, the phase variables i(t) are completely uncorrelated and, hence, there is a normal diffusion

233

234 13

in the random variable A<j>(t) = fa{t) - 4>i(t). If, however, a small coupling between the two oscillators is present, phase synchronization can occur in the sense that |A0(t)| is bounded between 0 and 2ir. When the coupling is strong, complete synchronization bewteen Xi(t) and x2(t) occurs, i.e., limt-Kx) |xi(t) - x 2 (i)| -> 0. However, before complete synchronization can be achieved, there can be a regime of values of the coupling parameter in which xi (t) does not synchronize with x 2 (t) but, instead, it synchronizes with X2(4 + r ) , where r ^ 0 is a time delay depending on some parameters that characterize the mismatch between the two oscillators. As such, lag synchronization cannot be observed if the two oscillators are completely identical. Since it is not possible to have identical nonlinear oscillators in realistic situations, it was speculated that lag synchronization would be typical in systems of coupled chaotic oscillators 14 . In a recent brief note, experimental observation of phase synchronization was reported 15 . In any experimental situations, noise is inevitable. One aim of this Review is then to address, quantitatively, how noise affects phase synchronization. We find: (1) additive white noise, a type of noise that encountered commonly in experimental situations, can induce phase slips in units of 2n between the coupled oscillators, which would otherwise be synchronized in phase in the absence of noise, and (2) the average time duration between successive phase slips appears to obey the following scaling law with the noise amplitude e:

r~exp(Ke-a),

(1)

where a > 0 is the scaling exponent depending on system parameters such as the coupling strength and K > 0 is a constant. An implication is that in the presence of only small noise, the average time duration to observe phase synchronization can be extremely long. Phase synchronization is robust in this sense. In this Review, we also address the observability of lag synchronization in laboratory experiments. Specifically, we construct electronic circuits that replicate the dynamics of the chaotic Rossler oscillator 17 to detect lag synchronization. What has typically been observed in experiments is that the time-delayed variables of one oscillator tend to follow the variables of another oscillator only intermittently in time in some range of the coupling strength. In particular, if one measures the difference Ax T (i) = |x2(i + r) - x i ( t ) | with r chosen to minimize the root-mean-square, normalized average of Ax T (i), one observes that Ax T (i) tends to exhibit intermittency with frequent large bursts away from zero. Increasing the coupling strength often leads to a transition to complete synchronization, i.e., |xi(t) - X2(i)| —• 0. We speculate that the inevitable presence of small random disturbance during the experiments may be a key factor that obstructs the observation of sustained lag synchronization, a phenomenon which relies on a precise timing between the dynamics of the coupled oscillators. We have also performed numerical computations to investigate the influence of small random noise on lag synchronization. Our analyses suggest that lag synchronization is destroyed when the noise level is larger than or comparable to the amount of the average mismatch between the two chaotic oscillators. At small noise levels, lag synchronization appears in an intermittent fashion over many orders of magnitude of the noise amplitude, a result that is consistent with our experimental observation. The implication is that it is perhaps difficult to observe lag synchronization in noisy environments, in contrast to the case of phase synchronization. In Sec. 2, we present numerical scaling results for the influence of noise on phase synchronization. In Sec. 3, we give two independent derivations for the scaling law. In Sec. 4, we present experimental measurements and analyses for lag synchronization. A discussion is presented in Sec. 5.

235 2

Effect o f n o i s e o n p h a s e s y n c h r o n i z a t i o n

We consider t h e following system of two coupled Rossler oscillators, the one t h a t was originally used by Rosenblum et al. 1 2 to first report phase synchronization: dXl,2/dt

= -^1,22/1,2 - 2l,2 + C(X 2 ,1 - Xl,l),

dyialdt

= w i,2^i,2 + 0.15j/i, 2 ,

dzla/dt

= 0.2 + (Xi, 2 - 10.0)21,2,

(2)

where C is t h e coupling strength, and we choose (U)I,LJ2) = (1.015,0.985) so t h a t the two oscillators are slightly different in order t o minic a typical experimental situation where t h e oscillators cannot be perfectly identical. T h e Rossler chaotic a t t r a c t o r 1 7 has t h e property t h a t its (x,y) variables represent a chaotic rotation with well-defined phase angles I 2 . To compute the phase angles associated with t h e two oscillators, we find it convenient to use the polar coordinates (r, 9) t o replace t h e (x,y) coordinates. In the cylindrical coordinate (r,0,z), the Rossler equations become: —~

= 0.157%2 sin 2
—r-2- = oil 2 + O.15sinf?i,2cos0i dt -21,2] sin 0i,2, ^

at

[C(r2 1 cos# 2 ,i - n

2

(3)

2cos#i,2)

r-i,2

= 0.2 + ( n , 2 cos61,2 - 10.0)zi, 2 .

W h e n there is no coupling, t h e phase angles 0i(t) and 02(t) are uncorrelated and, hence, t h e phase difference A0(t) = \02(t) — 0i(t)\ increases steadily with time. P h a s e synchronization occurs when C is increased through t h e critical value Cv ta 0.029, in which we have A6(t) < 2-K. T h e lower trace in Fig. 1 shows such a situation for C = 0.03, where A0(t) versus t is plotted. To model noise, we a d d different realizations of t h e t h e following t e r m s eoTieiZ(t) t o every of t h e six variables in t h e coupled Rossler system Eq. (3) at each step of integration, where e is t h e noise amplitude and cr's are r a n d o m variables uniformly distributed in [—1,1]. T h e upper trace in Fig. 1 shows A8(t) versus t for C = 0.03, where the noise amplitude is e ss 10~ 3 . We see t h a t noise induces occasional phase slips in units of approximately 27r in A8(t). These phase slips are, however, rare and become extremely infrequent as t h e noise amplitude is decreased further. To quantify t h e 2?r phase slips in Fig. 1, we compute how t h e average time interval r between successive phase slips ° changes as t h e noise amplitude is changed. For t h e p a r a m e t e r setting described above, we find t h a t r can be so prohibitively long t h a t numerical computation of it becomes infeasible when t h e noise amplitude e is small t h a n , say, 1 0 - 4 . Figure 2 shows log 1 0 r versus e~a for 1 0 ~ 3 ' 5 < e < 10~ 1 , 5 (approximately two orders of magnitude in e), where a » 0.31 is a fitting p a r a m e t e r . T h e approximate linear scaling behavior in Fig. 2 suggests t h e scaling relation (1), which implies t h a t t h e average time interval t o observe t h e 27r phase slips behaves like e°° as e —> 0. This is similar to the behavior of super-persistent chaotic transients observed previously 19,20,13 To qualitatively u n d e r s t a n d t h e scaling behavior in Fig. 2, we perform the following numerical experiment. First, we set e = 0 and plot, in t h e coordinate (r = \Jr\ + r\, Aff), t h e a t t r a c t o r s resulted from two different initial conditions with 0 < AB < 2-rr and 2n < A6 < 4TT, respectively, °We find numerically (data not shown) that at large noise levels, say e > 10~ 2 . the probability distributions of the time intervals of temporal phase synchronization are exponential. At smaller noise levels, the distributions cross over to become Lorentzian 18 . In both bases, an average time interval r can be defined.

236

(b)

Ae 1 s

(a) .»»^«.i»iit»i»'»Wii*)iV»*(»i'*»<)il'H*>l>»i>»'W'Wtm'iw<
Figure 1. For the system of two coupled Rossler oscillators: phase synchronization without noise (lower trace); and 27r-phase slips induced by noise of amplitude e fts 1 0 - 3 (upper trace).

Figure 2. For the system of two coupled Rossler oscillators at C = 0.03: log 1 0 r versus e fitting parameter. Each point represents the average over 100 time intervals.

a

, where a fa 0.31 is ;

as shown in Fig. 3(a). Note that the variable A0 is in fact a lifted angle variable 21 by which differences of integer multiple of 27r's are considered distinct. We see that initial conditions with 2TT difference in A9 result in attractors that live in different basins of attraction. Depending on the initial conditions, there is an infinite number of these attractors separated from each other by 2ir in A#. In the absence of noise, these attractors are completely isolated, corresponding to the situation of phase synchronization where A0 remains within 2w if they start with a value less than 2-K. Next, we examine the influence of noise on the phase-space structure in Fig. 3(a), as shown in

237

Figure 3. Using the lifted phase variable A0 for the system of two coupled Rossler oscillators: (a) two isolated phasesynchronized attractors in 0 < A0 < 27r and 27r < A0 < 47T in the absence of noise; (b) tunneling between the previously isolated attractors due to noise.

Fig. 3(b) for e = 10~ 2 . We see that the basins of attraction of the previously isolated attractors are now connected. There is now a nonzero probability that a trajectory can switch to different attractors separated by 2ir in AS, corresponding to the 2K phase slips observed in Fig. 1. The switch occurs when the trajectory falls into an open "tunnel" connecting the basins. The widths of these tunnels must be exponentially small so that the probability for the trajectory to fall into a tunnel is extremely small, leading to the super-persisitent transient behavior observed in Fig. 2. The numerically observed scaling law, as in Fig. 2, is only indicative of the dynamical characteristic of the noise-induced phase slips. It is difficult to extend the range of numerical computations because of the extremely long transient behavior between the phase slips. It is thus important to be able to derive heuristic theories to account for the scaling law. In the sequel, we provide two independent theories, one based on the dynamical system approach and another on statistical mechanical methodology. Both theories yield the same scaling law.

238 3 3.1

Derivation of the noise scaling law for phase synchronization Dynamical systems approach

Note that in Eq. (3), the scales of time variation of the amplitude variables ri,2(£) and phase variables #i,2(t) are generally distinct. Since, on average, we have Oit2(t) ~ uot, we see that the phase angles #1,2 (*) are "fast" variables. The amplitudes n,2(*) are > however, slow variables because the Rossler chaotic trajectories have approximately a circularly rotational structure. Thus, one can average over rotations of the phase angles to separate out the dynamics of the slow variables. Letting "1,2M = ^o* + 0i,2(*) and performing averaging in the time interval t 6 [0,2?r/a;o] yield 14 : d$(t) ss 2(5w + CG(ri,ri)

sin$(i) + white noise term,

(4)

where $(t) = i(t) = Oi{t) —#i(£), Su = OJI -0)2, and G(r\,r2) is a function that depends on the chaotic amplitudes rj 2 (i)- Equation (4) thus describes the dynamics of a chaotically driven limitcycle oscillator. While the specific form of Eq. (4) is for the system of coupled Rossler oscillators, we notice the general feature of the phase-synchronization problem: limit-cycle oscillator driven by chaos. To facilitate analysis, we construct the following model of two-dimensional maps incorporating the general dynamical features of phase synchronization 14 : Zn+l = / ( i n ) ,

(5)

+ g2{xn)$l + g3(xn)$3n,

*„+i = e + pgi{xn)§n

where f(x) is a chaotic map in which the variable x models the chaotic amplitudes in Eq. (4), e > 0 models the combination of the small noise and the slight parameter mismatch between the two coupled chaotic oscillators, 31,2,3(2:) are smooth functions, and p is a parameter that is proportional to the coupling strength. Assume that f(x) generates a chaotic attractor with an infinite number of unstable periodic orbits embedded in it, and phase synchronization occurs for p > pc. In the ^-direction, these periodic orbits can be stable or unstable. For p > pc, all periodic orbits are stable in the ^-direction in the absence of noise, so $ remains approximately constant (phase synchronization). Under the influence of noise, however, some of the periodic orbits become unstable in the ^-direction and, as such, a set of "tongues" opens at the locations of these periodic orbits, allowing trajectory to escape from one approximately constant $ state to another (27r-phase slips). Typically, these orbits have low periods and the sizes of the tongues are exponentially small 19 20 ' , which accounts for the extremely long time duration between the successive 27r-phase slips. Let A > 0 be the Lyapunov exponent of the x chaotic attractor and let T be the time for a trajectory to tunnel through one of the tongues. We have, for the typical size of the opening of the tongue, the following: 6 ~ e _ A T . The average time between the successive phase slips is then:

r ~ 1 ~ eXT.

(6)

0

The tunneling time T can be estimated by noting that when T is large, the map equation in $ in Eq. (5) can be approximated as: d$/dt f» e + \pgi{x) - 1]$ + g2(z)2 + g-i{x)&, which yields: Ttap

«

(7)

Jo e + \P9i ~ 1]* + S2* 2 + 93$ 3 The dependence of T on e thus depends on the specific functions 51,2,3(2:). For instance, since we know that most periodic orbits embedded in the x chaotic attractor are stable in the ^-direction, we have pgi(x) < 1. A possible condition for limit cycle oscillator is: 92(2;) « 1 and #3(2;) ss 0.

239 Under these conditions, we have: X ~ e 1//2 . If, however, we have gi(x) sa 0 and
Statistical mechanical approach

This derivation follows that by Stratonovich 22 for phase synchronization of limited-cycle oscillators. Note that Eq. (4), in the absence of noise, models the motion of a classical particle in a potential of the following form: V(§) = —2i5o)$ + CG(ri,r2)cos$. When the coupling strength is large enough, the potential function V($) possesses an infinite number of local minima separated by 2w in the phase variable $. On average, these minimum values of the potential function V($) decrease linearly because of the linear term — 2<5w. The chaotic amplitude factor G{ri,T2) models the fluctuations of the minimum potential values. When these minima are present, a particle starting near one of the local minima is trapped in its vicinity forever in a noiseless situation, signifying sustained phase synchronization. In the presence of noise, however, a particle originally in one of the local minima can be kicked into one of the adjacent minima, giving rise to a 2-K phase jump. The probability for this to occur is: P ~ e~AE/T, where A £ is the typical height of the potential barrier that separates neighboring minima and T is the "temperature" that is determined by the noise. Typically, we have T ~ ta, where a > 0. The average time for a 2-K phase jump to occur is thus given by: r ~ 1/P ~ exp (AEe~a), which is the scaling law Eq. (1). 4

Experimental observation of lag synchronization

4-1 Experimental setup Our experiments are conducted using a pair of electronic oscillators whose dynamics mimic that of the chaotic Rossler attractor. To have robust chaos for individual oscillators, we construct the circuits so that they contain components for which the voltage-current relations are piecewise linear 23 . Figure 4 shows a schematic diagram for one of the experimental circuits that we constructed: two unidirectionally coupled Rossler oscillators. To stipulate nonidentity of the two chaotic oscillators, Ri and i?2 in the circuit are chosen to have slightly different values of resistance. The typical oscillating frequencies of the circuits are in the audio frequency range. We use a simple linear scheme for coupling between the two oscillators, i.e., terms such as ±C(x2 - Xi), in the form of voltage, are applied to the derivatives of the x-variables in one or both circuits, where C is a parameter characterizing the coupling strength. In the experiments, C can be changed systematically with the accuracy of 0.5% of the change. The electronic components in each circuit are carefully chosen, and the circuits are assembled on high-quality printed-circuit boards in order to minimize the effect of internal and environmental noise. Both oscillators and the coupling circuits are operated by a low-ripple and low-noise power supply (HPE3631A). The voltages from x, y and z are recorded by using a 12-bit data acquisition board (DAS1800AO, Keithley) at the sampling frequency of 100kHz. The noise voltage of the circuit is measured by having the circuit operated in a steady state. The noise level is defined to be the ratio of the root-mean-square values of the noise to that of the chaotic signal. To quantify lag synchronization, we use the following similarity function defined with respect to one dynamical variable, say x, of the chaotic oscillators 14 : (l«»(Hr)-ii(()]») [<*2(4)>(*2(t)>]i/2 where r is the lag time. Let Smin be the minimum value of S(T) and let r m j n be the amount

240

Figure 4. Schematic diagram of two unidirectional coupled chaotic piecewise linear Rossler circuits, where we set Ri = 75kSl and B 2 = 67&0, so that the two circuits are nonidentical. The variable resistor RFi is used to change the coupling. The operational amplifiers are type 741. All resistors are metal-film type with tolerance 1% and all capacitors are polyester type with tolerance 5%. The circuit is run by a ±15 volts source.

of lag when Smi„ is achieved. Lag synchronization between the two oscillators is characterized by the conditions S m ;„ = 0 and Tmin ^ 0, while complete synchronization is by Smi„ = 0 and rmin = 0. Numerically, in the absence of noise, as the coupling strength is increased, one observes the transition from asynchronous chaos to lag synchronization and then to complete synchronization

241 14

. Let Cs be the critical value of C at which Smin reaches zero, and let CT be the C value at which Tmin becomes zero. In order to be able to observe lag synchronization, one must have C, < CT, so that lag synchronization occurs in the parameter interval [CS,CT]- However, if C T < C„, no lag synchronization can be observed because the lag time has already become zero before synchronization occurs (S m ;„ = 0). 4-2

Experimental Results

We report here measurements using the unidirectionally coupled circuits in Fig. 4. The unidirectionally coupling scheme is actually quite representative of coupled nonlinear oscillators in general, because there always exists a mathematical change of coordinates to transform a pair of mutually coupled (bidirectionally coupled) oscillators into a pair of unidirectionally coupled ones, at least locally near the state of synchronization 24 . The differential equations describing the circuit are: dxi

-7x1 -ayi

- zi +C(x2 -xi),

(9)

dyi

dt dz\

~dl dx2

IT

g(xi)

-zu

-7x2 - ay2 - z2,

dy2 /3i 2 + a2y2, dt dz2 g(x2) - 22, ~dt where g(x) = 0 if x < 3, g(x) = n{x — 3) if x > 3. The parameters in Eq. (9) are as follows: a = 0.5, 0 = 1, 7 = 0.05, a\ = 0.113, a 2 = 0.129, and fi = 15. The uncertainties in these parameters are about ±5%. The resistors R\ and R2 in the circuit are chosen to be 75fcO and 67fc£), respectively, to ensure a systematic parameter mismatch between the two Rossler circuits. This difference corresponds to approximately 10% difference in the parameters a\ and a 2 in Eq. (9). When the coupling is week, the dynamics of both oscillators are uncorrelated so that there is no synchronization. As the coupling is increased, the dynamical variables of the two chaotic circuits tend to follow each other, and lag synchronization begins to appear. Figures 5(a) and 5(b) show x\(t) versus x2(t) and x\{t) versus x 2 (t + r m j n ), respectively, for C = 0.023. The elliptical shape in the plot of x\ (t) versus x2 (t) in Fig. 5(a) indicates that there is a lag synchronization between x\ (t) and x2(t). This can readily be seen by examining two time-lagged periodic signals: x(t) = sint and y(t) = sin (t + T), which represent the following ellipse in the (x,y) plane: ax2 + 2bxy + cy2 = 1, where a = c= 1/ sin2 T and b = — cos T / sin2 r. The apparently random spread of points around the elliptical pattern in Fig. 5(a) is due to two factors: (1) the oscillations are chaotic; and (2) there is noise present. The effect of chaos is hidden in the plot of xi(t) versus x2(t + Tmin) because such a plot would be a line along the diagonal for a perfect chaotic lag synchronization 14 . The spread of points about the diagonal in Fig. 5(b) thus signifies the influence of noise. We now ask: what is the influence of noise on lag synchronization? To gain intuition, we consider the case of complete synchronization where Xi(t) = x2(t) (asymptically). To realize synchronization, the synchronization manifold xi(t) = x 2 (t) must be stable with respect to small perturbations transverse to the manifold 4 . Since, however, xi(t) and x 2 (t) are chaotic, there are an infinite number of unstable periodic orbits embedded in the synchronization manifold. Typically, even when

242

(a)

jrtl-

2

j&p*

^p

<6$r^

1

jf

if X

VI 0

Jp

#

-2

•&

t^"

- ^

Wt^^

Figure 5. For C — 0.023 in the unidirectionally coupled Rossler circuits: (a) xi(t) versus x^it), and (b) x\{t) versus X2(t + Tmin). The elliptical pattern in (a) suggests that there is lag synchronization.

the chaotic synchronization state xj(i) = X2(t) is transversely stable, there are unstable periodic orbits embedded in it which are transversely unstable 25 . Thus, when a chaotic trajectory moving in the synchronization manifold falls into the small neighborhood of one of those transversely unstable

243

1 0.8 0.6 0.4

_ X

0.2

-' 0

<

-0.2

-0.4 -0.6 -0.8

~ 0

0.02

0.04

0.06

0.08

0.1

t (Sec)

Figure 6. Intermittent behavior corresponding to the lag synchronization observed for C — 0.023.

periodic orbits, it can be kicked away from the synchronization manifold by noise. This induces a temporal desynchronization between the two oscillators 26 . But since the synchronization manifold is transversely stable, a deviated trajectory will eventually come back to the neighborhood of the synchronization manifold and stays there until it is kicked off again by noise. As a consequence, if one plots the time series Ax(£) s |xi(t) — x 2 (i)|, one typically observes an intermittent behavior: epoches of synchroonization state where Ax(t) as 0 interspersed by bursts for which Ax(£) ^ 0. The key observation about lag synchronization is that it is characterized by the presence of a stable lag-synchronization manifold: Xi (t) = x 2 (t + r ) . A similar argument suggests that noise will cause intermittency in the synchronization state. In particular, we expect the plot of Ax Tmjn (t) to exhibit an intermittent behavior. Figure 6 shows such a behavior observed in experiments for C = 0.023. To assess the parameter range in which lag synchronization occurs, we have performed a large number of experiments to measure the behavior of the coupled circuits at systematically increased values of the coupling parameter C. For each C value, we compute the quantities Smin and r m j n from the similarity function. Figures 7(a) and 7(b) show 5 m j„ and r m j„ versus C, respectively. We see that Smin becomes approximately zero b for C > Cs RJ 0.02, indicating synchronization for C > Cs. However, r m j n does not become zero until C exceeds Cc ss 0.06. For this particular experimental configuration, intermittent lag synchronization thus occurs for 0.02 < C < 0.06. 4-3

Numerical confirmation

We now present numerical confirmation for the experimental observation by using Eq. (2). Rosenblum et al. used CJI^ = 0.97 ± 0.02 to stipulate nonidentity of the two chaotic oscillators 14 . In order to mimic the influence of random disturbances and to maintain nonidentity (in the average sense) between the two oscillators, we choose UJI = 0.99 + Sai^(t) and ui^ = 0.95 + 6crx 2(i), where 5 is the noise amplitude, ffi(t) and o~2(t) are random numbers uniformly distributed in [—1,1]. In b Due to the inevitable random noise such as the thermal noise of the circuit components, small time delay (3/*s) and the finite resolution of the data acquisition device, the plot of xi(t) versus x 0 in the synchronization regime.

244

0

0.05

0.15

0.1 £

X10"3

:

f

PVw 0

0.05

t

0.1

(b)

I

0.15

Figure 7. For the unidirectionally coupled Rossler circuits: (a) Smin versus C, and (b) rmin versus C. We see that lag synchronization occurs in the following parameter range: C £ (0.02,0.06).

numerical experiments, noise is added at each integration step when the differential equations are integrated. Under the influence of noise, the average system mismatch is thus Au> = 0.04 and we vary S from 1 0 - 5 to 10°, a wide range that covers the magnitude of the system mismatch. We find that, at small noise levels, lag synchronization is only temporal and appears in an intermittent fashion, while when the noise level is comparable to Aw, the bursts occur so frequently that lag synchronization disappears practically. To quantify this behavior, we choose 50 noise levels uniformly distributed on a logarithmic scale in [10~5,10°] and for each noise level, we compute the average time interval T between adjacent bursts by setting a threshold Ax = ±0.08. The distributions of the time intervals T are apparently exponential so that T is well defined. Figure 8 shows T versus log10 S. For <5 -C Au, T remains at a constant, indicating an almost unchanged behavior of intermittency at small noise levels. As <5 increases, we see that T drops quickly to zero when 6 exceeds Au. Thus, at larger noise levels, bursts occurs more and more frequent, causing a practical disappearance of lag synchronization. Adding noise to other parameters or to dynamical variables of the system yields similar results. 5

Discussion

In summary, we have investigated experimental manifestations of phase and lag synchronizations in coupled chaotic systems. Our principal results are: (1) Under the influence of noise, indefinite phase synchronization is no longer possible c. Instead, 2ir phase slips between the oscillators occur. When the noise amplitude is small, these phase slips are extremely rare. Thus, we expect to be able to still observe phase synchronization for long time in well-controlled laboratory experiments where noise is small. (2) Lag synchronization can only occur intermittently in laboratory experiments. This is C A dynamical quantity that characterizes the onset of phase synchronization is the Lyapunov spectrum 1 2 . Under the influence of noise, the spectrum as a function of the coupling parameter is typically shifted by an amount that is proportional to the noise amplitude so that a larger coupling strength is required for phase synchronization to occur. There appears, however, no direct correspondence between the Lyapunov spectrum and the super-persistent transient scaling law.

245

12 10 8 lf-i

6 4

average system mismatch

2-

\

0 -4

-3

-2

^«»».^,

log 5 Figure 8. For C — 0.5 in the numerical model, the average time interval between bursts T versus the noise amplitude. The vertical line indicates the amount of the average system mismatch.

true even when the noise is small compared with the amount of mismatch between the systems. At large noise level, lag synchronization is no longer possible and one observes a direct transition to complete synchronization at sufficiently large coupling strength. Intuitively, this can be understood by noting that lag synchronization is a phenomenon which depends on a precise timing between the two chaotic oscillators, but such a timing is usually destroyed by inevitable random factors present in the environment. Our results suggest that one should be cautious when attempting to observe or to utilize lag synchronization in laboratory experiments or in practical systems. Acknowledgments This work was supported by AFOSR under Grant No. F49620-98-1-0400 and by NSF under Grant No. PHY-9722156. References 1. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983). 2. V. S. Afraimovich, N. N. Verichev, and M. I. Rabinovich, Radio Phys. and Quantum electron. 29, 747 (1986). 3. L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990). For a recent review, see the entire Chaos Focus Issue 7 (4), 1997. 4. J. F. Heagy, T. L. Carroll, and L. M. Pecora, Phys. Rev. E 50, 1874 (1994). 5. L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996). 6. L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar, and J. F. Heagy, Chaos 7, 520 (1997). 7. U. Parlitz, L. O. Chua, L. Kocarev, K. S. Halle, and A. Shang, Int. J. Bifurcation and Chaos 2, 973 (1992). 8. K. M. Cuomo and A. V. Oppenheim, Phys. Rev. Lett. 71, 65 (1992). 9. K. M. Cuomo, A. V. Oppenheim, and S. H. Strogatz, Int. J. Bifurcation and Chaos 3, 1629 (1993). 10. K. M. Short, Int. J. Bifurcations and Chaos 4, 957 (1994).

246 11. K. M. Short, Int. J. Bifurcations and Chaos 6, 367 (1996). 12. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996). 13. A. S. Pikovsky, G. Osipov, M. G. Rosenblum, M. Zaks, and J. Kurths, Phys. Rev. Lett. 79, 47 (1997). 14. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 78, 4193 (1997). 15. U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996). 16. S. Taherion and Y.-C. Lai, Phys. Rev. E (Rapid Communications) 59, R6247 (1999). 17. O. E. Rossler, Phys. Lett. A 71, 155 (1979). 18. K. J. Lee, Y. Kwak, and T. K. Lim, Phys. Rev. Lett. 8 1 , 321 (1998) 19. C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 50, 935 (1983); Ergodic Theory Dynam. Syst. 5, 341 (1985). 20. Y.-C. Lai, C. Grebogi, J. A. Yorke, and S. C. Venkataramani, Phys. Rev. Lett. 77, 55 (1996). 21. E. Rosa, E. Ott, and M. H. Hess, Phys. Rev. Lett. 80, 1642 (1998). 22. R. L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1963). 23. T. L. Carroll, Am. J. Phys. 63, 377 (1995). 24. K. Josic, Phys. Rev. Lett. 80, 3053 (1998). 25. Y. Nagai and Y.-C. Lai, Phys. Rev. E 56, 4031 (1997). 26. J. F. Heagy, T. L. Carroll, and L. M. Pecora, Phys. Rev. E 52, R1253 (1995).

Experimental synchronization of chaotic oscillations in two separate Nd:YVO, microchip lasers

A. Uchida, M. Shinozuka, T. Ogawa, and F. Kannari Department of Electrical Engineering, Keio University, 3-1-1-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

ABSTRACT Synchronization of chaotic oscillations generated in two separate Nd:YVOj microchip lasers is experimentally demonstrated with master-slave coupling types. For synchronization of chaos, precise locking of the optical frequency is required by injection locking. The variance of the correlation plot between two laser outputs decreases as the injection power of the master laser is increased under the threshold of injection locking range. Over the threshold, the variance is almost kept to constant. The pump modulation parameters in the slave laser do not always need to match those in the master laser for synchronization.

1.

Introduction

Synchronization of chaos has recently attracted great interest because of its potential for application in secure communications.' , 2 In secure communications that use chaos, messages mixed with a chaotic carrier are transmitted. Then, in the receiver system the messages are recovered from the received chaotic signals by subtraction of a local chaotic carrier that is precisely synchronized with the master chaotic carrier. 3 " 7 Recently, optical encryption systems that use laser chaos were experimentally d e m o n s t r a t e d 8 ' ; in these systems original chaotic oscillations in a transmitter were reproduced at a receiver, not in a synchronized chaotic laser but in a passive nonlinear medium. It is more difficult to realize synchronization of chaos in two separate chaotic lasers because both systems possess positive Lyapunov exponents, which is different from Pecora and Carroll's synchronization scheme.' However, from a security viewpoint, since the tolerable range of mismatch in various system parameters between two chaotic lasers during chaos synchronization can be much smaller than the range that is allowed for passive devices, more-secure encryption systems are attainable. Chaos synchronization in two lasers can be categorized into master-slave (one-way coupling of the laser field) and mutually coupled types. In experimental studies both master-slave-type'" " and mutually-coupled -type 12 ' 5 synchronization have been demonstrated. The chaos-synchronization 247

248 principle is different in the two coupling types. Chaos synchronization for the master-slave type is based on suppression of the original chaos of a slave laser and amplification of the injected master chaos into the slave cavity, whereas chaos synchronization in the mutually coupled type tends to generate a new dynamic system that creates two new chaotic attractors as a result of the mutual interaction of the two lasers. It is worth investigating the characteristics of chaos synchronization of the master-slave type since it is well suited for use in secure communication applications in which one wishes to connect distant chaotic lasers and maintain the original chaotic carriers. Chaotic outputs in microchip solid-state lasers are easily obtained by modulation of the injection current of a laser diode that used for pumping (pump modulation) near sustained relaxation oscillation frequencies of the order of a few megahertz. The relatively low relaxation-oscillation frequencies in microchip solid-state lasers compared with those of laser diodes (gigahertz order) make it easy to observe chaotic waveforms experimentally and to evaluate quantitatively the accuracy of synchronization performance. The order of the relaxation-oscillation frequencies is still sufficient for sending relatively slow signals, such as voice messages, with chaotic carriers. In this paper we experimentally demonstrate synchronization of chaos of the master-slave type with two separated diode-pumped NdrYVO., microchip lasers. To investigate the requirements in the slave laser for accurate synchronization we categorized master-slave synchronization schemes into two types. 1 6 A master-slave type I (MS-1) scheme consists of a chaotic master laser and a chaotic slave laser with a positive Lyapunov exponent. A master-slave type II (MS-2) scheme is composed of a chaotic master laser and a nonchaotic slave laser with a negative Lyapunov exponent.

2. Experiments 2.1.

Experimental Setup Figure 1 shows our experimental setup for chaos synchronization. Each Nd:YV0 4 microchip crystal (wavelength: X = 1064 nm) was 1 mm long and was pumped by a fiber-coupled laser diode (X = 809 nm). Both ends of the crystals were coated with dielectric mirrors (Rl = 9 9 . 8 % at 1064 nm, R2 = 9 9 . 1 % at 1064 nm). The oscillation frequencies of the microchip lasers and of the laser diodes were adjusted by control of their operating temperatures with thermo-electric coolers (resolution, 0.01°C). A fraction of the master-laser output was injected into the slave-laser cavity for chaos synchronization. An optical isolator and a A./2-wave plate were used to achieve one-way coupling from the master to the slave lasers. Chaotic temporal waveforms were detected by photodiodes and digital oscilloscopes. The carrier frequencies of the lasers were measured by an optical spectrum analyzer (resolution, 30 GHz) and a scanning Fabry-Perot interferometer (free spectral range, 7.5 GHz; resolution, 30 MHz).

249

Spectrum analyzer

Temperature control syslem

] Oscilloscope

PM

LI) M C L

Opiical spectrum analyzer

1.

Slave Laser M Z _ u

*

F-P

_MZ. I cmperalure

i E*** ©M control system

Master Laser

CD«

Fahry-Pcrot opiical spectrum analy/cr

—|^—Oscilloscope pr-i

Fig. 1 Experimental setup for chaos synchronization in two Nd:YV0 4 microchip lasers (with MS-1). BS's: beam splitters, L's: lenses, M's: mirrors, VA's: variable attenuators, LD's: laser diodes, MCL's: Nd:YV0 4 microchip lasers, PL's: Peltier devices, IS: isolator, PD's: pholodiodes, FC: fiber coupler, PM's: pump modulations, X/2 WP: A./2-wavo plate, F-P: Fabry-Perot etalon.

2.2. Injection locking Achievement of synchronization of chaos is highly dependent on the injection-locking performance of the slave oscillator. The two laser frequencies are adjusted by control of the temperature of the microchip crystals so that their laser-oscillation frequencies are within the injection-locking range. The laser frequencies of Nd:YV0 4 microchip lasers can be tuned linearly as a function of the temperature of the crystals at a rate of 1.57 G H z / ° C . " When the spectrum peaks of the two lasers are matched on the optical spectrum analyzer and the Fabry-Perot interferometer, we can observe a frequency beat at less than 30 MHz on the digital oscilloscope. When the beat frequency is settled within 2 MHz with more-accurate temperature control, injection locking is achieved. Consequently, the frequency of the slave laser is pulled to that of the master laser, and then the two laser frequencies are perfectly matched. In our experiments the temperatures of the microchip crystals were set to 50.00 and 27.55 °C for the master and the slave laser, respectively. The output powers of the microchip lasers were set to 5.4 and 5.5 mW (measured in front of the slave-laser cavity) for the master and the slave laser, respectively. We found that frequency locking of the sustained relaxation oscillation between the two nonchaotic microchip lasers is also essential for chaos synchronization. I S When the two sustained relaxation-oscillation frequencies were brought close to each other (master, 5.60 MHz; slave, 5.54 MHz) by adjustment of the pumping power of the laser diodes under the locking condition of the carrier frequency, the sustained relaxation-oscillation

250 frequency of the slave laser was pulled to that of the master with these pumping and temperature parameters, amplification master laser was achieved in the slave-laser cavity, and relaxation-oscillation frequency component of the master laser of the slave laser.

laser. Finally, of the seeding the sustained overcame that

2.3. Synchronization of chaos Under the same conditions but with the two lasers isolated, we modulated the injection currents of the laser diodes in the two lasers and adjusted the current modulation to generate similar chaotic spectra from both the lasers. When the pump modulations, which were precisely resonant with the relaxation-oscillation frequency, (master, 5.60 MHz, with a modulation depth of 5 3 % ; slave, 5.44 MHz, 50%) were applied to the two microchip lasers, individual chaotic oscillations were obtained from the two lasers, as shown in Fig. 2(a). Figure 2(b) shows a 5000-point plot of the correlation between the two laser outputs, exhibiting no linear correlation. When a fraction of the master-laser output was injected into the slave-laser cavity, the chaotic oscillations were synchronized, as shown in Fig. 2(c). In contrast with the correlation in Fig. 2(b), a linear correlation between the two laser outputs in Fig. 2(d) shows synchronization. In the present setup this synchronization can be maintained for several hours as long as injection locking of the two laser frequencies is maintained. This locking is destroyed mainly by temperature fluctuation in the lasers which causes slight shifts in the gain spectrum.

Master Intensity [arb units) Master Intensity [arb. units] Fig. 2 Experimentally obtained chaotic temporal waveforms and correlation plots for lhe two laser outputs (a) and (b) without synchronization and (c) and (d) with synchronization.

251

3. Accuracy of synchronization

We tried synchronization of chaos with other coupling types, MS-2 and a mutually coupled type without the optical isolator. We compared the quantitative accuracy of chaos synchronization for the three coupling types by use of the variance of the correlation plot from the best-fit linear relation a2 = 1/N X(I s -I m ) 2 from 15,000 points sampled from the measured temporal waveforms. The laser outputs were normalized by the maximum peak value in each time sequence. We found that the variances for three coupling schemes are closely similar (a2 - 0.0003). It should be noted that the chaotic spectrum obtained by chaos synchronization of the mutually coupled type, however, varies from the original chaotic spectrum of the master laser since new laser dynamics are generated when the two laser oscillators are coupled. The power level of injection of the master laser into the slave laser affects the accuracy of the synchronization. The variances of MS-1 and MS-2 are plotted as functions of the injection power in Fig. 3. It was found that both MS-1 and MS-2 can be used to obtain chaos synchronization with injected master powers in excess of 1.3 mW. Thresholds can be seen for both types, and these thresholds coincide with the thresholds of injection locking.

0.07 0.06

<

i

•

i

i

•

'

• ' • I

; -•-MS1 -^->-MS2

\

0.05

"

-

0.04 CM

b

1

0.03 0.02

-

V

-

&>4-^

z ^ -

^ ^ ^

-n

•

0.01 0 0

1

2

3

4

5

6

Master power [mW] F i g . 3 V a r i a n c e s of t h e c o r r e l a t i o n p l o i s of llie i w o l a s e r o u t p u t s from t h e b e s t - f i t l i n e a r r e l a t i o n as a function of t h e injected p o w e r of t h e m a s t e r l a s e r .

252

Next we investigated the slave-laser conditions required to achieve accurate synchronization in MS-1. The shape of the chaotic spectrum in the slave laser strongly depends on the pump-modulation frequency and amplitude. Figure 4 shows the variance as a function of the pump-modulation frequency and amplitude of the slave laser. The pump-modulation frequency of the master laser was fixed to 1.5 MHz in Fig. 4(a). In Fig. 4(b), the modulation amplitude of the master laser was set to 10%. Variances always kept low even when the modulation frequency and amplitude are shifted from those of the master laser. Therefore, the pump-modulation parameters in the slave laser do not always need to match those in the master laser. The variances levels for MS-1 and MS-2, which corresponds to the MS-1 with no pump modulation in the slave laser, are almost similar at various modulation parameters. Therefore, we speculate that the principle of synchronization of chaos in lasers is simple amplification of chaotic master laser in the slave cavity via injection locking, not synchronization between the original chaotic master and slave lasers.

(b)

(a) 0.01

0.01

0.008

0.008

........... -

1

1

1

1

•

.

1

.

1

.

1

.

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-

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o
I 0.004

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: I

.. • . .

. t . ••• . f . .'. . T . .*. . t . ,'. 0* 0 1 2 3 4 5 Modulation Frequency in Slave [MHz]

•

•

•

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•

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•

•

i

i

0 10 20 30 40 50 Modulation Amplitude in Slave [%]

Fig. 4 Variances as a function of (a) frequency and (b) amplitude of • lie pump modulation in the slave laser with MS-1.

253

4. Conclusion We have experimentally demonstrated synchronization of chaos in two separate diode-pumped Nd:YV0 4 microchip lasers with different master-slave coupling types. Synchronization is accomplished with a certain level of injected power of the master laser, where the threshold of synchronization is coincident with that of the injection-locking range.

5.

Acknowledgements

We gratefully thank Prof. R. Roy, Prof. K. A. Shore, Dr. L. Larger, and Prof. T. Taira for helpful discussions.

6.

References 1. 2. 3. 4.

L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64 (1990) 8 2 1 . K. M. Cuomo and A. V. Oppenheim, Phys. Rev. Lett. 7 1 (1993) 65. P. Colet and R. Roy, Opt. Lett. 19 (1994) 2056. C. R. Mirasso, P. Colet, and P. G.- Fernandez, IEEE Photon. Technol. Lett. 8 (1996) 299. 5. V. Annovazzi-Lodi, S. Donati, and A. Scire, IEEE J. Quantum Electron. 32 (1996) 953. 6. P. M. Alsing, A. Gavrielides, V. Kovanis, R. Roy, and K. S. Thornburg, J r . , Phys. Rev. E 56 (1997) 6302. 7. A. P. Napartovich and A. G. Sukharev, Quantum Electron. 28 (1998) 81. 8. G. D. VanWiggeren and R. Roy, Phys. Rev. Lett. 81 (1998) 3547. 9. L. Larger, J.- P. Goedgebuer and F. Delorme, Phys. Rev. E 57 (1998) 6618. 10. T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu, Phys. Rev. Lett. 72 (1994) 3502. 11. D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, Phys. Rev. E 57 (1998) 5247. 12. R. Roy and K. S. Thornburg, Jr., Phys. Rev. Lett. 72 (1994) 2009. 13. Y. Liu, P. C. de Oliveira, M. B. Danailov, and J. R. Rios Leite, Phys. Rev. A 50 (1994) 3464. 14. D. Y. Tang, R. Dykstra, and N. R. Heckenberg, P h y s . Rev. A 54 (1996) 5317. 15. A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, Phys. Rev. Lett. 78 (1997) 4745. 16. A. Uchida, T. Ogawa, and F. Kannari, Japanese J. Appl. Phys. 37 Part 2 (1998) L730. 17. T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, Opt. Lett. 16 (1991) 1955.

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255 AMPLITUDE DEATH EM COUPLED OPTO-THERMAL OSCDLLATORS R.HERRERO, M.FIGUERAS, J.RIUS, FPL G.ORRIOLS Departament de Fisica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain ABSTRACT The amplitude death phenomenon has been experimentally observed with a pair of nonlinear oscillators based on the thermo-optic effect and on the linearly coupled by heat transfer. A parametric analysis lias been done both experimentally and numerically, pointing out that the death effect occurs for strong couplings and similar input powers. 1. Introduction A large amount of theoretical and numerical works have been devoted to the coupling of similar nonlinear systems with a few number of variables. Systems with two variables are the less complex ones in order to obtain sustained oscillations but their coupling presents complex and unexpected behaviours. A variety of phenomena can already occur in the simple case of two coupled oscillators, although few experimental results have been apported1'2'11. One of the predicted effects is the so called amplitude death, dealing with the absence of oscillations for the whole system while each subsystem oscillates when isolated. This phenomenon is associated with the Hopf bifurcations of the coupled and uncoupled systems. It has been predicted for systems with weak nonlinearities and strong couplings both instantaneous or time delayed'' . Some numerical works have found the amplitude death for identical or slightly different oscillators5 while other authors predict this effect for different enough subsystems6. It has been also pointed out that the oscillatory dynamics of the coupled system is affected by significant changes in the stationary solution5. The amplitude death is also expected for a large amount of coupled oscillators with either global or neighbour couplings. On the other hand, a different way for achieving the steady state in coupled oscillators is the appearance of a saddle-node pair of points on a limit cycle11. We present the experimental observation of the amplitude death phenomenon with two coupled opto-thermal oscillators, located on the same interferometric device and thermally coupled. 2. The BOITAL system The nonlinear system is based on the so-called opto-thermal bistability with localized absorption (BOITAL) and consists of a Fabry-Perot cavity where the input mirror is a partially absorbing film, the rear mirror is a high reflection dielectric coating, and the spacer between mirrors is constituted by N transparent layers with alternativelly opposite thermo-optic effects 7. Nonlinearity is exclusively provided by the interferometer absorption A(v|/), where v|/ is the round-trip phase shift of the cavity. The nonlinear feedback loop when the

256 interferometer is illuminated, involves the light absorption in the input mirror and the heat propagation through the cavity spacer producing refractive index variations and the consequent changes in v|/. Competition and time delay between the optothermal contributions of the various layers to the light phase shift produce instabilities and sustained oscillations. The effective dynamical dimension of the BOITAL system is determined by the number N of layers in the cavity spacer and it can exhibit up to N-l oscillation modes7. Here we consider the two layer case, in which the system can oscillate with a single frequency and its behaviour can be sumarized as following8: Increasing the input power, the system destabilizes at the first stable branch through a Hopf bifurcation generating a limit cycle. This cycle grows approaching to the neighbouring saddle point and a homoclinic orbit occurs when the limit cycle becomes tangent to this point. After this connection the limit cycle is destroyed and the flow escapes towards an upper node branch where a similar behaviour takes place. For a transversally extended optical device, different oscillators can be created by using parallel ligth beams separated by a certain distance. The nonlinear elements are coupled by heat propagation through the cavity spacer and the separation distance may be used to adjust the coupling degree. The oscillators have the same cavity and spacer parameters, although slightly differences in the optical path must be taken into account due to cavity misalignments. Significant differences between the oscillators may be introduced with the incident light powers. This work describes results for two oscillators only. Experiments were performed with a two-layer device made of glass and sunflower oil with thicknesses 400(im and 100|im respectively. In the case of glass, thermal expansion works as a positive effect (10"5 K"1) while oil produces a negative phase-shifting effects due to refractive index changes (-3.4 10"4 K"1). The device is irradiated with a continuous-wave Ar+ laser beam splited in two beams, and each beam focalized to a 50u.m diameter spot on the device. The ligth power of the input beams can be separately adjusted. The BOITAL cavity is thermally stabilized by means of a thermo-electric device to assure no fluctuations on the initial round-trip phase shifts of each oscillator. 3. Experimental results The amplitude death has been observed when the oscillators are separated by short distances, lower than 200p.m. The upper part of Fig. 1 presents an example of the death effect. Both oscillators are illuminated with similar input powers above the Hopf bifurcation of each oscillator but, nevertheless, the system remains in a stable steady state. Blocking anyone of both input beams, oscillations appear in the reflected intensity of the illuminated interferometer while they disappear when both beams are replaced again. The transient to the stable steady state is through anti-phase damped oscillations. In this case, the whole system is near to the Hopf bifurcation since by a slightly increasing the input power sustained oscillations appear corresponding to an anti-phase state. The lower part of Fig. 1 presents a similar behaviour that could be called amplitude semi-death, that takes place for more asymetric input power values of both oscillators. In

257

a

b

a+b

nm\ p a pb (arb.unit.)

111!

11

*

•

•

v

...

. r — — a and b subsystems . • Fig. 1: Time evolutions of the uncoupled and the coupled a+b system showing the amplitude death effect for y=0.49, P„=62.1mW and the amplitude semi-death fory=0.73, Pe=38.3inW.

this case, the system response is a stable steady state when the highest power input beam is blocked and a limit cycle when the lowest power input beam is blocked. The oscillations disappear when both beams are replaced. Both systems of Fig. 1 have almost the same parameters, and only differ on the input power values of both oscillators Pca and P e . Scanning these two parameters, the induced death effects can be localized in the parameter space of the system. The diagram of Fig. 2 shows local and global bifurcations observed when the input power of both oscillators Pea and Peb are varied. Actually, the represented parameters are the total input power Pe=P(."+P<.1' and y=Pc'/Pc. The last parameter relates the input power of both devices, and its values y=0 and y=l indicate the uncoupled subsystems a and b, respectively. Each uncoupled subsystem presents a Hopf bifurcation at different input power values and the sustained oscillations created in them disappear with a homoclinic connection as explained above. The total input power Pe corresponding to the Hopf bifurcation of each uncoupled subsystems can be calculated for each y value, discribing two hyperbolic curves in the diagram. Each one of these curves determine the necessary total input power to achive the Hopf bifurcation of one of both devices for a given y while the input power of the other device is blocked. The bifurcations of the whole system have been obtained scanning P e for each y value. One characteristic of coupled systems is the appearance of new fixed points in the stationary solution5. In the diagram it is pointed out by the appearance of a tongue shaped bistable zone. It is located in the center of the diagram and vertically delimited by two saddle-fiode bifurcation points for each y value. We can distinguish three saddle-node curves designed in the diagram as 1,2,3. The curve pairs 1,2 and 2,3 join together in two codimension-2 cusp bifurcations at y=0.2 and y=0.8, which horizontally confine the bistable region. The curve described by the saddle-node bifurcation points vanishes at y=0.36 and y=0.4. Further from these points, the saddle-node bifurcations also exist but are experimentally hidden by the earlier appearance of a homoclinic bifurcation. Figure 3 shows schemes of the Pe scan for three different y values where the position of the three

258

saddle-node bifurcations are indicated. The same kind of diagrams are numerically observed representing the total phase-shift of one of both coupled oscillators. In figure 2, the oscillation zone is divided in two balloons, each one limited below by a Hopf bifurcation. At both sides of the diagram, these oscillatory ranges are above limited by a homoclinic bifijrcation. On the countrary, at the center of the diagram, the Pe intervals exhibiting oscillations become smaller, any homoclinic connection appears and the sustained oscillations disappear with an inverse Hopf bifurcation (Fig. 3). Note that both oscillatory ranges can appear for the same y value, belonging in this case to different stationary branches. The inverse Hopf bifurcation is necessary to close the oscillatory range of the stationary branch, and in the diagram this range must vanish before the cusp bifijrcation that marks the branch disappearance. Thus, the creation of an intermediate branch originated by the coupling induces the partition of the ©dilating area in two balloons and the consequent reduction of this area, making possible the amplitude death interval. In this way, the amplitude death and the tendence of a more complex stationary solution are two effects of coupled oscillators that seem to be related. For larger enough 150 —i

P(mW)

100

Fig. 2: Pe-y space diagram for two BOITAL oscillators with the parameters already given in the text and separated 200}im. + indicates Hopf bifurcations, A Saddle-Node bifurcations and a homoclinic bifurcations. Bright grey denotes the amplitude semi-death zone and the dark grey the amplitude death zone.

259

distances between both oscillators both balloons are glued in a singular interval, no additional branches appear and any amplitude death zone takes place. The amplitude death effect can be now localized in figure 2 as the non oscillating zones above both hyperbolic curves, and the amplitude semi-death effect as the non oscillating zones above only one hyperbolic curve. In this way, these regions are not only delimited by the Hopf bifurcation points but also by the saddle-node bifurcations where stationary branches disappear. An easier view of both effects can be seen in figure 3 where the death and semi-death intervals are localized. Figure 3c shows the input power scan for the same y value of figure 1. The creation due to the coupling of additional stationary branches, makes possible multistable situations and particularly the bistability of the amplitude death state with other system states 5-6. This bistability is experimentally presented in figure 2 by the intersenction between the amplitude death region and the tongue shape bistable zone. It presents two different types of regimes: the a zone presents two bistable amplitude death states, while in the P zones the amplitude death state appears bistable with oscillating states. Both bistabilities have been predicted with ODE's models without time delays5'6. 4. The model and numerical results Under some simplifying assumptions, the PDE system discribing the BOITAL cavity can be reduced to a Nth-order ODE model involving the partial phase shifts \\i; of each layer as variables9:

-^f-=-lb,J[¥,-a,A(¥)¥El

./= 1,2,..., N,

N

with i// = i//0 + £ V j where v|/0 is the round-trip phase shift in the absence of laser heating and v|/E represents the incident light intensity normalized in such a way that Zai=l. The by and ai coefficients are function of the physical parameters and their exact form is given in ref.9. The only nonlinear function of the system A(v|/) describes the input mirror absorption by including interference effects and its expression is a function of the mirror parameters7. The response of the system is given by the interferometric reflection R(\|/). y=0.38

' P,(mW)

y=0.49

+

A:+:+- A

+

A

™.P,(mW) pop

P,(mW) death interval semi-death interval

Fig.3: Schemes of the Pe scan projected to a characterizing variable 9 for three different y values. The Hopf bifurcations of the uncoupled oscillators are also ploted. Thick horizontal lines denote the death and semideath intervals.

260 Two BOITAL oscillators coupled by thermal diffusivity can be easily approximated by two sets of the former equations and coupling terms with time delays, which takes into account the finite speed of heat propagation. The strength of the couplings will depend on the temperature differences between the spacer layers. i|/j is proportional to the average temperature of the given layer and then the coupling terms can be written as differences of v|/ values. The following model only considers coupling terms associated to the heat transfer from the first layer of each device. This point is justified by the proximity of the first layer to the absorber and the much larger absolut values of temperature in this layer. Denoting the oscillators by superindex a and b, the ODE model can be written as = b

- nM / i° — t>12v)/2a +G,A(v|/ a +v|/»)v)/* +c,v|/, ( t - x , ) - ( c , +c 2 )v|/,'

Vi

V2 = " b 2iVi" " b22v}/2a + G 2 A(y" + v j ) ^ + c 2 ^-v|/, b (t - x 2 ) Til b

Vi

b12v|/2 +G,A(v|/ +v|/ 0 )v l / e +c l v l /, ( t - x , ) - ( c , + c 2 ) n / ,

=-b,,Vi b

b

b

b

b

b

a

v|/2 =-b 2 ,v ) ' 1 °-t)-b„v|/, +G 2 A( M / + M / ) H / e +c 2 ^M' 1 (t-T 2 ) 2 2 v|/ 2 Til

where G, = ^ b-a, and the coupling coefficients Ci have the form c, =f K,/d 2 and c 2 =f K,/(d2 +g 2 ) where f is a constant factor to express the heat transfer in the form of phase variations. The coupling terms describe the heat diffusion from the two first layers,

a+b

AAAAAAAAAA/L

20

0.2

Fig. 4: Numerical simulation of the fig.2 diagram with adimensional spacer parameters: gi=l/0.25, V(=l/-10, Ki=l/0.1, Dj= 1/0.1, hF=hB=0.5 and d=0.4. Bifurcations and death zones are denoted by the same symbols of fig.2. Time evolutions for the uncoupled a,b and coupled a+b systems for a set of parameters into the amplitude death zone.

261 and both time delays present different values due to the different distance between the implied layers, x , = d 2 / D , and t 2 =(d 2 + g 2 ) / D , . Using parameters similar to the experimental ones, Hopf and saddle-node bifurcations, have been analitically located and the homoclinic bifurcations numerically determined in the same parameter space of fig.2. All the parameters are considered to be equal in both oscillators, aside of the input intensity and the initial phase-shift. The bistable and amplitude death zones appear smaller than in the experimental case, but both diagrams qualitatively follow the same structure. Figure 4 also shows time evolutions of both reflected powers for the whole system and both uncoupled subsystems for a parameter set included in the amplitude death interval. Former works have been discused the necessity of time delays and differences between both subsystems in order to obtain the amplitude death effect. Works using the Hopf bifurcation normal form with additional coupling terms manifest the lack of the amplitude death effect when identical oscillators and no time delays are considered6'10. However, they predict this effect for coupled oscillators only differing on their frequencies and also for identical oscillators but time delayed equations4. All these results contrast with works dealing with more complex models used to describe real systems, where the amplitude death effect is possible without time delays and for identical subsystems7. In our case, frequencies remain very similar for both BOITAL subsystems with the same parameters but different initial phase-shifts. For instance, in the diagram of Fig.4 their values are co=3.693 (y=0) and co=3.697 (y=l). Time delays can be removed in the model used here, considering instantaneous heat transfers between both devices but only slightly variations are observed in the parameter space still appearing the amplitude death area. The death effect also appears in simulations considering identical subsystems. Moreover, the same qualitative results have been obtained with a more simple model considering only one time delay in the equations. 5. References 1. E.Sismondo, Science 249 (1990) 55. 2. Elson, A.I. Selverston, R.Huerta, N.F. Rulkov, M.I. Rabinovich, H.D.I. Abarbanel, Phys. Rev. Lett. 81 (1998) 5692. 3 S.H.Strogatz, Nature 394 (1998) 316. 4. D.V. RamanaReddy, A. Sen, G.L. Johnston Phys.Rev.Lett. 80 (1998) 5109. 5. K.Bar-Eli, Physica 14D (1985) 242. 6. G.B. Ermentrout, Physica D 41 (1990) 219. 7. J.I.Rosell, J.Farjas,R.Herrero, F.Pi, G.Orriols, Physica D 85 (1995) 509 8. J.I.Rosell, F.Pi, F.Boixader, R.Herrero, J.Farjas, G.Orriols,Optics comm.82(1991) 162. 9. J.Farjas, J.I.Rosell, R.Herrero, R.Pons, F.Pi, G.Orriols, Physica D 95 (1996) 107 10. D.G.Aronson, G.B.Ermentrout, N.Kopell, Physica D 41 (1990) 403. 11. M.Crowley, I.Epstein, J.Phys.Chem. 93 (1989) 2496. 12. P.C.Matthews, S.H.Strogatz, Phys.Rev.Lett. 65 (1990) 1701.

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VIII. Banquet Talk

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Case Study in "Experimental Complexity" — An Artificial-Life Approach to Modeling Warfare Andy Ilachinski Center for Naval Analyses Alexandria, VA 22302 [email protected] http://www.cna.org/isaac/ Artificial-life techniques—specifically, agent-based models and evolutionary learning algorithms-provide a potentially powerful new approach to understanding some of the fundamental processes of combat. This paper takes a step toward this goal by introducing two simple artificial-like "toy models" of land combat called ISAAC and EINSTein. These models are designed to illustrate how certain aspects of land combat can be viewed as emergent phenomena resultingfromthe collective, nonlinear, decentralized interactions among notional combatants. Their bottom-up, synthesist approach to the modeling of combat stands in stark contrast to the more traditional top-down, or reductionist approach taken by most conventional models, and represents a preliminary step toward developing a complex systems theoretic analyst's toolbox for identifying, exploring, and possibly exploiting self-organized emergent collective patterns of behavior on the battlefield. "War is... not the action ofa livingforce upon lifeless mass... but always the collision of two living forces." — Carl von Clausewitz

BACKGROUND In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations—now commonly called the Lanchester Equations (LEs)--as models of attrition in modern warfare (Lanchester, 1995). Similar ideas were proposed around that time by Chase (1902) and Osipov (1995). These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations (Hofbauer and Sigmund, 1998). The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. Taylor (1983) provides a thorough mathematical discussion. LEs are very intuitive and therefore easy to apply. For the simplest case of directed fire, for example, they embody the intuitive idea that one side's attrition rate is proportional to the opposing side's size. However, LEs are applicable only under a strict set of assumptions, such as having homogeneous forces that are continually engaged in combat, firing rates that are independent of opposing force levels and are constant in time, and units that are always aware of the position and condition of all opposing units. LEs also contain a number of significant shortcomings, including modeling combat as a deterministic process, requiring knowledge of "attrition-rate coefficients" (the values of which are, in practice, very difficult to obtain), inability to account for any suppressive effects of weapons, and failure to account for terrain. LEs are unable to account for any spatial variation of forces; no link is established, for example, between movement and attrition. Moreover, they do not incorporate the human factor in combat; i.e., the psychological and/or decision-making capability of the human combatant. 265

266 While the LEs are particularly relevant for the kind of static trench warfare and artillery duels that characterized most of World War 1, they are too simple and lack the spatial degrees of freedom to realistically model modern combat. The fundamental problem is that they idealize combat much in the same way as Newton's laws idealize physics. There have, of course, been many extensions to, and generalizations of, the LEs over the years, all designed to minimize the deficiencies inherent in their original formulation (including reformulations as stochastic differential equations and partial differential equations). However, most existing models remain essentially Lanchesterian in nature, the driving factor being forceon-force attrition. We believe that these models of land warfare are insufficient for assessing the advanced warfighting concepts being explored by the Marine Corps. In particular, the Lanchesterian view of combat does not adequately represent the Marine Corps' vision of combat: small, highly trained, well-armed autonomous teams working in concert, continually adapting to changing conditions and environments. To address these shortcomings, we are exploring developments in complex systems theory—particularly the set of agent-based modeling (ABM) and simulation tools developed in the artificial life community (Langton, 1995)—as a means of understanding land warfare in a fundamentally different way. Land Combat as a Complex Adaptive System? In recent years there has been a rapid growth in the study of complex adaptive systems; see Cowan, et al. (1994), Gutowitz (1990), Kauffman (1993), Langton (1995), Mainzer (1994) and Maes (1990). A complex system can be thought of, generically, as a dynamical system composed of many nonlinearly interacting parts. Much of complexity is rooted in the fundamental belief that much of the overall behavior of ostensibly diverse complex systems (natural ecologies, economic webs, immune systems, etc) in fact stems from the same basic set of underlying principles. The motivation for this study is to explore alternative non-Lanchestrian descriptions of combat. The central thesis (as developed in Ilachinski, 1996a and 1996b) is that land combat can be modeled as a complex adaptive system. Military conflicts, particularly land combat, possess the characteristic features of complex adaptive systems: combat forces are composed of a large number of nonlinearly interacting parts and are organized in a command and control hierarchy; local action, which often appears disordered, induces long-range order (i.e., combat is self-organized); military conflicts, by their nature, proceed far from equilibrium; military forces, in order to survive, must continually adapt to a changing combat environment; there is no master "voice" that dictates the actions of each and every combatant (i.e., battlefield action effectively proceeds according to a decentralized control); and so on. There have also recently appeared a number of papers discussing the fundamental role that nonlinearity plays in combat; see Beyercehn (1992), Hedgepeth (1993), Miller and Sulcoski (1995), Saperstein (1995) and Tagarev and Nicholls (1996). The general approach of this study is to extend these largely conceptual and general links that have been drawn between properties of land warfare and properties of complex systems into a set of practical connections. ISAAC, and its follow-on, EINSTein, were developed to address the basic question: "To what extent is land combat a self-organized emergent phenomenon?" As such, their intended use is not as full system-level models of combat but as interactive toolboxes (or "conceptual playgrounds") in which to explore high-level emergent behaviors arising from various low-level (i.e., individual combatant and squad-level) interaction rules. The idea behind developing these toolboxes is

267 emphatically not to model in detail a specific piece of hardware (Ml 6 rifle, Ml 01 105mm howitzer, etc.). Instead, the idea is to explore the middle ground between-at one extreme-highly realistic models that provide little insight into basic processes and—at the other extreme—ultraminimalist models that strip away all but the simplest dynamical variables and leave out the most interesting real behavior; i.e. to explore the fundamental behavioral tradeoffs among a large number of notional variables. ISAAC ISAAC (Irreducible Semi-Autonomous Adaptive Combat) was developed as a "proof-ofconcept," the concept being that land combat can be modeled as a complex adaptive system; see Uachinski (1997). The dynamics underlying this model is patterned after mobile cellular automata (CA) rules, and are somewhat reminiscent of Braitenberg's Vehicles (Braitenberg, 1984). Specifically, ISAAC takes a bottom-up, synthesist approach to the modeling of combat, rather than the more traditional top-down, or reductionist approach. Mobile cellular automata have been used before to model predator-prey interactions in natural ecologies (Boccara, et al., 1994). They have also been applied to combat modeling (Woodcock, et al., 1988), but in a much more limited fashion than the one ultimately envisioned for this study. More recently ABMs have been applied successfully to traffic pattern analysis (Barrett, 1997) and social evolution (Epstein and Axtell, 1996, Gilbert and Conte, 1995, and Prietula, et al., 1998). The long-term goal is for the follow-on to ISAAC (see EINSTein below) to become a fully developed complex systems theoretic analyst's toolbox for identifying, exploring, and possibly exploiting emergent collective patterns of behavior on the battlefield. Models based on differential equations homogenize the properties of entire populations and ignore the spatial component altogether. Partial differential equations—by introducing a physical space to account for troop movement—fare somewhat better, but still treat the agent population as a continuum. In contrast, ISAAC consists of a discrete heterogeneous set of spatially distributed individual agents (i.e., combatants), each of which has its own characteristic properties and rules of behavior. These properties can also change (i.e., adapt) as an individual agent evolves in time. In ABMs, the final outcome of a battle—as defined, say, by measuring the surviving force strengths—takes second stage to exploring how two forces might co-evolve during combat. Such models are designed to allow the user to explore the evolving patterns of macroscopic behavior that result from the collective interactions of individual agents, as well as the feedback that these patterns might have on the rules governing the individual agents' behavior. ISAAC Agents The basic element of ISAAC is an ISAAC Agent (or ISAACA), which represents a primitive combat unit (infantryman, tank, transport vehicle, etc.) that is equipped with the following characteristics: • • • •

Doctrine: a default local-rule set specifying how to act in a generic environment Mission: goals directing behavior Situational Awareness: sensors generating an internal map of environment Adaptability: an internal mechanism to alter behavior and/or rules.

268 The putative "combat battlefield" is represented in ISAAC by a two-dimensional lattice of discrete sites. Each site of the lattice may be occupied by one of two kinds of agents: red or blue. The initial state consists of either user-specified formations of red and blue agents positioned at diagonally opposite corners of the battlefield or of a random distribution of red and blue agents occupying the central square region (of user-specified dimension). Red and blue "flags" are also typically (but not always) positioned in diagonally opposite corners: a red flag in the red corner and a blue flag in the blue corner. A typical "goal," for both red and blue agents, is to successfully reach the "flag" positioned in the diagonally opposite coiner. ISAAC also has the provision of defining a notional terrain. Each agent exists in one of three states: alive, injured, or killed. Injured agents can (but are not required to) have different personalities and offensive/defensive characteristics from when they were alive. By default, an injured agent's ability to shoot an enemy is equal to 1/2 of its ability when alive. Up to ten distinct groups of personalities, of varying sizes, can be defined. Each agent has associated with it a set of ranges (sensor range, fire range, communications range, etc.), within which it senses and assimilates simple forms of local information (see below), and a personality, which determines the general manner in which it responds to its environment. ISAAC "Personalities" Each agent is equipped with a user-specified personality-or internal value .system—defined by a six-component personality weight vector, ,, (O2,..., «k), where -1 < (Q < 1 and Ej |coi| = 1. The components of the personality weight vector specify how an individual agent responds to distinct kinds of local information within its sensor and threshold ranges. The personality weight vector may be state-dependent. That is to say, a w need not, in general, be equal to (Qnjured. The components of co can be also negative, in which case they signify a propensity for moving awayfrom,rather than toward, a given entity. The default personality rule structure is defined as follows. Since there are two kinds of agents (red and blue), and each agent can exist in one of two states (alive and injured), each agent can respond to effectively four different kinds of information appearing within its sensor range rs: (1) the number of alive friendly (i.e., like-colored) agents; (2) the number of alive enemy (i.e., different colored) agents; (3) the number of injured friendly agents; and (4) the number of injured enemy agents. Additionally, each agent can respond to how far it is from both its own (like-colored) "flag" and its enemy's "flag." The i* component of
269 o»= (-1/6,-1/6,-1/6,-l/6,-l/6,-l/6)--wants to move away from, rather than toward, every other agent and both flags; i.e. it wants to avoid action of any kind. Move Selection An agent's personality weight vector is used to rank each possible move according to a penalty function. The penalty function effectively measures the total distance that the agent will be from other agents (which includes both friendly and enemy agents) and from its own and enemy flags, each weighted according to the appropriate component of the personality weight vector, OJ. An agent moves to the position that incurs the least penalty; i.e., an agent's move is the one that best satisfies it's personality-driven desire to "move closer to" or "farther away from" other agent's in given states and either of the two flags. A "penalty" is computed for each possible move: Zi, Z2,..., Zfl. If the movement range r M =l, N=9; if rM=2, N=25. The actual move is the one that incurs the least penalty. If there is a set of moves (consisting of more than one possible move) that incur exactly the same minimum penalty, an ISAAC randomly selects the actual move from among the candidate moves making up that set. Mcta-Rules An agent's default personality may be augmented by a set of meto-rules that tell it how to alter its default personality according to various environmental conditions and contexts. The three simplest meta-rule classes effectively define the local conditions under which an agent is allowed to advance toward enemy flag (class 1), cluster with friendly forces (class 2), and engage the enemy in combat (class 3). For example, a class-1 meto-rule prevents an agent from advancing toward the enemy flag unless it is locally surrounded by some threshold number of friendly agents. A class-2 meto-rule can be used to prevent an agent from moving toward friendly agents once it is surrounded by a threshold number. Finally, a class-3 meto-rule can be used to fix the local conditions under which an agent is allowed to move toward or away from possibly engaging an enemy agent in combat Specifically, an agent is allowed to engage an enemy if and only if the difference between friendly and enemy force strengths — locally — exceeds a given threshold. Other meta-rule classes include retreat, pursuit, support and hold position. A global rule set determines combat attrition (see below), communication, reconstitution, and (in future versions) reinforcement. ISAAC also contains both local and global commanders, each of which is equipped with its own unique command-personality and area of responsibility, and obeys an evolving command and control hierarchy of rules (Ilachinski, 1997). Combat Combat is adjudicated in the simplest possible manner. During the combat phase of an iteration step for the whole system, each agent X (on either side) is given an opportunity to "fire" at all enemy agents Y that are within a fire range rF of X's position. If an agent is shot by an enemy agent, its current state is degraded either from alive to injured or from injured to dead. Once "dead," that agent is permanently removed from further play. The probability that a given enemy agent is "shot" is fixed by user specified single-shot probabilities for red-by-blue and blue-by-red.

270 By default, all enemy agents within a given agent's fire range are targeted for a possible hit. However, the user has the option of limiting the number of simultaneously engageable enemy targets. If this option is selected, and the number of enemy agents within an agent's fire-range exceeds a user-defined threshold number (say N), then N agents are randomly chosen from among the agents in this set. This basic combat "logic" may be enhanced by three additional functions: (1) Defense, which adds a notional ability to agents to be able to withstand a greater number of "hits" before having their state degraded, (2) Reconstitution, which adds a provision for previously injured agents to be reconstituted to their alive state, and (3) Fratricide ("friendly fire"), which adds an element of realism to ISAAC combat by making it possible to inadvertently "hit" friendly forces. EINSTein includes an enhanced combat adjudication engine that includes weapon and ammunition selection, suppression effects, lethality contours, and individual- and squad-based targeting strategies. Run Modes ISAAC is designed to allow the user to explore the evolving patterns of macroscopic behavior that result from the collective interactions of individual agents, as well as the feedback that these patterns might have on the rules governing the individual agents' behavior. ISAAC can be run in three different modes: (1) Interactive Mode, in which ISAAC'S core engine is run interactively using a fixed set of rules. This mode, which allows the user to make 'on-the-fly' changes to the values of any (or all) parameters defining a given run (including the "decision-making personality" of individual agents), is particularly well suited for quickly and easily playing simple "What if?" scenarios. (2) Data-Collection Mode, in which the user can generate time series of various changing quantities describing the step-by-step evolution of a battle (including averaging over multiple runs of a single scenario) and/or keep track of certain measures of how well mission objectives are met at a battle's conclusion. (3) Genetic Algorithm "Breeder" Mode, in which a genetic algorithm is used to breed a personality for one side that tailored for performing some well-defined mission against a fixed personality or the other. This last mode illustrates how programs such as this can eventually be used to evolve real-world tactics and strategies. Sample Behavior Despite its simple local rule base, ISAAC has an impressive repertoire of emergent collective behaviors: forward advance, frontal attack, local clustering, penetration, retreat, attack posturing, containment, flanking maneuvers, "Guerrilla-like" assaults, among many others. Moreover, behaviors frequently arise that appear to involve some form of "intelligent" division of red and blue forces to deal with local "firestorms" and skirmishes, particularly those forces whose personalities have been "evolved" (via a genetic algorithm) to perform a specific mission. It is important to point out that such behaviors are not hard-wired but rather an emergent property of a, decentralized and nonlinear local dynamics. A small sampling of behaviors is shown below. Sample # 1: Fluid-like "Collisions" Figure 1 shows several frames of a simulated engagement that is reminiscent of a collision between two viscous fluids. The initial state (not shown) consists of 90 red and 90 blue agents

271 occupying random positions within 20-by-20 squares near the lower left and upper right regions of an 80-by-80 lattice, respectively.

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The blue force has a personality defined by the weight vector Wbiuc = (1/15, 4/15, 1/15, 4/15, 0, 1/3). Each of the red agents is assigned a random personality (which is constrained only in giving zero weight to moving toward the red flag located in the lower left corner of the battlefield). Red and blue forces collide head-on but are dispersed and aligned along two narrow columns at time t=50. The two sides continue battling each other in this manner, with neither side gaining an advantage or advancing closer to its enemy's. By time t=75, several blue agents have "found" a way to sneak around the bottom of red's column-like formation. This group is able to advance toward red's flag unchallenged because it is unseen by the red agents making up red's central column. By time t=125, a cluster of red agents breaks away from what used to be the central column and advances towards blue's flag. Meanwhile, blue forces continue advancing toward red's flag at the bottom of the frame.

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272 Sample # 2: Red "Encircles" Blue Figure 2 shows snapshots of an evolution in which red effectively "encircles" blue agents. An interesting question to ask is "How should blue alter its personality (i.e., its "tactics") — during the course of the battle — in order to prevent being encircled by red forces?" While ISAAC can be used to explore the behavioral consequences of matching alternative fixed blue personalities against the same red force, it does not have the flexibility to explore the consequences of a dynamically changing personality during a given run. EINSTein's updated dynamic engine uses a more generalfinite-state-machine.Current design plans also include realtime adaptive learning (see EINSTein below). Sample # 3: Non-Monotonic Behavior Figure 3 shows an instructive example of non-monotonic behavior. The three rows contain snapshots of three separate runs in which red's sensor range is systematically increased in increments of two: rs.rai = 5 for the top sequence; rs.ra
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In each of the runs, there are 100 red and 50 blue agents. Red is also the more the aggressive force: blue will engage in combat if the number of friendly and enemy agents is locally about even, while red will do so even if outnumbered by four enemy combatants. Both sides are endowed with the same fire range (rF = 4), the same single-shot probability (p = 0.005) and can simultaneously engage the same maximum of 3 enemy targets. Note that the flags for this sample run are located near the middle of the left and right edges of the notional battlefield.

273 The top row of figure 3 shows screenshots for when red's sensor range is equal to blue's. Here the red force effectively "barrels" its way through the blue defenses into blue's flag. As red advances toward blue's flag, a number of agents are "stripped" away from the central redflag:blue-flag axis as they respond to the presence of nearby blue agents. The snapshots for the middle row of figure 3 show that when red's sensor range is two units greater than blue's, red is not only able to mass almost all of its forces on the blue flag (a later snapshot would reveal blue's flag completely enveloped by red forces by time t= 100), but to defend its own flag from all blue forces as well. In this instance, the red force knows enough about, and can respond quickly enough to enemy action such that it is able to march into enemy territory effectively unhindered by enemy forces and "scoop up" blue agents as they are encountered. What happens when red's sensor range is increased still further? One might intuitively guess that red can only do at least as well; certainly no worse. However, as the snapshots for bottom row of figure 3 reveal, when red's sensor range is increased to rs,red = 9, red does objectively worse than it did in any of the preceding runs. "Worse" here means that red is less effective in (a) establishing a presence near the blue flag, and (b) defending blue's advance toward the red flag. This example illustrates that when the resources and personalities of both sides remain fixed in a conflict, how "well" side X does over side Y does not necessarily scale monotonically with X's sensor capability. As one side is forced to assimilate more and more information (with increasing sensor range), there will inevitably come a point where the available resources will be spread too thin and the overall fighting ability will therefore be curtailed. ABMs such as ISAAC are well suited for providing insights into more operationally significant questions such as How must X's resources and/or tactics (i.e., personality) be altered in order to ensure at least the same level of mission performance? Sample # 4: An "Autopoietic" Skirmish Figure 4 shows a sample evolution reminiscent of an autopoietic system (Maturana and Varela, 1974). Autopoiesis means, literally, self-creation ("from the Greek auto=self and poiesis = creation/production) and refers to a process that both defines and sustains itself. A well-known, and provocative, example is Jupiter's Great Red-Spot.

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274 In this sample run, a cluster of red and blue agents undergoing intense combat self-organizes near the center of the battlefield and persists for a very long lime despite agents continually flowing in and out of it. This cluster also appears driven by an emergent dynamics in which it first rotates counterclockwise and drifts slowly toward the red goal, comes to a stop, reverses direction and moves toward the blue flag, then reverses direction once again and moves back toward the red flag before finally disintegrating. Many interesting questions naturally suggest themselves: What are the conditions for an such an autopoietic structure to emerge? How stable is this structure? Are there conditions under which the one structure splits into two or more autopoietic structures? etc... Sample # 5: Genetic-Algorithm-Bred "Tactics" Genetic algorithms (GAs) are a class of heuristic search methods and computational models of adaptation and evolution based on natural selection. GAs mimic and exploit the genetic dynamics underlying natural evolution to search for optimal solutions of general combinatorial optimization problems. They have been applied to the Traveling Salesman Problem, VLSI circuit layout, gas pipeline control, the parametric design of aircraft, neural net architecture, models of international security, and strategy formulation. An excellent recent overview of GAs is given by Mitchell (1996). Figure 5 shows an example of a red personality that has been "bred" to perform a specific mission using a GA. The parameters defining the blue side arc fixed (except for initial spatial disposition, which is averaged over during the run), and the GA is used to find the "best" red personality that performs the following mission: get as many red forces near blue's flag as possible while minimizing red casualties. There are 70 agents on each side and the blue force is stationed in a semi-circle 25 units away from its flag in the upper right corner of the battlefield.

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275 Red's "tactic"—which derives solely from the personality found by the GA to be best "suited" for this mission—is to exploit a few red agents at the front of the advancing red force, using them to split apart blue's forces in order to temporarily "weaken" the center region of blue's defense. As soon as this center region is sufficiently weakened, red quickly penetrates through to the blue flag. (This particular personality was bred using a pool of 75 red personalities and averaging over 25 random initial conditions for each generation; for details see llachinski, 1997). What is most surprising about many of the runs using GA-derived personalities is that the red force appears to task different agents with different missions, despite the fact that each red agent is endowed with exactly the same personality! Thus, a higher-level tactic—such as use the two forward positioned agents to weaken the enemy's center-emerges out of the collective interactions of the same low-level decision rules; i.e., an apparently centralized order induced by decentralized local dynamics. Sample # 6: Mission-Fitness Landscapes Figure 6 shows an example of ISAAC'S built-in ability to sample a 2D "slice" of what is ostensibly a very large N-dimensional parameter space. The scenario is one that probes red's offensive capability. In this scenario, the blue force consists of 50 randomly wandering agents that attack all enemy agents within their fire range, and red's mission objective is to maximize the number of red agents near the blue flag. The fitness function,/,, is defined so that red's fitness ~ 1 only when all red agents get to within a distance d=10 of the blue flag within a specified elapsed time.

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Figure 6 shows a three-dimensional plot of the fitness 0 < fm (x,y) < 1 as a function of (x,y), averaged over 50 initial states. The number of red agents (= N„d) is the x-coordinate of the 2D "slice" (which ranges from 5 to 35) and the combat aggressiveness (= Yci see Afeta-Rules above) as the y-coordinate (which ranges from -20 to +20). Figure 6 shows that the red force's ability to perform its mission is a strong function of the x and y parameters chosen for this plot. In particular, red's overall mission fitness f~ 0 unless the red force is relatively small (N„d < 20) and yc< 0. Maximum mission fitness is achieved near Hd = 10 and ye"-10.

276

The "experiment" at me far right of figure 6 shows the benefit of adding a communications capability to the red force: a region of relatively low fitness for a non-communicating mediumsized force (Nred » 20) can be increased to effectively match that of the previous fitness maximum by allowing red force agents to communicate. In this simple example, the dynamics are easy to understand. A force that is too large and/or too timid (i.e. with a large, positive y) will tend to disperse more and thus be less successful in seizing the blue flag. In addition, allowing agents to communicate fosters a tighter red formation that can more easily defend itself as it marches toward the enemy flag. For more complex scenarios, where the number (and kinds) of local decisions that are made by each agent may be very involved, mission-fitness landscapes such as the one shown in figure 6 are indispensable for gaining useful insights into how all of the different parameters "fit together" to shape the overall flow of battle. PAYOFFS TO USING MULTIAGENT BASED MODELS Most traditional models focus on looking for equilibrium "solutions" among some set of (predefined) aggregate variables. The LEs are effectively mean-field equations, in which certain variables (i.e. attrition rate) are assumed to represent an entire force, the equilibrium state is explicitly solved for and declared the "battle outcome." In contrast, ABMs focus on understanding the kinds of emergent patterns that might arise while the overall system is out of (or far from) equilibrium. The payoff of this multiagent-based approach is a radically new (and decidedly nonLanchesterian) way of looking at fundamental issues of land combat. Models such as these are emphatically not to be used for prediction; rather they are best used to enhance understanding. Specifically, ABMs help analysts... •

• •

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Understand how all of the different elements of combat fit together in an overall "combat phase space:" Are there regions that are "sensitive" to small perturbations, and, if so, might there be a way to exploit this in combat (as in selectively driving an opponent into more sensitive regions of phase space)? Assess the value of information: How can I exploit what I know the enemy does not know about me? Explore tradeoffs between centralized and decentralized command-and-control (C2) structures: Are some C2 topologies more conducive to information flow and attainment of mission objectives than others? What do emergent forms of a self-organized C2 topology look like? Provide a natural arena in which to explore consequences of various qualitative characteristics of combat (unit cohesion, morale, leadership, etc.) Explore emergent properties and/or other "novel" behaviors arising from low-level rules (even combat doctrine if it is well encoded): Are there any universal patterns of combat behavior? Provide clues about how near-real-time tactical decision aids may eventually be developed using "natural selection" (via genetic algorithms).

277 •

Address questions such as "How do two sides of a conflict co-evolve with one another?" and "Can one side exploit what it knows of this co-evolutionary process to compel the other side to remain out of'equilibrium1?"

ABMs provide a natural arena in which to explore the Clausewitzian "fog-of-war," including the effects of uncertainties and/or inaccuracies of intelligence data and of time-delays in reporting information. More important, from an Information Warfare perspective, ABMs provide a framework for quantifying the "value" of information on a battlefield. ISAAC can, for example, be used to explore the consequences of given (personality-defined) force and/or weapon mixes. It can also be used to re-examine traditional measures of combat effectiveness and define requirements for what might loosely be called nonlinear data collection, which refers to data that capture the continuously evolving relationships among all of the interdependent components of combat (as compared with more static measures-such as force attrition—commonly used by conventional models). ElNSTein To facilitate the examination of these issues, ISAAC is currently being enhanced in several ways. Preliminary work has started on developing an ambitious follow-on to ISAAC, called ElNSTein (Enhanced ISAAC Neural Simulation Tool). An alpha test version of ElNSTein that runs under windows 95/98, may be downloaded from http://www.cna.org/isaac/einstein.htm. ISAAC was developed as a simple "proof-of-concept" model to illustrate how combat can be viewed as an emergent self-organized dynamical process. It introduced the key idea of building combat "up from the ground up" by using complex adaptive agents as primitive combatants and focusing on the global co-evolutionary patterns of behaviors that emerge from the collective nonlinear local dynamics of these primitive agents. ElNSTein builds upon and extends ISAAC into a bona-fide research tool for exploring selforganized emergent behavior in combat. Some of the features planned for ElNSTein include: (1) a fully integrated Windows 95/98 GUI front-end (see figure 7); (2) an object-oriented C++ code base (compared to ISAAC'S plain vanilla ANSI-C source); (3) context-dependent and userdefined agent behaviors (i.e. personality scripts); (4) on-line genetic algorithm, neural-net, reinforcement-learning, and pattern recognition toolkits; (5) on-line data collection and multidimensional visualization tools; (6) on-line chaos-data/time-series analysis tools; and (7) on-line mission-fitness co-evolutionary landscape profilers. ElNSTein is the centerpiece of an ambitious on-going project designed to weave together several inter-related methodological strands. First and foremost, ElNSTein will serve as an interactive "tool box" (or conceptual laboratory) for the general exploration of combat as a complex adaptive system. As such, analysts will have an opportunity to play multiple "What If?" scenarios and to experiment with fundamental issues of the dynamics of war. Secondly, ElNSTein will provide a collection of on-line data-collection/data-analysis and pattern recognition tools; this will mark the first time that many of the tools commonly found in the study of nonlinear dynamical, and complex adaptive, systems will be available in the context of military combat analysis. Thirdly, ElNSTein will have a data visualization package to facilitate the exploration of multiple high-dimensional co-evolutionary fitness landscapes, and to foster the development of an intuition of the overall combat phase space. With embedded genetic algorithm, neural-net, and

278 reinforcement learning based adaptive learning toolkits, ElNSTein will be an ideal testbed for developing future tactical decision aids. Finally, the long-term vision is to extend ElNSTein to become a remote dynamic real-time "combat engine" that assimilates, processes, and automatically analyzes the evolutionary outcomes and consequences of user-defined scripts, tactics and doctrine. In short, a fully realized intelligent-agent-based laboratory for exploring collective self-organized emergent behavior in combat. l/M^l|^IWIIII]

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DISCUSSION & SPECULATIONS The strength of ABMs lies not just in their providing a potentially powerful new general approach to computer simulation, but also in their infallible ability to prod researchers into asking a host of interesting new questions. This is particularly apparent when ISAAC (or ElNSTein) is run interactively, with its provision for making quick "on-the-fly" changes to various dynamical parameters. Observations immediately lead to a series of "What if?" speculations, which in turn lead to further explorations and further questions. Rather than focusing on a single scenario, and estimating the values of simple attrition-based measures of single outcomes ("Who won?"), users of agent-based simulations of combat typically walk away from an interactive session with an enhanced intuition of what the overall combat fitness landscape looks like. Users are also given an opportunity to construct a context for understanding their own conjectures about dynamical

279 combat behavior. The agent-based simulation is therefore a medium in which questions and insights continually feed off of one another. Two issues of fundamental importance that are currently being studied with EINSTein are (1) Command and Control (C2) topology, and (2) Relevance of Battlefield Information. Both ISAAC and EINSTein contain embedded code that hardwires in a specific set of command and control (C2) functions (i.e. both contain a hierarchy of local and global commanders), so that either program can be used to explore the dynamics of a given C structure. However, a much more interesting question is, "What is the best C2 topology for dealing with a specific "threat", or set of threats?" One can imagine using a genetic algorithm, or some otherfitness-landscapesearch tool, to search for alternative C2 structures. What forms should local and global command take, and what is the best-suited communication connectivity pattern among individual combatants, squads and their local and global commanders? An even deeper issue has to do with identifying the primitive forms of information that are actually relevant on the battlefield. Traditionally, the role of the combat OR analyst has been to assimilate, and provide useful insights from, certain conventional streams of battlefield data: attrition rate, posture profiles, available and depleted resources, logistics, rate of reinforcement, FEBA location, morale, etc. While all of these measures are obviously important, and will remain so, the availability of an agent-based simulation permits one to ask the following deeper question: Are there any other forms of primitive information—perhaps derived from measures commonly used to describe the behavior of nonlinear and complex dynamical systems—that might provide a more thorough understanding of the fundamental dynamical processes of combat? Finally, what lies at the heart of an artificial-life approach to simulating combat, is the hope of discovering some fundamental relationships between the set of higher-level emergent processes (penetration, flanking maneuvers, containment, etc.) and the set of low-level primitive actions (movement, communication, firing at an enemy, etc.). Wolfram (1994) has conjectured that the macro-level emergent behavior of all CA rules falls into one of only four universality classes, despite the huge number of possible local rules. While EINSTein's and ISAAC'S rules are obviously more complicated than those of their elementary CA brethren, it is nonetheless tempting to speculate about whether there exists—and, if so, what the properties are, of—a universal grammar of combat. ACKNOWLEDGEMENTS 1 would like to thank Rich Bronowitz, Dave Kelsey, and Igor Mikolic-Torreira for their interest in, and support for, this project. I would especially like to Lyntis Beard for many enjoyable discussions and for coining the memorable names ISAAC and EINSTein. REFERENCES Barrett, C. 1997. Simulation Science as it Relates to Data/Information Fusion and C2 Systems, Briefing Slides, Los Alamos. Beyerchen, A. 1992. "Clausewitz, Nonlinearity, and the Unpredictability of War," International Security, Volume 17, No. 3, 59-90. Boccara, N., O. Roblin and M. Roger, 1994. "Automata network predator-prey model with pursuit and evasion," Physical Review E, Volume 50, No. 6, 4531 -4541. Braitenberg, V. 1984. Vehicles: Experiments in Synthetic Psychology, MIT Press.

280 Chase, J. V. 1902. A mathematical investigation of the effect of superiority in combats upon the sea. (Reprinted in B. A. Fiske, 1988. The Navy as a Fighting Machine, Annapolis, MD: U.S. Naval Institute Press.) Cowan, G.A., D.Pines and D.Meltzer, 1994. Complexity: Metaphors, Models and Reality, Addison-Wesley. Epstein, J. M. and R. Axtell, 1996. Growing Artificial Societies: Social Science From the Bottom Up, MIT Press. Gilbert, N. and R. Conte, editors, 1995. Artificial Societies: The Computer Simulation of Social Life, UCL Press. Gutowitz, H. A., editor, 1990. Cellular Automata: Theory and Experiment, Elsevier Science Publishers. Hedgepeth, W. O. 1993. "The Science of Complexity for Military Operations Research," Phalanx, Volume 26, No. 1,25-26. Hillebrand, E. and J. Stender, editors, 1994. Many-Agent Simulation and Artificial Life, IOS Press. Hofbauer, J. and K. Sigmund, 1998. Evolutionary Games and Population Dynamics, Cambridge University Press. Ilachinski, A. 1996a Land Warfare and Complexity, Part I: Mathematical Background and Technical Sourcebook, Center for Naval Analyses Information Manual CIM-461, Unclassified. Ilachinski, A. 1996b. Land Warfare and Complexity, Part II: An Assessment of the Applicability of Nonlinear Dynamics and Complex Systems Theory to the Study of Land Warfare, Center for Naval Analyses Research Memorandum CRM-68, Unclassified. Ilachinski, A. 1997. Irreducible Semi-Autonomous Adaptive Combat (ISAAC): An Artificial-Life Approach to Land Warfare, Center for Naval Analyses Research Memorandum CRM 97-61. Kaufrman, S. 1993. Origins of Order: Self-Organization and Selection in Evolution, Oxford University Press. Lanchester, F. W. 1995. Aircraft in Warfare, Lanchester Press (Originally published in 1916 by Constable & Co, London, England). Langton, C. G., editor, 1995. Artificial Life: An Overview, MIT Press. Maes, P., editor, 1990. Designing Autonomous Agents: Theory and Practice from Biology to Engineering and Back, MIT Press. Mainzer, K., 1994. Thinking in Complexity: The Complex Dynamics of Matter, Mind, and Mankind, Springer-Verlag. Varela, F. J., Humberto R. Maturana, and R. Uribe, 1974. "Autopoiesis: The organization of Living systems, its characterization and a model," Biosystems, Vol. 5, pp. 187-196. Varela, F. J., 1979. Principles ofBiological Autonomy, New York: Elsevier, North Holland. Miller, L. D., and M. F. Sulcoski, 1995. Foreign Physics Research with Military Significance: Applying Nonlinear Science to Military Programs (U), Defense Intelligence Reference Document, NGIC-1823-614-95. Osipov, M. (translated by R. L. Helmbold and A. S. Rehm), 1995. The Influence of the Numerical Strength of Engaged Forces in Their Casualties, Naval Research Logistics, Volume 42, No. 3, April 1995,435-490. Prietula, M.J., K.M.Carley and L.Gasser, editors, 1988. Simulating Organizations: Computer Models of Institutions and Groups, MIT Press.

281 Saperstein, A. 1995. "War and Chaos," American Scientist, Volume 83, November-December, 548-557. Tagarev, T. and D. Nicholls, 1996. "Identification of Chaotic Behavior" in Warfare, Nonlinear Dynamics in Human Behavior, edited by W. Sulis and A. Combs, World Scientific, 126-143. Taylor, J. G. 1983. Lanchester Models of Warfare, Volumes I andII, Operations Research Society of America. Wolfram, S. 1994. Cellular Automata and Complexity: Collected Papers, Addison-Wesley. Woodcock, A. E. R., L. Cobb and J.T. Dockery, 1988. "Cellular Automata: A New Method for Battlefield Simulation," Signal, January, 41-50.

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IX. Optics

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ADAPTIVE CONTROL OF STRONG CHAOS F.T. ARECCHI Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Florence, Italy and Department of Physics, University ofFirenze, Largo E. Fermi 2, 50125 Florence.Italy

ABSTRACT The standard control methods of chaos have been tested for decorrelation (Lyapunov) times longer than the period of the orbit to be stabilized and for low-dimensional systems. We introduce the notion of strong chaos, corresponding either to a decorrelation faster than the orbital period or to an orbit embedded in a large dimensional space. In these cases a recently introduced adaptive control method is mandatory; however an approximate version of it, easily implementable, is sufficient. We present an optical demonstration of this method.

1. Weak and strong chaos For the sake of convenience, in most cases, chaos has been identified with the occurrence of UPO's (unstable periodic orbits) characterized by the following limitations: i) ii)

The decorrelation (or Lyapunov) time is longer than the orbital period; the system is low dimensional, that is the orbit is embedded in a D-dimensional space with (D > 3) so that it makes sense to refer to a 2-dimensional Poincare section.

The two established control techniques, OGY and Pyragas2 require the simultaneous fulfillment of the two above conditions. We call "strong" chaos the breakdown of either i) or ii). To show what happens in these cases, we recall sequentially the arguments of Refs. 1 and 2. In OGY we start from an unstable orbit. In three dimensions the three expansion rates are respectively X0 = 0 along the trajectory and X+ > 0,/l_ < 0 on the Poincare plane. The instability manifests itself with a perturbation of the position P on that plane after one period T, that is 5P = P(t + T)-P(t)

(1)

If we call M the return map to the Poincare section and a the set of control parameters P(t + T)=M(P(t);a) then, application of a parameter perturbation 8a induces the shift 285

(2)

286

8P(t + T) =

5a

(3)

da and a suitable choice of 8a allows to compensate (1) . The assumption of small perturbation is crucial for this argument, and it requires the necessary condition T,=/K>T

(4)

Furthermore, to my knowledge no extension of this argument is available for the case of a large-dimensional Poincare section. In Pyragas , we refer to a continuous dynamics ruled by i = f(x,a)

(5)

near an orbit of period T. Take a scalar component y of the vector x and consider its variation over a period T due to the orbital instability 8y = y{t)-y(t-T)

(6)

If we perturb the original dynamics (5) with a feedback signal proportional to (6), we can correct the instability. Calling K a suitable weight constant and U = K8y

(7)

the feedback signal, we add t/to (5). As the orbit stabilizes, U goes to zero. The effectiveness of Pyragas is based on the embedding theorem. Indeed, adding (7) to (5) is like a repeated application as compared to the singular sampling at the Poincare section of OGY. Also Pyragas is a perturbative technique, which requires condition i). Let us see what happens when ii) does not hold. If D » 3 , then the Poincare section is a hyperplane of dimension D-l. We may consider the situation in which condition i) still holds. In this case, having a large number of positive Lyapunov exponent A+ , by Pesin formula3 we consider the Kolmogorov-Sinai entropy as the sum

(A5) = £V> and define TA

=l/(KS).

(8)

287

Even if T^»T, a (D-l) dimensional, Poincare section is not manageable by OGY technique. As for Pyragas a fundamental difficulty occurs. Collecting a time sequence of difference data 8y over a period as Eq. (6) and using this signal as a correction with a fixed weight implies a critical proviso, namely, even though a dense sampling of 8y is as powerful as an embedding technique, limiting it to a single scalar is in general not sufficient; in fact, even when (KS) is not too high, the instantaneous local direction of maximum expansion may pertain to any one of the scalar components of x different from y. Recently, Farini et al4 have introduced the concept of rotation rate Q, which is the speed of change of the direction of maximum expansion on the plane transverse to the trajectory. If Q » ( K S ) , then beyond a time \C. = Tn « TA the sequence of sampled values of a single component are no longer a good embedding reconstruction. Furthermore, Pyragas requires a precise localization of T and it is not robust for fluctuations in T5. 2. The adaptive control6'7 We have recently proposed an adaptive control method which has resulted efficient even for strong chaos, and which furthermore is robust for inaccurate assignments of T. With the help of fig. 1, it can be visualized as an extension of Pyragas method by introduction of an adjustable weight in the feedback term. Precisely, let us consider the difference of a scalar component >> over a period T 8n=y(t„)-y(t„-T)

Figure 1. Case of an unstable periodic orbit of period 7*. Multiple sampling of the displacement 8 over T of the y coordinate of the crossing point with a transverse plane. The time location tn of the n-th plane is separated from K-\ by a n amount Tn which depends upon the previous evolution of 5.

(9)

289 Here the observation times t„ form a sequence tH =?„_ 1 +r„ where the separation intervals T„ are selected with the following criterion. Let us consider the difference in S„ over two consecutive observations and take the following logaritmic variation rate

(10)

P„ = log

ju„ is positive or negative depending on whether 5n ><5„_,, or 8n <<5„_,, that is, on whether the two pieces of orbit separated by a period T are departing or approaching during the time lap zn. Based on Eq. (10), one can reduce or increase the next separation between two consecutive observations via the correction *„+1 =T„(l-tanhcrAO ST„(l-CTAt„)

where o" is a suitable sensitivity factor. The tanh in Eq. (11) introduces a saturation, to avoid negative T'S; however for small corrections the linear approximation is sufficient. The set {tn} is distributed around its average (T) . By adjusting a, (T) represents a suitable time scale intermediate between the minimal time step (the instrumental resolution in an experiment, the Runge-Kutta step in a numerical simulation) and the period T. Over a single inspection step we can resort to a linear expansion as y(t„^) = y(tJ+<*>y(t„)

(12)

where y is the time derivative. It then easily follows that 5„+1=5„+5„ ,

(13)

=^-

,

(14)

5

'

f

8 ^

= < T > 1-CT < T > —

(15)

288 Eq. (15) derives from Eq. (11) by use of Eq. (14) and replacement of xn with <x>. Now, let us introduce a correction like Eq. (7) but use as weight the reciprocal of the inspection interval, that is, at each updating step, replace Eq. (7) with C „

+

(16)

, = — •

In the linear approximation, by Eq. (15) UK+l=-^-

+ a-SH

.

(17)



Eq. (17) consists of two contributions, the first one being Pyragas with a suitable constant weight y

which has now a physical origin and which eventually depends on the

sensitivity a of the data collection, the second one is proportional to the local time derivative. In its full expression, Eq. (16), the adaptive control succeeds in those cases where Pyragas fails,but in many practical cases the linear approximation is sufficient, as discussed later below.

3. Adaptive control of a delayed dynamical system The essential problems arising in the passage from concentrated to extended systems are already present in delayed dynamical systems, i.e., systems ruled by y = F{y,yd)

(18)

where y = y(t) e Rm, dot denotes temporal derivative, F is a nonlinear function, and y
t=-n+6T

(19)

290 where 0 < TJ < T is a continuous spacelike variable and 6 e N plays the role of a discrete temporal variable. By such a representation the long range interactions introduced by the delay are reinterpretated as short range interactions along the 0 direction, since now yd = y{r\,6 -1). In this framework, the formation and propagation of space-time structures, as defects and/or spatiotemporal intermittency, can be identified9. When T is larger than the oscillating period of the system, the behavior of a delayed system is analogous to an extended one with k=l. In particular, it may display phase defects, i.e., points where the phase suddenly changes its value and the amplitude goes to zero. The adaptive method applies successfully to models as the one in Ref. 10 for a CO2 laser with delayed feedback, which indeed displays phase defects as those reported on Fig. 2 (from Ref. 9).

f

f0.4

I

I

0.6

1.0

t (ms) (a)

0.Z5

0.27

0.29

0.31

0.33

0.35

t (ms) (b) Figure 2. (a) Intensity signal within a delay unit (T=1 ms); (b) expanded view of the signal between the arrows exhibiting a phase jump (solid line) and reference signal (dashed line) translated from a regular region without noticeable intensity variations (from Ref. 9).

Let us see how phase defects emerge. We adjust the pump and delay parameters so that the system enters the chaotic regimes. For low T values, chaos is due to a local chaotic evolution of the phase, whereas no appreciable amplitude fluctuations are observed. We call this regime phase turbulence (PT). By increasing T, we observe a transition toward

291 amplitude turbulence (AT), wherein the dynamics is dominated by the amplitude fluctuations, and a large number of defects is present. Both PT and AT have counterparts in a one-dimensional complex Ginzburg-Landau equation. Here the parameter space shows a transition from a regime of stable plane waves toward PT (Benjamin-Fair instability), followed by another transition to AT with evidence of space-time defects. We succeed in controlling both regimes by the adaptive technique. This technique adds iteratively a small correction U(t) to Eq. (18), as follows. At time tn+, =tn+xn (T„ being an adaptive observation time interval to be later specified), the observer defines the variation y(tn+l -TH)- y(tn+l) between the actual value of y and the value delayed by the period of the UPO to be controlled (TH being the Hopf period). The corresponding variation rate, once linearized, is given by Eq. (14) and the correction reduces to Eq. (17) that we rewrite as

U(t) =

with K, =

Kl[y(t-TH)-y(t)] + K2[y(t-TH)-y(t)]

and K2 = a . The consequences of this approximation are interesting.

First of all, for K2 =0 one recovers the Pyragas control method. However, in our case, Ki and K2 can be independently selected, and this introduces an extra degree of freedom with respect to Ref. 2. Now, the control is more active when the error is increasing and vice versa, so reducing oscillations. Indeed, Eq. (18) performs as a proportional derivative controller, the more usual action for stabilizing feedback linear systems, due to its effect which consists of increasing the phase of the compensated system in a suitable frequency band11. In Fig. 3 we report the application of our method. The desired oscillation, which in the space-time representation gives rise to a roll set, is controlled in PT and in AT. Going back to the above discussion, the results show that, while the choice K^-K2 = 0.2 assures the roll stabilization for small perturbations, fixing K2 = 0 as in the Pyragas case would have implied prohibitively large Ki values for obtaining the same stabilization (in our tests, if K2 = 0, K, should be 10), resulting in very large perturbations of the system, which eventually give rise to relevant distortions of the roll amplitudes. The stabilization consists in suppressing the defects present in the AT regime. Suppose, indeed, that some defects are present at the beginning of the controlling procedure. The spontaneous lifetime Ta of a defect can be evaluated in a free running (no control) situation. The scaling behavior of Ta as a function of the delay time T depends on the nature of the turbulent process. Namely, in AT, Ta scales quadratically with T. When a control is applied, we expect it to be effective after a transient time Tt of the order of Ta. Thus a measurement of T, provides an estimate of the lifetime Ta. In Fig. 4 we have reported the scaling behavior of TjT as a function of T/TH . The quadratic scaling confirms that control is achieved within the AT regime.

292

a

%M

jr

0 •

b

Figure 3. Space (cj)-time (0) representation of the controlling process for a delayed system with delay T (ml 11), (a) T=15, PT regime. The chaotic dynamics results hi a local turbulent phase of the Hopf oscillation which is corrected by the controlling algorithm. Kx = K2 = 0 . 2 . Arrow indicates the instant at which control is switched on. (b) T=50, AT regime. The dynamics is dominated by amplitude fluctuations, with the presence of defects. The algorithm (Kt = K2 = 0 . 2 ) suppresses the defects and restores the regular oscillation. Arrow indicates me instant at which control is switched on. (c) Pyragas' method. T=50» AT regime. The dynamics is first perturbed with

Kt=0.2,K2=0

)

(first

arrow). To achieve control with K2 = 0 it is necessary to select K1 = 1 0 (second arrow), which, however, produces a large amplitude distorsion (the amplitude of the controlled oscillation is now one half of the amplitude of the Hopf one).

293

10 Figure 4. Plot of the ratio TJT

15

20

25

30

as a function of TITH (see text for definitions). The quadratic scaling of Tt (T) confirms

that control is achieved within AT. For all cases Kx = K2 = 0.2. (from Ref. 11).

4. Experimental control of chaos in a laser with delayed feedback12 The adaptive control demonstration reported in sec. 3 was only numerical. In order to apply it "in vivo", we refer to a single mode C0 2 laser with delayed feedback on the cavity losses (Fig. 5).

CHAOTIC SYSTEM

CONTROL LOOP

Figure 5. Experimental setup. M: electro-optic modulator; LT: laser tube; D: HgCdTe detector; T: delay line; WF: washout filter; d/dt: derivative block; VGA: variable gain amplifier; HVA: High voltage amplifier; B: bias input.

294 A photodetector yields a voltage proportional to the laser output intensity. This voltage, after a suitable delay and amplification, drives an intracavity electro-optic modulator. The detector is a fast Hg-Cd-Te photodiode and the delay line is realized using fast (2 MHz) and accurate (12 bits) A/D (analog-to-digital) and D/A converters allowing variations of the delay time x up to 130 ms with 0.5 (is resolution. The high-voltage amplifier adds to the delayed signal a continuous voltage level B which, once x is fixed, acts as the control parameter. Even at zero delay, by increasing the bias B, the system undergoes a Hopf bifurcation, with the fundamental period around 20 (j.s, and eventually it enters a chaotic regime. If the delay x is of the order of the oscillating period of the system, the fractal dimension of the chaotic attractor, reached through quasiperiodicity, remains around 3 13 . Since the aim of the present work is to implement a control strategy for a high dimensional chaotic regime (characterized by more than one positive Lyapunov exponent), we have explored a delay range from x=50 |is to x=600 (is. By a simple argument13, the number of extra degrees of freedom due to the delay is given approximately by the ratio between x and the correlation time of the intrinsic chaotic motion which is around 100 (is. Evidence that such a delayed system has more than one positive Lyapunov exponent for x larger than 50 lis has been recently given with a new method of analyzing delayed dynamics' 4 ' 15 . A typical time sequence of the laser intensity is shown in Fig. 6 for x=150 (is, together with the corresponding power spectrum. For such a delay time, four positive Lyapunov exponents have been found15, with a Kaplan-Yorke dimension of about 8.5. A further increase of B leads to a collapse of the chaotic attractor into another stable limit cycle. In our delayed system, condition i) of Sec. 1 is fulfilled, but condition ii) breaks down. Indeed, by measuring the Lyapunov exponent spectrum, for the chosen delay we have four positive Lyapunov exponents, with a maximum value15 Xl = 7 - 1 0 V

Time (msec)

Frequency (kHz)

Figure 6. (a) Laser intensity signal for T = 150 us and B = 275 V. (b) Corresponding power spectrum. f,=l/T=43 kHz is the Hopf frequency, f2 = 1/T. = 6.67 kHz is the reciprocal delay time.

295 Therefore, by Pesin formula

(A5) = 5 > + S 3 - 1 0 V \ which is much less than both peaks of the chaotic spectrum (Fig. 6b) that are respectively \L = 43kHz and V = 6.61kHz. /T /T However, the Kaplan-Yorke dimension is 8.5 and therefore the Poincare section is more than 7-dimensional. As a consequence, by what said in Sec. 2, Pyragas based on the feedback of a single scalar quantity should not be sufficient. Indeed, it has been experimentally verified that this is the case. For the sake of simplification, rather than introducing a delayed control, we operate with a so-called "washout filter" in the feedback loop16. All unwanted frequencies present in the chaotic spectrum of the output signal are transmitted as corrections through a selective filter. In this way, the system is allowed to oscillate at the only frequency which is not subtracted, namely that of the unstable orbit to be stabilized. The observation of the power spectrum reported in Fig. 6 can help in defining the filter characteristics. The peak structure is due to the competition between the Hopf frequency fi = 43 kHz and the inverse of the delay time 1/T = fi = 6.67 kHz. Thus, following a strategy already tested18, we use a washout filter whose transfer function presents a zero of amplitude at fi and a zero of phase at ii (see Fig. 7). In this way, the feedback loop provides minimum and maximum rejection at fi and f2, respectively. These features could in principle stabilize the unstable limit cycle with frequency fi, while canceling the effects of the delay at frequency f2. The transfer function of Fig. 7 can be represented analitically by the following formula (s=ia>)17

fo(*2+^2) (S)=

Q2

where Q=27ifi, k is the gain, C, and |i suitable parameters. It is important to observe that the above control method presents several analogies with Pyragas2. Pyragas re-injects into the system the difference between the output signal and its delayed value with a certain amplification factor K. The corresponding transfer function is C'(s)=K(l- e - s T ). In the low frequency domain, the transfer function C'(s) is similar to that of the washout filter C(s), since both present a zero of the amplitude at fi, provided fi=l/T (Fig. 7).

296

1.0 0.8 -g

0.6

1i

0.4

e "

0.2 0.0 100 50

0

3

0


nj

£

-50 -100 0

40

80

120

160

Frequency(kHz)

Figure 7. Amplitude and phase of the two transfer functions C(s) (solid line) and C'(s) (dashed line). For both functions fi=l/T=43 kHz.

On the other hand, the Pyragas transfer function amplitude goes to zero also for all fi harmonics (thus ensuring exact zeroing of the control signal when the stabilization is obtained), while the phase necessarily crosses zero at fi 12. Several experimental applications of the Pyragas method have been reported in the literature18 but all working in the "weak" chaos regime. Recently, Just et al. analyzed the mechanism of time delayed feedback control from a theoretical point of view, also addressing the problem of the mismatch between the delay T and the period T of the unstable orbit to be stabilized, and the influence of the control loop latency5. At variance with the results on low dimensional chaotic dynamics, the application of a control loop based only on the washout filter fails when t>50 us. The presence of a large delay and, consequently, of high dimensional chaotic signals suggests that more information on the state of the system is needed in the control loop in order to achieve stabilization. To solve this problem, we implement an analog version of the adaptive algorithm. This technique would require a computer-aided real-time analysis of the temporal signal. Anyway, the adaptive control can be approximated at the first order in time with the control signal Eq. (17) which, considering the close analogy between C(s) and C'(s), leads to the following transfer function C*(s) = (Kj + sK2) C(s).

297 Such a control can be easily realized if the output voltage from the washout filter is added with its time derivative (see Fig. 5). Obviously, the amplification factors for the two signals contributing to the global control must be regulated separately. With the use of this improved scheme we have converted the chaotic motion into a limit cycle with period 1/f; the time sequence of the stabilized output laser intensity is reported in Fig. 8, for the same parameter values of Fig. 6.

0.5 I

0,4

1

ri i ii i n 11 ii i ii 11 II i n 11 II i II 11111 till 11 in HI 11 nil II Hum HI i J

?3 °' 3 l H

2

•2- °' I x LI ! 1 I I I I I I 1 1 I I I V I I I I I I I I I I I 1 I 1 I I I I I I I I ! I I I V 1 I I V I U V I V V I I V 0.1 I O.o I 10.0

i

1 10.5

•

' 11.0

•

1 11.5

Time (msec)

Figure 8. Stabilized Hopf oscillation of the laser intensity for the same parameter values of Fig. 6.

It is important to stress the following experimental results: (i) the control signal compared with the unperturbed delayed feedback is of the order of a few percent; (ii) the control works for K2 > 0.8 Ki thus confirming the relevance of the derivative correction. For K.2 < 0.8 Ki, long delays are not stabilized by the washout filter. In such a case, an increase of the amount of perturbation Ki above a few percent changes drastically the nature of the dynamics, forcing the system into another periodic orbit no longer coinciding with the original Hopf orbit.; (Hi) the control loop has been entirely realized with analog circuitry and consequently can be very fast; (iv) finally, stabilization is effective independently of the delay time x. We have thus experimentally demonstrated that in a delayed dynamical system, until delays are relatively short, (i.e. comparable with the de-correlation time of the non delayed chaotic system ), it is possible to stabilize periodic behaviors by using Pyragas, however, for a long delay implying a high dimensional chaos, it is necessary to recur to the adaptive control, based on the combination of a washout filter with its time derivative in the feedback loop.

298 References [I] E. Ott, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 64, 1196 (1990). [2] K. Pyragas, Phys. Lett. A 170, 421 (1992); 206, 323 (1995). [3] J.B. Pesin, Russ. Math. Surv., 32, 55 (1977). [4] A. Farini, S. Boccaletti and F.T. Arecchi, Phys. Rev. E, 53, 4447 (1996). [5] W. Just, D. Reckwerth, J. Mockel, E. Reibold, and H. Benner, Phys. Rev. Lett. 81, 562 (1998); W. Just, D. Reckwerth, E. Reibold, and H. Benner, Phys. Rev. E (1999). [6] S. Boccaletti and F.T. Arecchi, Europhys. Lett. 31,127 (1995) [7] F. T. Arecchi and S. Boccaletti, Chaos 7, 621 (1997). [8] F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, Phys. Rev. A 45, 4225 (1992). [9] G. Giacomelli, R. Meucci, A. Politi, and F. T. Arecchi, Phys. Rev. Lett. 73, 1099 (1994). [10] G. Giacomelli and A. Politi, Phys. Rev. Lett, 76, 2686 (1996). [II] S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, Phys. Rev. Lett. 79, 5246 (1997). [12] M. Ciofini, A. Labate, R. Meucci and F.T. Arecchi, European Phys. Journ. D., (1999), to be published. [13] F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, Phys. Rev. A 43, 4997 (1991). [14] R. Hegger, M. J. Bilnner, H. Kantz, and A. Giaquinta, Phys. Rev. Lett. 81, 558 (1998). [15] M. J. Bilnner, M. Ciofini, A. Giaquinta, R. Hegger, H. Kantz, R. Meucci and A. Politi, European Phys. Journ., submitted. [16] A. Tesi, E. H. Abed, R. Genesio, and H. O. Wang, Automatica 32, 1255 (1996); M. Basso, R. Genesio, and A. Tesi, Systems Control Lett. 31, 287 (1997). [17] R. Meucci, M. Ciofini, and R. Abbate, Phys. Rev. E 53, R5537 (1996); M. Ciofini, A. Labate, and R. Meucci, Phys. Lett. A 227, 31 (1997); R. Meucci, A. Labate, and M. Ciofini, +Phys. Rev. E 56, 2829 (1997); A. Labate, M. Ciofini, and R. Meucci, Phys. Rev. E 57, 5230 (1998); R. Meucci, A. Labate, and M. Ciofini, Int. J. Bif. and Chaos 8, 1759 (1998). [18] S.Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. E 49, R971 (1994); T. Pierre, G. Bonhomme , and A. Atipo, Phys.Rev.Lett. 76, 2290 (1996); T. Hikihara and T. Kawagoshi, Phys. Lett. A 211,29 (1996);C. Simmendinger, and O. Hess, Phys. Lett. A 216 , 97 (1996); D. W. Sukow, M. E. Bleich, D. J. Gauthier, and J. E. S. Socolar, Chaos 7, 560 (1997)

Chaotic Behavior and Multi-Stable Oscillations in the Visible Lightwave from Semiconductor Lasers and Their Applications to Novel Optical Communications Wakao SASAKI Naoya NAKASHITA and Kouta INAGAKI Department of Electronics, Doshisha University, Kyo-Tanabe City, Kyoto, 610-0321 JAPAN Phone: +81-774-65-6354 Fax: +81-774-65-6783 E-mail: [email protected] ABSTRACT We demonstrate a simple method for controlling the nonlinear oscillations by only using a semiconductor laser and photo detectors. The nonlinearity necessary for exhibiting period-doubling bifurcations and chaotic phenomenon was realized by appropriately superposing the threshold characteristics of semiconductor lasers on the saturation characteristics of photodiodes in their electric current vs. light intensity properties. An electro-optical NDFS (nonlinear delayed feedback system) has been composed by making use of this nonlinearity. A low-pass filter was inserted into this NDFS as an electric feedback element, which gave variety to the behavior of the nonlinear oscillations. With this system, multi-stable oscillations and chaotic transitions with various patterns have been observed. In this scheme, an optical dynamical memory function has been demonstrated to perform binary data writing. In addition, we have applied an external light input to the NDFS, for example, some sinusoidal wave modulations with acoustic frequencies. If the external input light effects to the oscillation waveform are to be reproducibly derived, new application to a waveform recognition system and cryptography will be expected. Keywords: chaos, optical bistability, electro-optical hybrid systems, period doubling bifurcations, optical dynamical memory, pattern recognition, optical cryptography.

1. Introduction Chaotic lightwave generation has attracted many researchers in the field of technology and science so that a number of contributions to this subject have been published so far '"3. As a result, various features concerning chaotic phenomena have been revealed4'5. In this paper, we 299

300 demonstrate a new method particularly simple method for chaotic lightwave generation based on a nonlinear delayed feedback system only using a commercially available semiconductor laser diode (LD), photo-detectors and very ordinary electronics such as low-pass filters and so on. The nonlinearity necessary for exhibiting period-doubling bifurcations and chaotic phenomenon was realized by appropriately superposing the threshold characteristics of semiconductor lasers on the saturation characteristics of photo diodes in their electric current vs. light intensity properties. An electro-optical NDFS (nonlinear delayed feedback system) has been composed by making use of this nonlinearity. A low-pass filter or a band-pass filter was inserted into this NDSF as an electric feedback element, which gave variety to the behavior of the nonlinear oscillations. With this system, multi-stable oscillations and chaotic transitions with various patterns have been observed. In this scheme, an optical dynamical memory function has been demonstrated to perform a 47-bit binary data writing. In addition, some characteristic routing sequences from period-doubling bifurcation to chaos have been reproducibly observed when some sinusoidal-wave modulations with acoustic frequencies, for example, are applied to the NDFS. This means that applying some lightwave signals, modulated by somebody's voice or by video image signal of somebody's face, for example, to the photo detector of this NDFS system as an external light input will lead the system to generate a certain, reproducible output code pattern in the output light of the semiconductor laser of the NDFS. Eventually, the derived output code pattern from this system is only one of the innumerable scenarios by which a nonlinear oscillation mode of this NDFS leads to chaos. As a result, the discovered chaotic mode transition characteristics for different modulation frequencies of the present work will open up a novel method for voice/picture pattern recognition and cryptography in optical communications. 2. Experiments and results Figure 1 is a block diagram of the present nonlinear delayed feedback system. The nonlinearity of the V-shaped function is composed by combining the optical input versus electric current output characteristics including saturation of these two photo-diodes. And the resultant output current of the photo-diodes is to be fedback to the laser diode as its driving current after passing through the low-pass filter and delay line. And an analog multiplier is used for the loop gain controller. The external input included in this figure will be mentioned in the followings. Fig.2 is to explain how to compose the V-shaped nonlinear function. Two photo-diodes with opposite bias voltage are connected with each other so that a V-shaped output voltage 1 can be performed. The saturation of a photo-diode output is made by using a voltage limited.

301

External Input

w Nonlinear Function ~\r

Offset Voltape V„,.

Digital Delay

Fig. 1. Block diagram of the experimental setup.

Fig.2. Schematic explanation for composing a V-shaped nonlinear function.

Figure 3 is the examples of the oscillation waveform of our nonlinear delayed feedback system with low-pass filter. For a small loop gain u. that is low optical power from the laser diode, the stable state of the loop is a constant intensity signal. At higher u., this state this state becomes unstable and there arises stable self-oscillation in the loop. The period of the fundamental oscillation is about 2Tr (Tr is the feedback delay time of the system). With increase in u., there occur successive period-doubling bifurcations of the order m=l, 2, .... , finally reaching to chaos (m=C). So, we label the modes with harmonic number n and bifurcation order m as (n,m). There are also odd harmonic modes, as shown in Fig.4, which are multistable with the fundamental mode. The odd harmonics have period about T/n. where, harmonic number n is an odd integer, and T is the period of the fundamental mode, which is twice Tr.

302

f H ((Iff if '•(1 (III ftflf, .111y.;•u 11111111 I, Hill 1

;; •£- ~\

r

*— Tr

'

-P-

u r

^

_

"— Tr - ** - Tr —**— Tr

L.

1 X

*•

i t=—=r

Trn.ni)=

I>

-Tr-

Tr —

- Tr

t H M J, if\v IJ

|

•

•

Tune t lima/div]

jj^fl VAAJl/U

W

-£\&

-J^UJJH >WV

I

wdlxOT

I' IIt ffflffiMiui Tims t jZnuAliv]

Fig. 3. Oscillation waveforms of LD driving voltages of NDFS with low-pass filter (harmonic number n=l and feedback delay time Tr=5ms).

1, 0, 0, 0

Fig.4. Examples of odd harmonic modes seen in LD driving voltage with multistable levels (n=9, Tr=5ms).

(1, 0, 0, 0, 0, 1, 1, 1, 0)

0, 1, 1, 1, 0

Fig.5. Schematic explanation for the binary-bit coding of (9,2) mode by a seed (external clock) signal.

From these figures, we can see that the waveform of an (n,2) mode oscillation is such that there are 4 levels, two peak levels and two valley levels corresponding to the period-4 point of the period-doubling bifurcation model. It is observed from the experimental data that (n,2) modes are stable enough to be used for storing information encoded in peak level. At a given value of (j., there can be many different stable (n,m) modes with different peak modulation patterns. With an external reference clock to determine the phase of the pattern, the memory

303 capacity of (n,2) modes is n bits, corresponding to 2 distinct stable oscillation patterns (Fig.5). So far, stable harmonic (n,2) modes was observed in our system, and up to (47,2) mode oscillations were stably excited. We demonstrated phase-locking of the (47,2) oscillations to obtain phase stability and 47-bit capacity. We then executed writing and reading of 47-bit binary data based on the methods of waveform coding5'6, which Dr. Peter Davis, et al. of ATR, originally proposed7. The phase of the (47,2) mode could be locked to that of an external reference clock by weakly modulating the light power of the laser diode at a clock frequency n/2Tr =1.1 kHz, near the frequency of the free-running (47,2) mode oscillation. This allowed stable determination of the phase of the oscillation. Now, we have proved that the present system can be efficient for the dynamical optical memories and bit-data coding for optical communications since it has sufficient reproducibility and stability of self-oscillation mode waveform for the applications of binary data bit coding. Now, we have proved that the present system can be efficient for the dynamical optical memories and bit-data coding for optical communications since it has sufficient reproducibility and stability of self-oscillation mode waveform for the applications of binary data bit coding. 3. Discussion for applications In this section, we would like to mention of the possibilities for more advanced applications, which can be regarded as analog data processing rather than digital one, based on our present method. If the produced scenario by the self-oscillation mode transition to chaos in our system is reproducible enough, we will be able to utilize this system for optical data recognition and cryptography for optical communications. So, for this purpose, we have carried out some preliminary experiments in which a sinusoidal waveform was applied as an externally input optical signal to the present nonlinear delayed feedback system. Then, we have measured how the external input signal can affect the behaviors of self-oscillation mode patterns observed in this nonlinear delayed feedback system. Figure 6 shows the examples for external input signal waveform. We have tried two types of waveforms as shown in Figs. 6 (a) and (b). The type (a) waveform is a sinusoidal wave of 294 Hz with limited duration of 50 ms. The application of this type of waveforms to the external input signal is intended to stimulate or trigger the NDFS system to transit into some different states. And type (b) is a continuous sinusoidal wave. In the experiments, these two types of waveforms with different frequencies were separately applied to the LED of the external signal light source for the system.

304 f=294fH7l

:: Awwyvwvwwl1 '

'"

Time 10[ms/div]

(a) f=l.l[kHz]

Time 10[ms/div] (b) Fig.6. Examples for external input signal waveform applied to NDFS.

The diagram in Fig. 7 represents a series of oscillation mode can be controlled by the external input signals of type (a). In this case, we can control the oscillation harmonic numbers by varying the frequency of the external input signal, 294Hz for (3,1) mode, 490Hz for (5,1) mode, 686Hz for (7,1) mode, if we preliminary set an adequate loop gain value and delay time of the system. In this system, we have confirmed that oscillation harmonic numbers can be directly chosen from 1st order up to 9th by setting the frequency of the external input signal according to their characterized values for the harmonic numbers, for example, in this case all these values are the multiples of 98Hz. As for the results with type (b) waveform, resonance phenomena can be seen. Figure 8 is the amplitude signal derived from the driving current signal of the laser diode in the system, which is directly reflected from the feedback signal of the present NDFS system, and this driving current signal generates the optical output waveform of the laser diode. Now, the amplitude of this signal shows a clear, resonance like dependence upon frequencies of the external input signal with respect to their harmonic numbers seen in the case of mode-transition behavior with type (a) waveform. Now, our results can be applicable to further advanced techniques something wiser than binary data coding or dynamical memories. If the reproducible correspondences between characteristics of an external input signal, such as its frequency, amplitude, etc. and transition scenario pattern of self-oscillation mode to chaos, are to be proved, we will be able to recognize the input data by only referring to the scenario pattern.

305

Amplitude [ V j

|98Hz

294H

f

490H

I

.

11

686H

|

.l\J

400

686 Hz

Fig.7. Schematic diagram showing mode transitions induced by external input waveforms of type (a) with different frequencies, which are the harmonics of 98 Hz.

I

1078Hz

A A 1 AA 'LIV _A /

as

>

882H

600

1000

1200

Frequency [ Hz ] Fig.8. Resonant frequency characteristics seen in the driving current amplitude for the laser diode in the NDFS, which is reflected from the feedback signal.

4. Conclusion We have demonstrated a simple electro-optical nonlinear delayed feedback system for optical chaos generation and have observed multi-stable oscillations and transitions to chaos in this nonlinear feedback system. We have also demonstrated this system can perform optical dynamical memory and binary data coding up to as much as 47 bits. Lastly we have carried out a preliminary experiment suggesting that the present system may have reproducible response of self-oscillation mode transition scenario to chaos for the externally input sinusoidal waveform. This can prove the potential applicability to something wise communications method such as optical pattern recognition and cryptography.

REFERENCES 1. K.Ikeda and H.Daido, Phys. Rev. Lett. 45 (1980) p.709. 2. H.Gibbs, F.A.Hopf, J.L.Kaplan and J.V.Shoemaker, Phys. Rev. Lett. 46 (1981) p474. 3. K.Ikeda and K.Kondo, Phys. Rev. Lett, vol.49 (1982) p. 1467. 4. T.Aida and P.Davis, IEEE J. Quantum Electron 28 (1992) p.686. 5. W.Sasaki and N.Otani, Sci. andEng. Rev. ofDoshisha Univ..37 (1996) p.58. 6. W.Sasaki and K.Nakamuta, Proc. SPIE 3468 (1998) p.279. 7. T.Aida and P.Davis, Jpn. J. Appl. Phys. 29 (1990) p. L1241.

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OPTICAL IMPLEMENTATION OF CHAOTIC MAPS WITH MACH-ZEHNDER INTERFEROMETERS KEN UMENO YOSHINARI AWAJI Communications Research Laboratory Ministry of Posts and Telecommunications Koganei, Tokyo 184-8795, Japan

Department

KEN-ICHI KITAYAMA of Electronics and Information Osaka University Suita, Osaka 565-0871, Japan

Systems

Abstract We propose an optical device implementation of chaotic maps. The key design principle is given by combining a simple mathematical fact that various maps with an ideal chaotic property can be interpreted as multiplication formulas of periodic functions such as sin functions with a simple physical fact that Mach-Zehnder interferometers produce a s'm2() profile of output lightwave intensity.

1

Introduction

Photonic security of fiber-optic communication systems is an unexplored area of research, which can have a large impact on our future telecommunications. One clear bottleneck of realizing such a demand is constructing all-optical devices of transmitting high bit-rate signals with a properly guaranteed security. After the pioneering works about synchronization of chaos [1] and chaotic lasers for communication systems [2], such a quest for applicability of chaos to high-speed communication networks is stimulated to be active. In this paper we explore a novel approach towards a construction of optical devices of generating chaotic waveforms. In Section 2, we describe a special characteristic of chaotic maps to be implemented in optical devices. In Section 3, we introduce multi Mach Zehnder type interferometers which implement a class of the chaotic maps introduced in Section 2. Experimental results in planar lightwave circuits (PLC) devices are reported in Section 4. Conclusion is given in Section 5.

2

Chaotic Maps

In 1947, S. Ulam and J. von Neumann [3] discovered that the one-dimensional logistic map Xj+i = F(Xj) where

Y = F{X) = 4X(1-X) 307

(1)

308

is ergodic with respect to the invariant probability measure

M<&0 =

/ **

,

(2)

irJxyl — x) over the unit interval [0,1]. This remarkably simple dynamical system generates stochastic behavior from a purely deterministic system. Furthermore, it is known that this map F(x) has a mixing property with respect to the invariant measure in Eq. (2). Consequently, map F(x) is ergodic and we can calculate the time average of quantities A{x) by using the ergodic principle

Q = & if £

QiXj) =

Io QWpWdx-

(3)

The last equality holds almost everywhere with respect to the initial condition Xi 6 [0,1]. Thus, we can evaluate time correlation functions Q(Xj)Q(Xj+T) by the formula Q(Xj)Q(Xj+T)

= lim ^J2Q(xi)Q(xi+r) AT—voo 1\

= f1Q(x)Q[F^x)]p(x)dx,

(4)

Jo

where j Fj{x) ='Fo.'-oF(x). (5) It is noteworthy that a theoretical estimate of correlation functions is extremely important for designing architecture of cryptology and communication systems with the use of such chaotic random-number generators. Such kinds of chaotic maps with explicit ergodic invariant measures can be obtained from multiplication formulas of sin functions and elliptic functions. For example, the Logistic map can be interpreted as the duplication formula of sin2(-) because the following equality sin2(20) = F[sin2(0)] = 4sin 2 (0)(l - sin2(0)) (6) holds. In 1964, Adler and Rivlin obtained Chebyshev maps Fp(-) from the pth multiplication formulas of a cos function, which have the unique invariant measure fi(dx) = p(x)dx — ,*^ 2 [4]. For example, the Cubic map G(-) satisfies the triplication formula sin2(3#) = G[sin2(#)] where, Y = G{X) = X(3 - AXf.

(7)

Such chaotic maps are further generalized by multiplication formulas of elliptic functions [5, 6, 7] and tangent functions [8]. All of the chaotic maps obtained

309

from multiplication formulas has the following remarkable property: We can calculate Kolmogorov-Sinai entropy h{fi) which is the asymptotic rate of information creation by an iteration of a chaotic map F(-) by the Pesin identity [9]:

MM) = \ = fnH^Mdx) =

fnln\[^-]\p(x)dx

(8) = \og(p),

(9)

where 0 is the domain of the dynamical variables and A is the positive Lyapunov exponents of a map F(-). Thus, it can be said that the class of maps obtained from such multiplication formulas has an ideal chaotic property. Especially, they are useful as spreading sequences for code division multiple access (CDMA) communication systems [10, 11]. They can effectively serve as pseudo-random number generators for a general Monte Carlo computation [12]. Quite recently, more general multi-dimensional chaotic mappings with explicit invariant measures are successfully constructed by the skew product of such one-dimensional chaotic mappings [13].

3

Optical Circuit Design

Our circuits are composed of N Mach-Zehnder interferometers (MZIs). A MZ interferometer has a simple interference structure between the two optical paths which acquire different phase shifts as depicted in Fig. 1.

Input Ports

3dB coupler Figure 1:

Output Ports

3dB coupler

Mach-Zehnder interferometer.

The basic configuration of the optical circuit of generating chaotic sequences is shown in Fig. 2. In the circuit generating N chaotic values X(l), X(2), •• •, X(N), input signals are divided into N with equal powers after passing anlxN coupler. Such a coupler can be constructed by a combination of 3-dB couplers. At the

310

output of j-th. Mach-Zehnder interferometers with an optical path length difference AL(j), an intensity of light is measured by a power meter and the numerical value is denoted by X(j). Each AL(j) is assumed to satisfy the formula: AL(J + 1) = mAL{j),

(10)

where m > 2. The transfer function at the output port of the j-th MZ interferometer is simply given by the formula x(3) = s ^ ^ ) ,

(ii)

where A is the wavelength of the light source and n is the effective refractive index of the optical paths of the MZ interferometers. Thus, the output powers X(l), X(2), • • • ,X(n) satisfies the chaotic map obtained by the m-th multiplication formula of sin2(-) function. One can easily change the initial condition X(l) by changing the wavelength of the light source.

4

Experiments

The Mach-Zehnder interferometers are fabricated on a planar lightwave circuit (PLC), where planar optical wave-guides are formed on a silicon substrate. The monolithic integration ensures the precise control of the optical path length difference and the stability [14]. The wavelength tunable laser diode (modified Littman-Metcalf type, Photonetics [Model 3642]) is used as a light source near 1.55/tim . The output optical spectrum is measured by an optical spectrum analyzer. Fig. 3(a) shows the measured optical powers in the case of m = 2 and N = 2, where the Mach-Zehnder interferometers with FSR (free spectral range) being 80 GHz and 40 GHz were used for generating the values X(l) and X(2) respectively and the wavelength was changed from 1560.00 nm to 1560.50 nm by the step-size of 0.01 nm. This agrees well with the Logistic map. Fig. 3(b) shows the measured optical powers in the case of m = 3 and N = 2, where the Mach-Zehnder interferometers with FSR (free spectral range) being 240 GHz and 80 GHz were used for generating the values X(l) and X{2) respectively and the wavelength was changed from 1551.00 nm to 1552.95 nm by the step-size of 0.05 nm. This also agrees well with the Cubic map.

5

Concluding Remarks

Wide variety of chaotic sequences have been obtained by tuning the wavelength of the light source and programming the phase shifts of Mach-Zehnder interferometers. Measured results have agreed well with analytic expressions of chaotic maps discussed in Section 2. The proposed devices have an engineering

311 merit that they are simple and they can easily be fabricated in a planar lightwave circuit because no active device element is used throughout the circuit. In this experiment, the sensitive dependency on the initial condition, a key characteristic of chaos, is observed as the sensitive dependency on the wavelength for the devices. Thus in this context these devices of chaotic generators can match recently developed wavelength-division multiplexing (WDM) networks.

References [1] L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64 (1990) 821-824. [2] G. D. VanWiggeren and R. Roy,Science 279(1998) 1198-1120. [3] S. M. Ulam and J. von Neumann, Bull. Am. Math. Soc. 53 (1947) 1120. [4] R. L. Adler and T. J. Rivlin, Proc. Am. Math. Soc.l5(1964) 794-796. [5] S. Katsura and W. Fukuda,P/j)/Mcal30 A (1985) 597-605. [6] K. Umeno, Phys. Rev. E 55 (1997) 5280-5284. [7] K. Umeno, "Exactly solvable chaos and addition theorems of elliptic functions",RIMS Kokyuroku No. 1098 (1999) 104-117. (electronic preprint chaodyn/9704007). [8] K. \Jmeno,Phys. Rev. E 58 (1998) 2644-2647. [9] Y. B. Pesin, Russ. Math. Surv. 32(4)(1977) 55-114. [10] K. Umeno and K. Kitayama,£/ectron. Lett. 35 (1999) 545-546. [11] K. Umeno and K. Kitayama," Improvement of SNR with chaotic spreading sequences for CDMA",Proc. IEEE 1999 Information Theory Workshop (20-25 June 1999, Kruger National Park, South Africa) (1999) p.106.(electronic preprint chaodyn/9903001). [12] K. Umeno," Chaotic Monte Carlo computation: a dynamical effect of randomnumber generations", (electronic preprint chao-dyn/9812013). [13] K. Umeno," Method of constructing multi-dimensional exactly solvable chaos",Proc. 31th Symposium on Celestial Mechanics (3-5 March 1999, Kashima Space Research Center, Communications Research Laboratory, Ibaraki, Japan) (1999) 88-92. [14] N. Takato, K. Jinguji, M. Yasu, H. Toba and M. Kawachi, J. Lightwave Technol. 6 (1988) 1003-1009.

312

Multi Mach-Zehnder Interferometers

j ' Wavelength Tunable Laser Diode

j

A. 2(|)

4(j)

— •

X(1) X(2) X(3) • • •

2 N -'(|)

X(N) ' Chaotic Sequence

Figure 2: Schematic of an optical chaos generator composed of N MachZehnder interferometers.

313

(a)

10

20

30

X(1)[uW]

Figure 3(a): Measured optical powers X{2) are plotted against the corresponding measured optical powers X(l) when the conditions N — 2 and M = 2 are satisfied for the optical chaos generator in Fig. 2. (b): Measured optical powers X{2) are plotted against the corresponding measured optical powers X{\) when the conditions N = 2 and M — 3 are satisfied for the optical chaos generator in Fig. 2.

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SPLITTING OF ATTRACTORS INDUCED BY RESONANT PERTURBATIONS

V.N. CHIZHEVSKY 12 ' 3 , R. VILASECA1 , R. CORBALAN2 and A.N. PISARCHIK23 'Departament de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Colom 11, E-08222 Terrasa, Spain. 2 Departament de Fisica, Universitat Autonoma de Barcelona, E-08193 Bellaterra (Barcelona) Spain. 3 Stepanov Institute of Physics, National Academy of Sciences, 220072 Minsk, Belarus.

ABSTRACT We present experimental and numerical results on the dynamics of a loss-modulated C02 laser which demonstrate that weak resonant perturbations at subharmonic frequency (i) induce bistability in the system, splitting the primary attractor into two new ones, and (ii) shift all bifurcation points for these two coexisting attractors in opposite directions as the perturbation amplitude changes. Both perturbation-induced bistability and shift strongly depend on the phase difference between perturbation and fundamental frequencies. We also demonstrate well controllable switchings between new coexisting attractors using the targeting technique based on the action of a short-lived pulse perturbation. 1. Introduction Over the last years much attention has been paid to the effect of resonant perturbations on the dynamics of nonlinear systems. This is motivated by the fact that weak periodic perturbations at subharmonic frequency, used in the nonfeedback control technique, can suppress chaos1'2. Typically, the resonant perturbations at subharmonic frequency, depending on their amplitude and phase and the initial state, may produce stabilizing or destabilizing effects on the nonlinear system. As it has been experimentally shown, they may suppress chaos3"6, suppress periodic orbits, induce a chaotic behavior if the system is initially in a periodic state, and, finally, may induce crisis of strange attractors7. Up to now the main attention has been paid to the question of how resonant perturbations modify the dynamics of nonlinear systems. This raises another question, namely, whether the system itself is essentially modified by introducing the weak resonant perturbation. By essential modifications we mean the appearance of new attractors in the system induced by weak resonant perturbations. Recently, it has been theoretically shown8 from an analysis of a logistic map that feedback control9 may induce new attractors in optical systems. Here, we show experimentally and numerically that weak periodic perturbations at subharmonic frequency (i) induce bistability in the system so that instead of one primary (initial) attractor at least two attractors appear in the system and (ii) shift all bifur-

315

316 cation points belonging to these branches in opposite directions as the perturbation amplitude changes. We also demonstrate that bistability and shift of the bifurcation diagrams strongly depend on the phase difference between perturbation and fundamental frequencies. From the practical point of view, for example, in engineering, a new problem arises here: how to switch the motion from one attractor to another one? Recently, for this purposes the technique of targeting periodic orbits has been proposed and developed experimentally10"'2 which is based on the action of short-lived large-amplitude perturbations to the system that is equivalent to a change of initial conditions. There exists the optimal timing and the optimal amplitude of pulse perturbations which allow one to perform switching practically without transient12. In this context, the second aim of this work is to show that a combination of resonant perturbations which induces bistability in the system with the targeting technique makes the nonfeedback control of nonlinear systems (not only in a chaotic state) more flexible in the sense of fast switching between controlled orbits belonging to new coexisting attractors. For some initial states we demonstrate experimental examples of such a type of control. This technique does not require any feedback system and might be experimentally implemented in diverse nonlinear systems. 2. Experimental setup and laser model The experiments were carried out on a C0 2 laser with two acousto-optic modulators inserted in the laser cavity13'14. We performed two series of experiments. In the first ones the driving signal, Vd(t)\cos (27ft), had the frequency/= 160 kHz and the amplitude Vd(t) which was periodically changed by a triangle law with a frequency 30 Hz. The second series of experiments was performed with a constant driving amplitude Vfi. The perturbation signal, Vp cos(nft+cp), had the frequency/^, an amplitude Vp and a phase
317

calculations the following fixed parameters were used: T =3.5 x 10" 9 s, Y = l^VS^lO* Yo = 0.175, k„ = 0,17303. The other parameters were varied in the simulations. 3. Results and discussions First, let us describe the results of computer simulations. Figure 1 shows typical bifurcation diagrams of a modulated C0 2 laser obtained from Eqs. 1 and 2 with triangle sweeping (3) of the driving amplitude and stroboscopic sampling of the data. Fig. la corresponds to reverse sweeps of the driving amplitude without the perturbation (k„=0).

6

1-

r

•1 ; up- * i

0

1000 2000 3000 4000 Time (number of periods T)

Fig. 1. Numerical bifurcation diagrams of a C0 2 laser showing the appearance of the bistability. (a) Laser response without perturbation, (b) and (c) responses with perturbation. The parameters used in simulation: Ay varied between 1.24x10~4 and 4.4*10-5 (a) kp=0, (b) and (c) kp=9xl0\
0 0.5 1 Time (units of the shape period 1/F) Fig. 2.. Numerical bifurcation diagrams of C0 2 laser versus time for different values of the perturbation amplitude kp and phase f with a triangle shape of the driving amplitude shown in (a). £ = 22°and 120 , respectively.

When a weak resonant perturbation of a frequency ofJJ2 is introduced the primary solutions splits into two solutions, which are shown separately in Figs, lb and lc, respectively. From Fig. lb it is seen that as the driving amplitude reaches some critical value, which depends on the perturbation amplitude, the solution-1 undergoes an inverse 2T-T bifurcation and becomes an unstable T periodical solution. After that the system approaches rather quickly to the other coexisting stable solution (solution-2) with changing the phase in the response by n. This transition corresponds to a new location of the first bifurcation point shifted by the resonant perturbation. Thus, rather weak resonant pertur-

318 bations induce bistability just above the first shifted period doubling bifurcation point and therefore changes drastically the nature of the system. From a comparison of Figs, lb and lc it is also seen that the resonant perturbation pushes in opposite directions the bifurcation points belonging to these new solutions. Fig. 2 shows the effect of periodic perturbations with different amplitudes and phases on both coexisting solutions. Here one period of the slow periodic shape of the driving amplitude is shown (Fig. 2a). The system periodically alternates in time between two solutions. In order to simplify Fig. 2, the part of solution-2 which coexists with solution-1 is not shown. As the perturbation amplitude increases (from Fig. 2b to Fig. 2d), all bifurcation points of solution-1 are shifted to a higher value of the driving amplitude and chaos in solution-1 disappears (Figs. 2c and 2d). At the same time all bifurcation points of solution-2 shift in an opposite direction towards lower values of the driving amplitude as the perturbation increases. One can see in Fig. 2d the appearance of chaos. Thus, we can state that the resonant perturbation produces simultaneously stabilizing effects on one attractor and destabilizing effects on the other one. A further increase of the perturbation amplitude leads to a reduction of the parameter range of the driving amplitude where solution-1 exists. At some critical value of kp (this critical value essentially depends on the
Time (units of the shape period) Fig. 3. Time derivative of C0 2 laser responses for Fig. 4. Time derivative of C0 2 laser responses for different values of the perturbation amplitude Vp two values of phase W which differ approximately and with triangle shape of the driving amplitude K
319

The experimental investigations qualitatively confirm all main conclusions obtained from numerical simulations. Figure 3 shows experimental bifurcation diagrams versus time in units of the shape period 1/F for different values of the amplitude of the resonant perturbation at the frequency f/2. In order to reveal more clearly the location of all bifurcation points we represent the time derivative of the bifurcation diagrams. In the absence of the perturbation, the diagram in Fig. 3b shows delayed bifurcations at a forward sweep with respect to a reverse sweep. When we introduce the perturbation, one can see (i) the appearance of two branches (which for reverse sweep look very similar to the solutions obtained numerically (compare Fig. 2 and Fig. 3) and, correspondingly, the appearance of bistability, (ii) the transition between two coexisting branches with a n phase shift in the laser response, that corresponds to reaching the unstable T-orbit in the first branch, and (iii) opposite shifts of all bifurcation points belonging to these two different coexisting attractors, so that the first attractor is stabilized while the second one is destabilized as the perturbation amplitude increases (Fig. 3c, 3d and 3e). We also found experimentally that the shift strongly depends on the perturbation phase (Figs. 4a and 4b) and amplitude. For the given perturbation amplitude the maximal shift was observed at

320 2T orbit (Fig. 6c) and vice versa. The later case is very interesting from standpoint of a fast change of the state in the system between "chaos" and "order".

(a)

4. Conclusions To conclude we have shown that weak resonant perturbations fundamentally modify the phase ?2 space of nonautonomous systems by splitting a primary attractor into two new ones. We have also ) 8T 2T ^ turbations. Such a type of control of nonlinear sys4 _s—.—.—-,^ tems could be of interest in the context of control of the systems with fast response because it does not t 1 require any feedback systems. '(c) Chaos

«>

# 3

is

2T

f

0

5. Acknowledgments 1

1000 2000 Time (units of T) Fig. 6. Experimental stroboscopic C0 2 laser responses versus time (the laser intensity is sampled with the modulation period T). The unperturbed system is initially in a 4T state. The experimental parameters: Vd = 4.11, Vp = 3.52 V, (a) and (b) f = 215° , (c) and (d)

V.N.C. acknowledges support from the Generalitat de Catalunya. A.N.P. is indebted to the Ministereo de Educacion y Cultura (Spain) for a grant No. SAB94-0538. Financial support from the DGICYT (Project No. PB95-0778-C02) is also acknowledged.

6. References 1. 2. 3. 4.

R. Lima and M. Petttini, Phys. Rev. A41 (1990) 726. Y. Braiman and I. Goldhirsch, Phys. Rev. Lett. 66 (1991) 2545. A.S. Azevedo and S.M. Rezende, P/jys. Rev. Lett. 66 (1991) 1342. L. Fronzoni, M. Giocondo, and M. Pettini, Phys. Rev. A43 (1991) 6483.

321 5. W.X. Ding, H.Q. She, W. Huang, and C.X. Yu, Phys. Rev. Lett. 72 (1994) 96. 6. R. Meucci, W. Gadomski, M. Ciofmi, and F.T. Arecchi, Phys. Rev. E49 (1994) R2528. 7. V.N. Chizhevsky and R. Corbalan, Phys. Rev. E54 (1996) 4576. 8. A. Gavrilides, P.M. Alsing, V. Kovanis, and T. Emeux, Opt. Commun.115 (1995) 551. 9. E. Ott, C. Grebogi, and J.A. Yorke, Phys. Rev. Lett. 64 (1990) 1196. 10. V.N. Chizhevsky and S.I. Turovets, Phys. Rev. A50 (1994) 1840. 11. V.N. Chizhevsky and P. Glorieux, Phys. Rev. E 51 (1995) R2701. 12. V.N. Chizhevsky, E.V. Grigorieva and S.A. Kaschenko, Optics Commun. 133 (1997) 189. 13. V.N. Chizhevsky, R. Corbalan, and A.N. Pisarchik, Phys. Rev. E56 (1997) 1580. 14. V.N. Chizhevsky, R. Vilaseca, and R. Corbalan, IJBC (1998), to appear. 15. J.R. Tredicce, F.T. Arecchi, G.P. Puccioni, A. Poggi, and W. Gadomski, Phys. Rev. A31 (1988) 2073. 16. P. Glorieux, C. Lepers, R. Corbalan, J. Cortit, and A. N. Pisarchik, Opt. Commun. 118(1995)309.

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X. Quantum Chaos

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METHODS IN ACOUSTIC CHAOS

CLIVE ELLEGAARD AND KRISTIAN SCHAADT Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen 0, Denmark

ABSTRACT We study experimentally the eigenvalue spectra and the eigenfunctions of acoustics in solid objects (e.g. single quartz crystals). The experiments are inspired by quantum chaos, and the results are discussed within the framework of Random Matrix Theory. However the wave-equation in elastic media is very different from the Schrodinger equation and generates a variety of acoustic modes. For two-dimensional systems we show experimental methods of selecting the various types of eigen-modes associated with the different modes of sound propagation - and show how they mix. We show how we can (in contrast to quantum mechanics) scan the wavefunction itself.

1. I n t r o d u c t i o n Acoustic resonance spectroscopy is used as one of the analog systems to yield experimental data for discussing quantum chaos. Analog system means a system governed by a wave-equation analog to the Schrodinger equation. The first analog system studied in this way was microwave cavities 1 . In a two-dimensional cavity the Helmholz equation for the electric field becomes exactly analog to the Schrodinger equation in 2D. The point of using the analog system is that the data produced can be of much higher quality than for true quantum mechanical systems. The acoustic system presented here is an example of such very high quality data. Chaotic (or complex) quantum systems are often discussed in terms of Random Matrix Theory (RMT) 2 . Experiments on analog systems can test the predictions of RMT to levels of acuracy not possible before. Experiments on systems governed by wave-equations different from the scalar Schrodinger equation can test the range of validity of the ideas of RMT. The acoustic system is thus an ideal system. It yields very high resolution data and the wave-equation governing the displacement field is vectorial with the consequence that the wave propagation is much more complex than in quantum mechanics. 325

326 The first use of an acoustic system was in 1989 by Weaver 3 . More recently specific transitions between regularity and chaos and details of symmetry breaking have been studied by EUegaard et al. 4,5 . These first experiments stressed the similarity with the Schrodinger equation. The present paper will discuss some of the main differences and show some experimental techniques to separate out specific elastomechanical effects. In linear elasticity, three types of waves exist for three-dimensional infinite media: one longitudinal wave, called P (for 'primary' or 'pressure'), and two transverse waves, called S (for 'secondary' or 'shear'). Their wave velocities are: cP =

p{l + v)(\-2v)

'

cs =

\j2p{l + v

where E is Young's elastic modulus, v is the Poisson coefficient and p is the density. For a finite body, these waves propagate independently inside the medium, and couple only at the boundaries. In a thin plate the transverse waves become two distinct types depending on whether the displacement is perpendicular to or in the plane of the plate. These modes are named flexural and shear respectively. The flexural modes do not couple to other modes, whereas the in-plane shear modes can be expected to couple to the other in-plane modes, the longitudinal modes, through mode conversion. 2. E x p e r i m e n t The general experimental setup is: The solid object, in this case an aluminium plate, is suspended as freely as possible. We need to defy gravity and that is done by placing the plate on three hard points, in this case three small ruby spheres. This ensures minimal contact with the surroundings and we have as near as possible a freely vibrating plate. The three spheres are glued to piezo-electric transducers of which one acts as transmitter and the two others as receivers. The transmitter is fed the amplified sine wave from the source of a spectrum analyser and the signal from a receiver is amplified and sent to the input of the analyser. The analyser thus scans the acoustic response of the plates. The plates are thin in the sense that, at the frequencies used, there is not room for 1/2 wavelength in the thickness direction. The measurement usually takes place in vacuum (10~2 Torr) in order to keep the damping from air at a minimum. The effect of damping is part of the discussion in the next section. In vacuum Q-values are of the order of 104 . For typical spectra are see fig. 3. 3. S y m m e t r y We have conducted a series of measurements where we started with a quadratic plate where the spectra showed the Poissonian statistics expected for a regular sys-

327 tern. We then broke more and more symmetries until we ended up with the shape called a Sinai stadium with no visible symmetries. For this plate both the nearest neighbour distance distribution (NND) and A 3 -statistics indicate a remaining symmetry, fig. 1 left. The measured curves in both cases correspond very closely to two

Figure 1: NND (top) and A3 (bottom) statistics for the measured spectra of the 2mm thick quarter of a Sinai stadium plate. Left: with no cut, right: with 2 cuts. Comparison with the Poisson distribution, a single-GOE distribution and a double-GOE distribution. noninteracting GOE distributions. The source of this is to be found in the symmetry with respect to the midplane of the plate. The three different modes of the plate: the bending (flexural), the compressional and the in-plane shear modes fall into two classes as shown in fig. 2. The flexural mode is antisymmetric with respect to reflection in the midplane. The two others are symmetric with respect to this reflection. Furthermore the latter modes are expected to mix through mode-conversion at the surface (in this case the edge) of the plate. The modes from these two symmetry classes do not interact. To give additional proof of this interpretation we break the mirror symmetry by cutting a slit in the plate on one side. The result is shown in fig. 1 right, and we see that all levels are now strongly mixed and we get exactly one GOE distribution for both NND and A 3 .

328

Flexural

In-plane

v=0

v«u

v»u Vertical transverse

Horizontal transverse

Longitudinal

Figure 2: The vibrational modes can be either flexural or 'in-plane' modes. On this exaggerated figure, we illustrate the displacement vector (u,v), and show its properties concerning the mirror symmetry with respect to the middle plane of the plate. Flexural modes: u is antisymmetric and v is symmetric. In-plane modes: u is symmetric and v is antisymmetric.

4. M o d e selection w i t h width distribution What we will show now is a way of separating the two classes experimentally. Fig. 3 shows a section of a spectrum recorded with different surrounding pressure. One clearly sees that for some peaks the width is essentially unchanged whereas for others the width increases strongly with pressure.

Vacuum

0 08

0.06

0.02

J

I1 ii

Vacuum

i ju ^

0.04 0 02 n

(I,

r

1 ^

Frequency [kHz]

0.08

f[HZ] 0 i

Intermediate pressure

0 08

r l\ » 1 1 J V A Jl.J <

\

0 96

F

0.04

0.04

v iji

0.02

402

• 404 Frequency (kHz|

intermediaie pressure

0 06

0.02

A-A^-

,il,

/L

J"Ul\

1

r[Hzi 0 1 0 08

Atmospheric pressure

Atmospheric pressure

0.06 0.04

Ji_JL

0 02

A

Figure 3: A segment of the resonance spectrum at three different pressures. The resonance labelled I (F)'is an in-plane (a flexural) mode.

A.... n

^

^

•

Figure 4: T h e distribution of widths for the acoustic resonances at pressures similar to the ones in fig. 3.

329 The flexural modes have by far the largest coupling to the surrounding atmosphere, and fig. 4 shows how these modes can be cleanly separated from the in-plane modes. Plotting the NND and A 3 statistics we get the results shown in figs. 5 and 6. For the flexural modes we find exactly one GOE. For the in-plane modes we find something intermediate between one and two GOE, indicating that the symmetric modes are not just one class. The mixing induced by mode-conversion is evidently not that strong.

Figure 5: The A 3 statistics for the ilexural modes and the g=1.0 random matrix result (dashed curve).

Figure 6: The A3 statistics for the inplane modes and the g=0.1 random matrix result (solid curve).

To get an estimate of the coupling strength, we make a small model within RMT 6 . We let H be a random real symmetric NxN matrix with the following block structure: H

(DA + A 0 \ , „ f 0 C\ ~ { 0 DB+B) + S [CT 0 )

where A and D are random Ni x N\ matrices, and B and DB are random N2 x N2 matrices. The random matrix C is Ni x N2, and the coupling strength g is a real parameter. The elements of the diagonal matrices DA and DB are drawn uniformly on the interval [-0.5,0.5] and ordered in increasing order for each block. The elements of A, B and C are Gaussian distributed with zero mean. The variance of the distribution of the diagonal elements of A and B is set to 16/iV 2 . The variance of the off-diagonal elements of the matrices A and B and the elements C is set to half of this. With this definition of the random matrix ensemble the symmetry conserving and symmetry breaking matrix elements scale in an appropriate way, and the transition from 2 GOE (g=0.0) to GOE (g=1.0) takes place as a function of g, independent of the size of the matrix.

330 We t h e n fit t h e resulting calculated NND a n d A 3 t o t h e e x p e r i m e n t a l d a t a and o b t a i n a value of 0.1 for t h e coupling p a r a m e t e r g, see fig. 6. 5.

Wavefunctions

In this e x p e r i m e n t an e x t r a receiving t r a n s d u c e r (a pickup) is a d d e d . This pickup can be a c c u r a t e l y positioned in a fine grid bv m e a n s of t h e xv table. T h e transmitter is t u n e d to keep t h e p l a t e excited at one given resonance, a n d t h e pickup scans t h e whole surface of t h e plate to give t h e a m p l i t u d e l a n d s c a p e of t h e eigenfunction. T h r e e e x a m p l e s of wavefunctions are shown in fig. 7 t o g e t h e r with their a m p l i t u d e distributions.

Figure 7, left: Contour plots for three modes at 318.9 kHz, 425.1 kHz, and 510.b kriz, exemplifying eigenfunctions that are flexural, in-plane transverse, and in-plane longitudinal, respectively. To enhance contrast, the grayscale is logarithmic. Black represents maximum intensity, white is zero intensity. Right: plots showing probability distribution for the intensity, corresponding to each mode. The smooth curve is the Porter-Thomas distribution.

331 The amplitude distribution gives the Porter-Thomas distribution as expected from RMT. RMT 2 also gives a prediction for the autocorrelation function

W = (|*(ro)|W+r;)|') where ( ) means averaging over starting points and directions. The prediction is P2(r) = 1 + 2Jg (r). To fit this distribution, r has to be scaled according to a dispersion relation, and it turns out that the dispersion relation needed is either pure flexural, pure longitudinal or pure shear. This again indicates that the in-plane modes mostly retain each their character of longitudinal or shear respectively, in agreement with the small coupling parameter found from the eigenvalue spectra. 6. References

1. H.J. Stockmann and J. Stein, Phys. Rev. Lett. 64 (1990) 2215; U. Stoffregen, J. Stein, H.-J. Stockmann, M. Kus and F. Haake, ibid. 74 (1995) 2666; E. Doron, U. Smilansky and A. Frenkel, ibid. 65 (1990) 3072; S. Sridhar, ibid. 67 (1991) 785; A. Kudrolli, V. Kidambi and S. Sridhar, ibid. 75 (1995) 822; D. Graf, H.L. Harney, R. Hofferbert, H. Lengeler, A. Richter, P. Schardt and H.A. Weidenmiiller, ibid. 74 (1995) 62. 2. T. Guhr, A. Miiller-Groeling and H.A. Weidenmiiller, Physics Reports 299 (1998) 189 and references therein. 3. R.L. Weaver, J. Acoust. Soc. Am. 85 (1989) 1005. 4. C. Ellegaard, T. Guhr, K. Lindemann, H.Q. Lorensen, J. Nygard and M. Oxborrow, Phys. Rev. Lett. 75 (1995) 1546. 5. C. Ellegaard, T. Guhr, K. Lindemann, J. Nygard and M. Oxborrow, Phys. Rev. Lett. 77 (1996) 4918. 6. A. Andersen, C. Ellegaard, A. D. Jackson and K. Schaadt, to be published.

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XI. Mechanics

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STABILITY TRANSITIONS IN A NONLINEAR AIRFOIL LAWRENCE VIRGIN, STEPHEN TRICKEY AND EARL DOWELL School of Engineering, and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708-0300.

ABSTRACT This paper describes some experiments conducted in a wind tunnel on a twodimensional airfoil with a loose flap. The piecewise linear nature of the flap stiffness (incorporating a deadband) is a strong source of nonlinearity. It is also a good model of the inevitable and increasing looseness encountered in aircraft control surfaces, and in mechanical systems in general. In the absence of the freeplay nonlinearity the airfoil behaves in a classical manner (i.e., an eigenvalue problem) with aeroelastic flutter encountered at a critical flow speed, i.e., effective damping becomes negative at a Hopf bifurcation. When freeplay is added to the model a host of pre-flutter limit cycle behavior is observed together with sudden stability transitions. Quasi-periodicity and chaos is also encountered and many of the observed features are generic, in the sense that much of this behavior occurs in sequences familiar from the classical circle map. Numerical simulation is used to verify the results of the experiment using a standard aeroelastic analysis. 1. Preamble This paper describes certain stability transitions encountered in the oscillations of a nonlinear airfoil. A somewhat simplistic schematic preview of the main types of (hysteretic) transitions is shown in Figure 1.

n—r~

> Response

J\t

> Flow rate

Figure 1: A schematic of the generic bifurcations highlighting hysteresis in the response of an airfoil subject to changing flow conditions. The principal control parameter here is the flow rate. The system initially experiences a subcritical Hopf bifurcation. The resulting limit cycle response then experiences a saddle335

336 node bifurcation or a secondary Hopf bifurcation. Quasi-periodicity and chaos also appear in the vicinity of these areas of co-existing attractors. These instabilities occur in sequences that are very similar to those encountered in the circle map. Despite the inevitable uncertainty present in experiments (e.g., unsteady flow effects, measurement error, etc), and the relatively high-order nature of this fluid-structure interaction problem, it is encouraging to observe the good correlation between theory and experiments. Furthermore, it is also promising that simple, generic systems such as the circle map, can anticipated much of the behavior in a more complex system. 2. The Airfoil The potentially serious effect of a free-play nonlinearity on the response of airfoils is well established1, especially within the context of wear and maintenance where a degradation of moving joints is inevitable. Although the main emphasis in this paper is experimental behavior, some numerical simulations of the governing equations of motion are also conducted (with special consideration given to the subtle stiffness change as the free-play effect is encountered). The airfoil section whose aeroelastic behavior is under study is the NACA 0012 and it is shown in Figure 2(a). (a) (b)

Figure 2: The typical section NACA0012 with flap. The degrees of freedom are: • plunge, h (nondimensional, H = h/b8) • pitch, a (nondimensioanl, A =oc/8) • flap, p. (nondimensional, B = p75) The free-play is present at the axis of rotation of the aileron. The details of this modeling are quite involved2"4, consequently only an overview is given here. In terms of nonlinear dynamics we have 3 mechanical degrees of freedom and together with their rates of change give a six dimensional phase space. In addition the aerodynamics of the flow lead to an infinite dimensional state space that is approximated by a finite (two) state model. The structural system is modeled as a number of connected linear sub-systems which

337 switch abruptly due to a discrete change in stiffness. This is shown schematically in Figure 2(b) where the flap experiences no restoring force between certain contact angles. However, once the critical angle (±S) is exceeded then the system experiences a linear restoring force. Numerically, it is important to match accurately the switching point (since the error will tend to be cumulative)5. Previous experimental work in this area6"7 has suggested the possibility of finiteamplitude limit cycle oscillations prior to the onset of flutter. This study confirms the presence of quite complex nonlinear dynamics, for a relatively large range of flow velocity in both the numerical and experimental studies. 3. The Aeroelastic Model Theodorsen theory8 is used to model the aerodynamic effects on a typical section (NACA 0012) airfoil which leads to the following nondimensional equations of motion : IJk + (J„ + b(c - a)Sp)'$ + Sah + Caa = Ma

(1)

(Ip + b(c - a)Sp)a + Ip'0 + SJi + CpP = Mp

(2)

Saa + sJ + Mh + Chh = L,

(3)

where the aerodynamic modeling is contained in Ma,M„ and L, and is based on 2D, incompressible, potential flow, i.e., these forces are composed of linear combinations of cc,d,p, etc. The full modeling of this system includes Theodorsen aerodynamics consisting of Jones' approximation of Wagner's indicial loading function910. The spring stiffnesses, C, static moments, S, and moments of inertia, /, are fixed and (c - a) represents the distance between the flap axis of rotation and the overall center of rotation of the airfoil.

Re[Pitch] Re[Plunge] Re[Flap]

2

Re[X] 0

^ ~

^ ** ""

-2

_,— - " " "

---""""""""^

'••

-fi

I :

~~~

l

~~J><*^^

X v

~"" --.

-4

_ -

^ -^

.

•>*.

I

1

1.25

0 = U/IT

Figure 3: The characteristic eigenvalues of the linear system for the system with flap stiffness (continuous) and without (dashed). UF corresponds to 23 m/s, and is the nominal flutter for the linear system with flap stiffness.

338 Before encountering the full spectrum of nonlinear behavior, we make a preliminary study of the underlying linear system. The airfoil can be rendered linear in one of two ways: • Incorporating a purely linear flap spring stiffness (i.e., 8 —> 0) • Removing the flap stiffness altogether (i.e., 8 —> <*>). In both of these cases the airfoil experiences the onset of flutter (unbounded motion) at a Hopf bifurcation11. A pair of complex conjugate eigenvalues enter the positive-real half plane as the system experiences negative damping and completely loses stability. Both of these cases are illustrated in Figure 3 with the latter case exhibiting a critical flow speed about 27% that of the former case. In both cases it is predominantly the plunge mode that goes unstable. 4. Experimental Results A model airfoil conforming to the NACA0012 configuration was placed in the Duke University low-speed wind tunnel. Responses of the three degrees of freedom were measured using standard RVDT's, although some use was made of time-lag embedding of the flap (p) displacement, since this turned out to be the cleanest signal.

' -1'..

1

|lljlllj 1 §

:

0g|lB|H| 881 o

©o o o

s

—

•illiSooii t

'3

,

,

,

I

4

,

.

,

I

5

.

,

,

I

6

,

,

,

I

7

,

I

,

I

8

I

I

,

9

1 0 1 1

U(m/s)

Figure 4: A bifurcation diagram showing a variety of limit cycle behavior with coexisting oscillations (hysteresis) and abrupt stability transitions.

A bifurcation diagram is shown in Figure 4. Here, the flap position is used as a Poincare trigger, which is then plotted as a function of flow speed. This figure is the superposition of two sweeps, one with increasing flow rate and the other reversed. Since the major stability transitions occur at different flow rates we observe the presence of hysteresis. Focusing attention on the sweep up, the stable equilibrium position becomes

339 unstable at about 4.3 m/s. This response is typically quasi-periodic, as shown in Figure 5. The time series indicates that the motion consists of an oscillation which gradually grows, comes into contact with the flap stiffness (nondimensionalized to one) which seems to quench it somewhat, followed by a similar sequence. The power spectrum (linear scale) shows the side-bands present in this signal. The reconstructed phase projection is based on delay-embedding the flap position, and the closed nature of the Poincare section also suggests quasi-periodicity. Some numerical evidence suggests that this type of response depends quite strongly on a non-zero angle of attack.

4

6 time (sec) U = 4.5 m/s

-

0.04 RMS'

0.02

. .

, ..„MI

L., .

10

12

14

- 1 . 0

1

Frequency (Hz)

Figure 5: A summary of aperiodic behavior just after the loss of stability of equilibrium. The time series exhibits a beating effect. The power spectrum and Poincare section suggest this response is quasi-periodic.

340

On further increase in flow rate the systems settles in to a periodic oscillation which again transitions back into quasi-periodicity or chaos. It is important to point out that in this range of flow rates there are coexisting oscillations, which are not only due to hysteresis. It is possible to perturb the system from one motion into another at the same flow rate. At about 9 m/s the system jumps to a high-frequency 'buzz*-type motion dominated by the flap response (relative to the other degrees of freedom). Numerical simulation suggests that behavior persists until ultimately the airfoil goes catastrophically unstable at about 23 m/s but this is not confirmed experimentally because of the damage this would cause.

Figure 6: A contour plot showing the spectral content of the motion as a function of the flow speed. Figure 6 is a summary of the frequency content of the motion. These are spectrograms corresponding to the bifurcation sweeps of Figure 4. One of the important features here is that the sudden transitions occur at different flow rates: approximately 4.3 and 9.2 m/s during the sweep up, and 3.6 and 7.8 m/s during the sweep down. The hysteresis scenario sketched in Figure 1 is again confirmed although there are a number of subtleties involved. It is interesting to observe that the initial onset of limit cycle behavior is foreshadowed by a mild growth of energy near a characteristic frequency in the nominally stable conditions. It should also be noted that although conducting two sweeps provides a more complete picture than a bifurcation diagram generated from a single set of initial conditions, there may be other types of behavior which are simply not picked-up in a necessarily limited number of experimental runs. 5. Simulation Clearly there are some complicated dynamical processes at work here. We conclude by displaying some numerical results in order to shed light on the preceding experimental results. A bifurcation diagram based on numerical simulation is shown in Figure 7. Here, the pitch coordinate is used as the Poincare trigger, and since the response

341 measure is different from that used in Figure 4 we focus attention on the bifurcations from one type of behavior to another rather than a strict comparison of response.

0.015 Pitch, a

B

C D

0.01

W 15

n 20 U (m/sec)

Figure 7: The bifurcation diagram showing the loss of stability. We see the hysteresis associated with the initial onset of limit cycle behavior, although not the quasi-periodicity of Figure 5. The second region of hysteresis is a very complicated sequence of bifurcations involving both quasi-periodicity and chaos, prior to the evolution of the "buzz" periodic behavior for high flow rates observed in the experiments. A typical response within this region is shown in Figure 8.

3 4 Time (sec)

-0.01

-0.005

0

Figure 8: Typical time series and phase projection illustrating aperiodic behavior corresponding to "B" in figure 4 (U = 9.137 m/s). A closer look within this region reveals a variety of nonlinear behavior. Torus breakdown1213 occurs as the system transitions from quasi-periodicity to chaos. Poincare sampling can be used to highlight the transitions between different types of behavior. A typical example is shown in Figure 9 where Poincare projections are extracted from a very narrow range of flow rates (nondimensionalized with respect to the linear flutter velocity, UP), and plotted using the nondimensional pitch position and velocity. The first

342 part shows a closed curve corresponding to quasi-periodicity. This then transitions into a period nine oscillation (b), which is followed by (five-banded) chaos14, (nine-banded) quasi-periodicity, back to period-nine, and finally a more extended chaos. This type of behavior is not dissimilar to the stability transitions observed in the circle map. Of course, these are just projections from a high-order phase space onto the plane, but a variety of diagnostic techniques including power spectra confirm these subtle transitions. (a)

(d)

1.5

•-< 1.0

V

0.5

''0.106

0.107

0.108

0.109

;

•

0.110

A (b)

(e)

(c)

(f)

Figure 9: Some Pojncare sections taken from the pitch response over a very narrow range of flow rates: (a) U = 0.3944 to (f) U = 0.3951. 6. Conclusions This paper has shown that the response of a nonlinear airfoil subject to fluid flow exhibits rich bifurcational behavior. There is good correspondence between simulation and experiment, and a number of these features are quite generic. However, this example also points out the inherent difficulty in trying to extract good quality data from a

343 moderately high-order, relatively noisy experimental environment. The sequence shown in Figure 9, based on numerical simulation, occurs over a dimensional flow range of about 0.016 m/s, i.e., a range completely lost within the ability to control the flow rate, or even measure in the experimental model. However, the characterization, prediction and elimination of these generally undesirable effects is an important design consideration in practical aircraft development: a goal considerably assisted by the insight gained from nonlinear dynamical systems theory. 7. References 1. D.S. Woolston, H.L. Runyan and R.E. Andrews, An investigation of effects of certain types of structural nonlinearities on wing and control surface flutter, Journal of Aeronautical Sciences, 24, pp. 57-63, 1957. 2. M.D. Conner, Nonlinear aeroelasticity of an airfoil section with control surface freeplay, Duke University, 1996. 3. M.D. Conner, D.M. Tang, E.H. Dowell and L.N. Virgin, Nonlinear behavior of a typical airfoil section with control surface freeplay: A numerical and experimental study, Journal of Fluids and Structures, 11, pp. 89-109, 1997. 4. L.N. Virgin, E.H. Dowell and M.D. Conner, On the evolution of deterministic nonperiodic behavior of an airfoil, International Journal of Nonlinear Mechanics, 34, pp. 499-514, 1999. 5. M. Henon, On the numerical computation of Poincare maps, Physica D, 5, pp. 412414, 1982. 6. T. Theodorsen and I.E. Garrick, Mechanism of flutter, a theoretical and experimental investigation of the flutter problem, NACA Report No. 741, 1940. 7. D.M. Tang and E.H. Dowell, Damping prediction for stalled rotor flap-lag stability with experimental correlation, Journal of the American Helicopter Society, 40, 1995. 8. T. Theodorsen, General theory of aerodynamic instability and the mechanism of flutter, NACA Report No. 496, 1935. 9. E.H. Dowell, E.F. Crawley, H.C. Curtiss, D.A. Peters, R.H. Scanlan and F. Sisto, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 1994. 10. J.W. Edwards, H. Ashley and J.V. Breakwell, Unsteady aerodynamic modeling for arbitrary motions, A/AA Journal, 17, pp. 365-374, 1979.

344 11. H. Alighanbari and S.J. Price, The post-Hopf bifurcation response of an airfoil in incompressible two-dimensional flow, Nonlinear Dynamics, 10, pp. 381-400", 1996. 12. S. Ostlund, D. Rand, J. Sethna and E. Siggia, Universal properties of transition from quasi-periodicity to chaos in dissipative systems, Physica D, 8, 303, 1983. 13. T. Matsumoto, L.O. Chua and R. Tokunaga, Chaos via torus breakdown, IEEE Transactions on Circuits and Systems, CAS-34, pp. 240-253, 1987. 14. E. Ott, Chaos in Dynamical Systems, Cambridge University Press, 1994.

RAY CHAOS IN QUADRATIC INDEX MEDIA: A NON-MECHANICAL APPLICATION OF MECHANICS

RANDALL TAGG Department of Physics - Campus Box 157, University of Colorado at Denver, Denver, CO 80217-3364 and MASOUD ASADI-ZEYDABADI Department of Physics University of Colorado at Boulder, Boulder, CO 80309

ABSTRACT Hamilton's equations govern ray paths that describe waves traveling through media with slowly varying properties. A simple model of the variation of index of refraction with transverse and axial coordinates in a waveguide is shown to exhibit a combination of regular and chaotic dynamics. This suggests bench-top experiments that could explore chaotic ray propagation and its connection to problems in quantum chaos. 1. Introduction A remarkable analogy exists between mechanical motion of particles and the description of wave propagation through media with slowly varying properties. For electromagnetic or acoustic waves, the "geometric optics" approximation describes the wavefronts by tracing rays along the directions in which energy is transported. These rays act like particle paths and both rays and particle motions can be described by Hamilton's equations1,2. Indeed, even though we associate the Hamiltonian description with mechanics, this elegant formulation of the geometric approximation to wave propagation is due to Hamilton himself3. The existence of a Hamiltonian description of ray propagation suggests that chaos occurs in suitably chosen wave propagation problems in the geometric optics limit of vanishingly small wavelength. Questions arise about how features of such chaos will survive once the existence of finite (nonzero) wavelength is restored. This has direct connections to the problems of quantum chaos4'5. Ray chaos in inhomogeneous waveguides was reported by Chigarev and Chigarev6. Abdullaev and Zaslavski showed that key features of the nonlinear dynamics of Hamiltonian systems could be found in waveguide problems7'8.These include chaos and the nonlinear trapping of rays by resonances between the ray undulations and periodic variations in the waveguide properties. This and related work has been summarized in a recent review9 and book10. The existence of ray chaos in ocean waveguides was discovered by Palmer et al.11, and considerable interest developed over the chaos-induced exponential 345

346

proliferation of eigenmodes that connect a source and a receiverfixedat either end of a waveguide12'13. One signature of the ray chaos studied by several of the above authors is the introduction of a complex distribution of signal propagation times. Holm and Kovacic14 and Jiang et al.15 analytically found sufficient conditions, via the Melnikov method, for ray chaos to occur in certain models of the wave speed profile. As noted above, the problem of ray chaos is especially important because it connects with deeper issues of quantum chaos when the geometric approximation to wave propagation breaks down and diffraction effects must be explicitly considered. Recently, Sundaram and Zaslavsky analyzed the cross-over from the geometric to a full wave dynamics description in the context of the oceanographic models for sound speed variation16. Our entry into this problem shows how fundamental ideas recur in a variety of experimental contexts. We were interested in using sound as a diagnostic for vortexdominated flows1718 and in particular for cellular patterns like the stack of toroidal vortices formed between rotating cylinders in the Taylor-Couette experiment19. In order to gain insight into this problem, we looked at a geometric acoustics description10,20, and went back to the more general problem of deflection of sound waves in shear flows21. Here we uncovered the Hamiltonian structure for geometrical acoustics and guessed that suitably chosen flow profiles or sound speed profiles could generate chaotic ray trajectories. Thus we came across the literature reporting that such ray chaos indeed occurs, including examples of rays in moving fluids10'22'23. At this point, we set out to find a wave speed profile that is relatively simple, exhibits ray chaos, and might be experimentally achievable. Some plots of oceanographic sound speed11'13 show a single minimum across the waveguide. When models included periodic modulation of the sharpness of this minimum down the axis of the waveguide, chaos could result. These observations led us to the quadratic model discussed in Section 3 below, in which axially periodic modulations of the quadratic coefficient induce parametric instability and chaos for certain ranges of parameters. Literature on optical gradient index waveguides suggested making the square of the index of refraction depend quadratically on the transverse coordinate (see Eq. 31)24. In fact, the parametrically forced model was used earlier to describe the leakage of energyfromwaveguides in the ionosphere due to parametric instability of the ray paths25 and the nonlinear dynamics of ray paths in quadratic media was investigated by Abdullaev et al.26 and discussed in Abdullaev's book10. Thus our work underscores the idea that this model is a simple and useful prototype worthy of further consideration. Moreover, we demonstrate below that interesting dynamics, including chaos, occurs for parameter values that might be experimentally feasible in laboratory waveguides. 2. Derivation of the ray equations as a Hamiltonian dynamical system 2.1 Ray paths

Consider a medium in which the speed of light or sound varies slowly with position according to

*(*>-TV n(x),

(1)

347

where c 0 is a reference velocity and n(;t)is the local index of refraction. The following scalar wave equation is a model for light or sound propagating through this medium, where V could represent an electric field component or sound pressure: n(x) d ip

Vfy

T2

= 0.

(2)

To find a solution, try the form: ip « A(x)exp[ik0S(x)-cot], where a reference wavenumber is defined by

(3)

w

kh0*—. (4) c o The result of substitution of Eq. 3 into Eq. 2, after separating real and imaginary parts, is: V2A - fc02A|VSf + nzk02A = 0. 2VA-V5 + AV 2 5=0.

(5) (6)

In the geometric optics/acoustics limit, the wavelength 2JV I kg is vanishingly small compared to the scale over which the medium changes significantly, so that V2A kn A

-*0.

(7)

Then Eq. 5 becomes, asymptotically, what is known as the eikonal equation: |VSf-n(if) 2 . (8) Define the local wavevector by k * k„VS. (9) Surfaces of constant S are surfaces of constant phase, and thus k lies in the direction normal to the constant phase surfaces. The eikonal equation implies that the magnitude of k is determined by the local index of refraction: k = k0 n(x) . (10)

k(x(t))

k(x{t+

dt))

Figure 1. The ray geometry. The solid curve is the ray, the dashed curves are surfaces of constant phase, and the arrows are the local wavevectors.

348 Ray paths are defined to lie everywhere along the direction of energy propagation, which in this case is normal to the phase front (Fig. 1). In a time dt, a point on the ray path moves a distance c{x)dt in the direction of the unit vector k Ik. Thus: dx _ k ^-c(f)-, (11) dt k Substituting forc(x) and k from Eqs, 1 and 10, the i'th component of this equation is: dx, c0 k, 2 dt n(x") k0 The corresponding equation for the i'th component ks of the wavevector is obtained by recognizing that k = k(x(/)), so that: dkt -2, 6k, dx,

—L = y—-—-. dt

f& Ac j

dt

Since k is proportional to the gradient of a scalar function S(x) (see Eq. 9), dXj

(i3) (14)

dxt

This fact and substitution from Eq. 12 lead, after some manipulations, to:

4.fo_LIAtt2) Since k = V ( x ) (see Eq. 10),

fe0«(3r)22^

dt

(15)

dkt 1 dn(x) ~T-coK—— -r— • (16) dt n(x) dxt We now obtain the correspondence with Hamiltonian mechanics by noting that a>-c(Jc)Jt --^a-jfe. (17) n(x) From this the reader may verify that eqs. 12 and 16 for the ray position and wavevector are also obtained from: ^ dt

^

. dkt

(18) v

3L-*SL. (19) dt oKt These are precisely Hamilton's equations of motion for a system whose conjugate variables are x(t) and k(t) with Hamiltonianm(x,k) given by Eq. 17. This correspondence suggests that, through a suitable choice of index of refractionn(x), one could obtain ray trajectories that are chaotic. 2.2 One-Way Equations For the problem of a ray propagating down a waveguide, it is advantageous to change variables by assuming that the ray never turns back upon itself. Thus, if z is the direction down the waveguide, we require that dz — *0. (20) dt

349

Restricting to two dimensions and dividing the first (x) component of Eqs. 12 and 16 by dzldt gives: — .hi. (21) dz kz ^£. = ^ o l n * ! . (22) dz kz dx Note that

kt-JF^kJ-JiKxftf-k;.

(23)

Define p = kx lk0 as the new variable conjugate to x. Then the dynamical system for the ray trajectory is: ^ =__L=. (24) 1 dz ^ ' " -P * dn r,i — Q-- , dx °* . (25) Here the axial coordinate z has taken the role of time. The variable p is related to the angle 8 that the ray makes with the axis of the waveguide (the z-direction) by: p = nsinfl. (26) Finally, observe that eqs. 24 and 25 can be written in the form of Hamilton's equations for the conjugate variables x and p

T-fdz

op

?--?• dz

The Hamiltonian H is given by

dx

(27)

(28

>

H - -Jn(x,z? - p2. (29) Since z takes the role of time, the ray dynamics is generally non-autonomous when the index of refraction varies in the Z -direction. However, when this axial variation is eliminated the Hamiltonian is a constant of motion. Equivalently, - H = ncos6 (30) is a constant along the ray path. This is a generalization of Snell's Law of refraction for a medium whose index of refraction varies continuously in the transverse (x) direction. The presence of cosine rather than sine that is in the familiar form of Snell's Law is due to the definition of angle 8 relative to the waveguide axis.

350 3. The quadratic index profile and its closed-form solution In order to think about more commonly available optical waveguides, we change to cylindrical coordinates, expressing the transverse coordinate as r rather than x*. Only meridional rays (those lying in a plane containing the waveguide axis) are considered. There is a specific expression for the index of refraction that, in the absence of axial dependence, gives a closed-form solution for the ray path. (In fact, closed form solutions for the wave equation also exist and bear similarity to quantum harmonic oscillator solutions27.) Following the notation of Abdullaev10, this expression is the quadratic profile that approximates the form of real gradient-index optical waveguides24: r2 nir)1 =n02 -(r^2 -nJ)~Y for r s a. (31) a n r ( ) " n » for r> a. (32) Now rescale variables so that r'=*rla, t = zla, ri= n/nx, n0' = /% /n„and p'= plnx. (33) For convenience, drop the primes in the following, understanding that the variables are scaled in the remainder of this paper. Thus the scaled index of refraction has the form: n{rf = n02 -(n„ 2 - l)r 2 for r s l . (34) n(r) - 1 for r > 1. (35)

L

z

s

Figure. 2 Geometry of the dynamics of meridional rays in a cylindrical waveguide. The ray geometry is shown in Fig. 2. The dynamics (see Eqs. 24 and 25) is given by ^ =- ^ = = .

(36)

dn

*

V«

-P

Suppose that for z=0, the initial conditions are that the ray starts on the axis of the fiber, r = 0 , at an initial launching angle 8 =90. Then the ray has the trajectory: r=

JlAsm2?*

(38)

* Our notation, standard for optical fiber descriptions, is unfortunately the exact opposite of the notation used in the oceanographic literature. There, z is the transverse coordinate (depth) and r the axial coordinate (range).

351

p - ng sm00 cos

.

(39)

Here the ray oscillates with a wavelength Z that depends on the initial angle 80: z=2^cos0,

(4Q)

V«o 2 -l 4. Parametric forcing and instability Now introduce a perturbation that periodically modulates the strength of the quadratic term of the index of refraction profile in the axial direction z: II2

n

2

(n

2

11

r2

a(z) n(r) = 1 for r > a(z).

for r s a(z).

(41) (42)

Here we have defined (43) , A and interpret the perturbation as an axial variation in the characteristic scale ti(z) of the quadratic index of refraction profile. One could imagine, for example, that a gradient index optical "slug" is extruded at varying rates through a die in order to give an undulatory variation in the thickness of the resulting fiber. Since z acts like time in this dynamical system, Eq. 41 implies a parametric forcing of the system, which becomes a second-order system with periodic coefficients. Consider the stability of the axial ray, r = 0 , p = 0 ( 0 = 0 ) Vz, (44) that is a solution to the dynamical system of Eq. 36 and 37 even in the presence of the axial perturbation. The linearized equations are a{z) s 1+ecos

f - A dz n0

(45)

dz n0 a(z) An equivalent single differential equation in the variable r is: d\ (n 0 2 -l) 1 _ 0. (47) 1 dz n02 ~a(zf This is a linear second order equation with periodic coefficient and is an instance of Hill's equation28. Note that for the unperturbed system, the wavelength Z of sinusoidal oscillation of a ray about the axis for small initial angle 60 approaches the value Z

o--T^Tas0o^O,

(48)

as may readily be seen to be the period of the harmonic solution to Eq. 47 when e = 0 and thus a = 1. For definiteness, we will choose n0 -1.5 in the remainder of this article. In this case, Z0 = 8.430. One expects from Floquet theory of such systems to obtain two

352

types of solutions when the perturbation is turned on (e * 0): one that is quasiperiodic, and another that is a periodic function multiplied by an exponentially growing amplitude. The latter is the case of parametric instability. The type of solution obtained depends on the parameters s and A that respectively define the strength and wavelength of variation of a(z). The unstable solution occurs in "tongues" in the e -A parameter plane that come to points on the A - axis (where e - 0) at A„ - - Z 0 ,

where n - 1,2,3,- •

(49)

The case where A : = Z 0 / 2 is the well-known subharmonic instability, for which the period of the drive (ate = 0) is half the natural period of oscillation. This instability is the easiest to obtain because it occurs for the widest range of parameters £ and A. Figure 3 shows three phase portraits for the first three types of unstable solutions of the linearized system: n = 1,2,3 in Eq. 49. The heavy dots are plotted every time z increases by A and thus form a superimposed Poincare section. The dots anticipate the locus near the origin of the unstable manifold of the Poincare section of the fully nonlinear system (see below).

Figure 3. Phase portraits and superimposed Poincani sections (heavy dots) for the linearized dynamics starting with initial conditions r0=0, 0O=1°. For all three cases, no =1.5 so that the fundamental wavelength Zo=8.430. (a) Subharmonic instability: A=4.6; (b) Instability near the fundamental: /1=7.6; (c) Instability near 3/2 times the fundamental, A=l 1.1 In all cases a strong perturbation amplitude £=0.5 was used so that it was easier to find the instability bands for the higher harmonics. Notice that the number of cycles needed to spiral out and escape increases dramatically for the higher harmonics, indicating weaker instability.

S. The Fully Nonlinear System and Chaos The preceding analysis suggests choosing parameters in the subharmonic instability band in order to get strongly nonlinear behavior and to look for chaos near the axial ray. It is important to note that the instability ejects initially small-angle rays well away from the axial direction so that a paraxial (small angle) approximation cannot apply. Nonlinearity, however, is sufficient to keep the rays confined within the waveguide. The question then becomes whether or not the confined rays follow regular or chaotic dynamics.

353

Ray trajectories were numerically computed using a 4th-order Runge-Kutta algorithm with fixed step size Az = 0.05 (recall that the characteristic width of the waveguide has been scaled to a value 1). Calculations with variable step size and using other integration routines, including the numerical integrator built into Matlab, obtained similar results. Figure 4 shows a composition of Poincare sections computed for a range of initial conditions. The result shows many of the features one would expect from nonlinear Hamiltonian systems: periodic orbits, quasiperiodic dynamics, resonant islands, and chaos.

-60

Figure 4. Composition of Poincare" sections for «o=1.5, £=0.15, A=4. The sections that encircle the origin were computed for initial conditions r»=0 and 6b=l° to 37° in 1° increments. The "target patterns" that lie to either side of the origin were computed for initial conditions 6a=0° and /-«=0.1 to 0.6 in 0.05 increments. Also, a stable periodic orbit occurs for initial conditionfib=0°and r»=0.625±0.025. In all cases, the axial variable z varied from 0 to 5000.

Of particular interest is the apparent chaos that occurs for small initial angles (0O up to 2°). The subharmonic instability ejects these orbits out to angles approaching 45° before the orbits return close to the origin, where the return appears to create a chaotic "sea." 6. Discussion and conclusion We have demonstrated that a quadratic index medium with periodic axial variations exhibits many features of mixed regular and chaotic dynamics in the geometric

354 approximation. We believe the parameter values may be achievable in experimental optical or acoustic waveguides constructed on the "bench top." Important limitations may still make experiments difficult. As noted above, the application of the geometric approximation depends on the criterion a » A0, where A0 - 2jt lk0 is the reference wavelength of light or sound used in the experiment. In experiments, we anticipate choosing dimensions such that a IA0 = 1000. However, the axial range Z over which the geometric approximation remains valid is also limited; eventually diffraction effects would wash out interesting structures suggested by the Poincare sections. It has been shown29 that z must remain within the Fresnel zone defined by the width of the waveguide: zla «k0a -6000. (50) Thus the distance 5000 chosen for the Poincare sections in Fig. 4 is about at the limit for the scales of wavelength and waveguide width envisioned. Even tighter limitations on z will likely arise in attempting to investigate the waveguide with beams of width b intermediate between the wavelength and waveguide dimension: ideally, k « b « a . Thus, the possibility of bench-top experiments with perturbed quadratic index waveguides seems tantalizingly feasible, but remains subject to further analysis. 7. Acknowledgments We thank Daryl Holm, James Meiss, David Palmer, and Michael Shlesinger for helpful conversations. Significant portions of this work were done while enjoying the hospitality of the Center for Nonlinear Dynamics at the University of Texas in Austin and the Center for Nonlinear Science at Los Alamos National Lab. Funding for work on sound scattering from fluid flows was provide by the Research Corporation Cottrell Science Award C-3366 and funding for work on parametrically forced dynamical systems was provided by the Office of Naval Research contracts N00014-95-1-0185 and N0001499-1-0768. References 1. D. Marcuse, Light Transmission Optics, 2nded. (Van Nostrand Reinhold, New York, 1982) p. 94. 2. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993). 3. W. R. Hamilton, Mathematical Papers, Vol. /(Cambridge University Press, Cambridge, 1931). 4. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer Verlag, Berlin, 1990). 5. L. E. Reichl, The Transition to Chaos in Conservative Classical Systems: Quantum Manifistations (Springer-Verlag, Berlin, 1992). 6. A. V. Chigarev and Yu V. Chigarev, Akust. Zh. 24 (1978) 765 [Sov. Phys. Acoust. 24 (1978) 432]. 7. S. S. Abdullaev and G. M. Zaslavskii, Zh. Eksp. Teor. Fiz. 80 (1981) 524 [Sov. Phys. JEW 53 (1981) 265],

355 8. S. S. Abdullaev, Chaos 1 (1991) 212. 9. S. S. Abdullaev and G. M. Zaslavskii, Usp. Fiz. Nauk. 161 (1991) 1 [Sov. Phys. Usp. 34 (1991) 645]. 10. S. S. Abdullaev, Chaos and Dynamics of Rays in Waveguide Media, ed. by G. M. Zaslavsky (Gordon & Breach, Switzerland, 1993). 11. D. R. Palmer, M. G. Brown, F. D. Tappert, and H. F. Bezdek, Geophys. Res. Lett. 15 (1988) 569. 12. K. B. Smith, M. G. Brown, and F. D. Tappert, J. Acoust. Soc. Am. 91 (1992) 1939. 13. F. D. Tappert and X. Tang, J. Acoust. Soc. Am. 99 (1996) 185. 14. D. D. Holm and G. Kovacic, PhysicaD 51 (1991) 177. 15. J.-Y. Jiang, T. A. Pitts, and J. F. Greenleaf, J. Acoust. Soc. Am. 101 (1997) 1971. 16. B. Sundaram and G. M. Zaslavsky, Chaos 9 (1999) 483. 17. F. Lund and C. Rojas, Physica D 37 (1989) 508. 18. F. Lund, in Instabilities andNonequilibrium Structures, ed. by E. Tirapegui and D. Villarroel (D. Reidel, Dordrecht, 1987) p. 117 19. R. Tagg, Nonlinear Science Today 4, no. 3 (1994) 1. 20. D. S. Jones, Acoustic and Electromagnetic Waves (Oxford Univ. Press, Oxford, 1986) p.336. 21. C. S. Jones, "Sound Propagation in Moving Fluid," M A S . Thesis (University of Colorado at Denver, 1996). 22. S. S. Abdullaev, Chaos 3 (1993) 101. 23. S. S. Abdullaev, Chaos 4 (1991) 63. 24. W. B. Allan, Fibre Optics: Theory and Practice ((Plenum Press, London, 1973) p. 212. 25. N. T. Afanasyev, K. F. Svistunov, and M. V. Tinin, Izv. Vuzov Radiofizika 25 (1986) 133 [Radiophys. Quantum Electron. 25 (1986)]. 26. S. S. Abdullaev and Yu. A. Kravsov, B. A. Niyazov and M. V. Tinin, Izv. Vuzov Radiofizika 29 (1986) 1920 [Radiophys. Quantum Electron. 29 (1988)]. 27. Snyder, A. W. and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983) p. 263. 28. W. Magnus and S. Winkler, Hill's Equation (Dover, New York, 1979). 29. L. M. Brekhovskikh and O. A. Godin, Acoustics of Layered Media I: Plane and Quasi-Plane Waves (Springer- Verlag, Berlin, 1990) p. 167.

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ELECTRIC STEP MOTOR : NON LINEAR DYNAMICS AND ESTIMATION OF THE EMBEDDING DIMENSION Marie-Cecile PERA, Bruno ROBERT Laboratoire d'Automatique et de Microelectronique, Universite de Reims Champagne Ardenne, BP1039 Reims, 51687 Cedex2, France

Dominique GUEGAN Ecole Nationale de la Statistique et de VAdministration Economique, 3 av Pierre Larousse Malakoff, 92245 Cedex, France ABSTRACT Dynamic of a hybrid step motor is studied by tools of chaos theory, as Feigenbaum diagrams and Poincare maps. A route to chaos by quasiperiodicity and a way out by period doubling are described. This behavior is observed experimentally. The embedding dimension is evaluated and the attractor is rebuilt.

1

Introduction

The main advantage of an electric step motor is open-loop operation : position control can be achieved without shaft position feedback. The shaft can be stopped in any position with a high degree of accuracy, producing incremental displacements. It is used in many applications like printers, hard disks, toys, robots,... The motor is supposed to operate at synchronous speed, the rotation speed is proportional to the supply frequency. But when the frequency increases, erratic behavior occurs, leading to a loss of synchronism. Usual studies tend to elaborate motor control that avoids this phenomenon [1]. An original approach is proposed to study this problem. As a matter of fact, the dynamics of the motor are studied by chaos theory. First, the structure of the motor is described. Then the modeling of the system is presented. In section IV, Feigenbaum diagram and Poincare sections are used to analyze the dynamics of the motor, as the supply frequency is increased, showing a route to chaos by quasiperiodicity. In section V, the decomposition in singular values is applied in order to investigate the embedding dimension of the strange attractor. 2

Presentation of the motor

The principle of the motor is presented in Figure 1 [2]. The stator has windings, 1 and 3 in series fed by the voltage U a , 2 and 4 in series fed by Up. la is the current in the

a) la = In. Ip = 0

b) I a = 0, Ip = I„ c) I a = I„, Ip = I„ Figure 1 : Principle of the step motor

357

358 windings 1 and 3, Ip is the current in the windings 2 and 4. The rotor has permanent magnets. Torque is developed by the tendency of the rotor and stator magnetic fields to pull into alignment according to the sequential feeding of the phases. If phase a (windings 1-3) is fed, stator induction is horizontal and the rotor is also horizontal (Figure la). If phase P (windings 2-4) is fed, stator induction is vertical and the rotor turns of one step (Figure lb). If the two phases are fed simultaneously, induction produced by the stator has an intermediate position, the rotor turns of a half step. Phases are switched alternatively. Let's consider the following cycle : 1 (Ia=In, Ip= -In), 2 (Ia=In, Ip= In), 3 (I«=-In, Ip= In), 4 (I„=-In, Ip= -In) The rotor has four stable positions during the switch cycle, which are -7t/4, n/4, 3n/4, 57T/4. This supply mode is mostly used and is called mode 2. Torque has two origins : firstly, teeth on the stator and the rotor create a variable reluctance, secondly the magnetization of the rotor creates an interaction rotor magnets stator currents. According to the physical phenomenon responsible for the torque, motors can be classified in : variable reluctance motors, permanent magnet motors and hybrid motors. Variable reluctance motors have teeth on the stator and on the rotor, but no permanent magnet on the rotor, torque is due to the variable reluctance. Permanent magnet motors have rotors radially magnetized as described on Figure 1. The studied motor belongs to the third type, which is mostly used. The stator has salient poles, with two phases. The rotor includes a cylindrical axial permanent magnet, mounted on the shaft and placed between two iron disks with teeth (Figure 2). Due to the axial magnet, Z, teeth of the same disk have identical polarity, that is one disk bears Z, south poles, the other one bears Zr north poles. Teeth of disks are shifted of an electric angle 71, so a stator tooth faces alternatively north and south poles. The studied motor has Zr=12 teeth on each rotor disk, so the rotor has 2*12=24 poles. It is fed in mode 2 ; U a and Up are square voltages with phase shift of n/2. Then, the motor has 48 stable positions, it is a 48 steps/tour motor , with a stepping angle of 7.5°.

Figure 2 : Hybrid motor with Zr = 10 [3]

359 3

Motor model

Both phenomena described above contribute to the torque. Interaction magnet - currents leads to the torque : T, = - Kh I a sin (Z, GJ + K„ Ip cos (Z, 6m)

(1)

0 m represents the mechanical position of the rotor. Variable reluctance generates a detent torque, responsible for four stable positions when the motor is unexcited : T2 = -KhSin(4Zr9m) (2) F is a friction coefficient, T c a load torque, Q m the rotation speed. The electromechanical equation (3) is then obtained. J d£2 EL= T n T dt ' (3) Phase voltages are :

u a =RI a +L—^--K E sin(z r e m )n m dt din

/

•,

u,, = R I „ + L - £ - + K E cos(zrem)nm dt

(4)

R and L are phase resistance and inductance, mutual inductance is considered as negligible because phases have a shift angle of 7i/2. The third term represents the emf due to the magnet flux. Then we obtain a system of four non-linear equations [1] :

^i a ++^n m sin(z r e m ) + -^W I„ =-—!„

i p = _l I p _Jk n m C O s ( Z r e r a ) + Ml «m = - ^ - I a s i n ( Z r e m ) + ^ - I p c o s ( Z r e m ) - ^ . s i n ( 4 Z r e m ) - y £ 2 m

- ^

(5)

The motor is unloaded so Tc=0. Other parameters have been identified as follow : R=45Q, L=275 mH, Z.^12, J=18.10"6 kgm2, Kh=Ke=0.463 Nm/A, F=10' 4 Nms/rd, Kd=16mNm, Tc=0. 4 4.1

Motor dynamics Feigenbaum diagram

In order to investigate all possible behaviors of the motor according to the supply frequency f, we plot Feigenbaum diagram. A fifth order Runge-Kutta method with fixed step is used to solve numerically the system (5) for different values of f. Motor operating time equals 20 seconds. The first 5 seconds are not taken into account in order to eliminate points which don't belong to the attractor. Each simulation has at least 75000 points. Then, results are sampled at the frequency f. For each value of the parameter f, we get 4 series, [1,^, Ipk, On*, Qmk]k=i,n- Figure 3 is obtained by plotting (f,Iok)k=i,n- [4]

360

f(Hz) Feigenbaum diagram of I a versus f

4.2

As can be seen on figure 3, until 52.1Hz, the n points are superimposed, I« is periodic, at the supply period T=l/f. Beyond this point, the n points are gathered in a set of 5 values, the solution has changed to a 5T periodic steady state. Beyond 54.7Hz, for some value of f, the n points are scattered over a range of values and appear like continuous segments, the corresponding steady states are aperiodic. The aperiodic solutions alternate with periodic ones. In order to characterize aperiodic solutions, Poincare sections have been used.

Poincare sections

The non-autonomous system (5) is driven by periodic functions U a , Up at frequency f. Therefore, the Poincare section is performed by sampling the trajectory at the same frequency. The Poincare section is projected in the plan (la, Ip) in order to be studied. It is obtained by plotting (la, Ip)k=i,nBetween 54.8 Hz and 60.8 Hz, solutions corresponding to the segments seen on the Feigenbaum diagram have closed curves as Poincare sections (Figure 4). They are quasiperiodic. [5]. It can be seen on Figure 4 that the points are not evenly distributed on the curve ; arrows show where points accumulate. This is the beginning of a modelocking. The mode-locking is achieved at 56.3 Hz, where the solution is 4T-periodic. A wider window of 3T-periodicity can be seen on Figure 3 between 57.8 Hz and 60.8 Hz. Beyond 60.8 Hz, Poincare sections corresponding to lines on the Feigenbaum diagrams are not any more real cycles. The outline get further distorted (Figure 5-6). This might be due to a chaotic behavior of the system. As f increases, beyond 65.8 Hz, the Poincare section evolves to the Figure 7.

0.02 0.04 0.06

J

0.08

-0.3

-0.28

4..

-0.26

-0.24

-0.22

I„(A) -0.2

-0.18

Figure 4 : Poincare section in the plan (Ia,I,i), f=55Hz

Figure 5 : Poincare section in the plan ( U e ) , f=63Hz

361

lotAJ

If A .

-0.15 -0.155 -0.16

*

^

/

/

jSfi/ -0.165 -0.17

// .4'

/y

/

/

///

•

/

Jj/ y /0 S ,Mi4^

-0.175

_ UA)

I„(A) -0.146 -0.144 -0.142

-0.14

-0.138 -0.136

Figure 6 : Detail of Figure 5

-0.3

-0.25

Figure 7 : Poincare section in the plan(Ia,l3), f=73Hz

Sensitivity to initial conditions has been tested for f=63 Hz and 73 Hz. A slight change in the phase vector is quickly amplified after few periods of supply. Pi and P2 on Figure 7 are initial conditions, slightly different (l%o on I«). Trajectories, originated from Pi and P2, diverge rapidly from each other : P'i and P'2, the image of Pi and P 2 respectively, after only 20 periods of U a , are clearly separated. Beyond 74.5 Hz, Poincare sections are segment alternatively visited by the solution, always in the same order, but the value inside each segment changes. At 75 Hz, the Poincare section exhibits 4 segments, visited in the order indicated on Figure 8. At 75.1 Hz, the Poincare section exhibits 8 segments. Beyond 75.1 Hz, the solution is periodic.

-0.06

Experimental data

--

-?

-0.08 -0.1 -0.12 -0.14

-0.265

4.3

•

ET -0.26

-0.255

—

in

-0.25

UA)

-0.245

Figure 8 : Poincare section in the plan (I0,Ip), f=75Hz

Experiments have been carried out with an industrial step motor, Crouzet 82940 0. It is fed by a bipolar voltage supply 24V. The H-bridge is included in an integrated circuit Motorola SAA 1042. Current measurements are performed by Tektronix Hall probe with a 5MHz frequency band. A National Instrument PC card AT-DSP2200, with two synchronized inputs and 16 bits resolution, is used for real time acquisition of currents. Numerical treatment and result display are performed with Matlab 5.0. Figure 9 and 10 show two experimental Poincare sections, which can be compared to the simulated ones on Figures 5 and 7.

362

-0.01 -002 -0.03

<

-0.M -0 05 ; -0.06 -0 07 -0.08

\ I1

V

-""IK*.

-0 03

)

X *•>,

-Q.12 lelptia

lalpha

Figure 9 : Experimental Poincare section 5

Figure 10 : Experimental Poincare section

Embedding dimension and attractor reconstruction

At 73 Hz, the motor has an unused behavior. We want to reconstruct the attractor in that case, in order to determine if it is chaotic or not. The study is performed on the simulated current I a . Different methods have been proposed to evaluate the embedding dimension [6-7]. Singular value decomposition has been used [8]. 5.1

Singular value decomposition

In order to calculate Lyapunov exponent and the attractor fractal dimension, we need to reconstruct the attractor in the smallest embedding dimension. According to Takens, the attractor can be reconstructed by the method of delays [9]. Only sampled data related to one variable is needed, for instance (IaK)k=i,n- The sampling can come from either a numerical calculation at fixed step or from experiment with a constant acquisition frequency. Considering the delay x, a multiple of the acquisition or calculation period, a vector Xk can be built from the delayed values of la, as follows : x

k

=

Uak Ia(k+t)"'lafk+mi) J

/g)

The evolution of the vector Xk with time follows an attractor which have the same dimension and Lyapunov exponents than the original attractor. (m+1) is the embedding dimension, i.e. the smallest dimension of the space where the attractor can be seen with no ambiguity. The problem is to find the adequate values for (m+1) and x. We have applied a statistical using concepts of information theory and signal processing, which is the singular value decomposition [8]. X is the trajectory matrix built from vectors xk as follows :

x

*ro

= # r X can be expressed a s : X=SSCT (8) where S is a (Nxm+1) matrix, C = (ci, C2,...,cm+i) is a (m+lxm+1) matrix, built from the singular vectors of X. The elements of the diagonal matrix E are the singular values of X. The number of significant singular values gives the relevant value of m+1, the others are related to numerical (or experimental) noise. It can be established that XC is the projection of the trajectory matrix X onto the basis Cj. The attractor is then reconstructed

363 in the basis c,. Singular values of X o"i have been calculated for different values of x and n, maintaining roughly constant the window length nx~ 13.7 ms. Figure 11 shows the spectrum of singular values for different values of n and x. Singular values are not influenced greatly by the variations of n and x. The embedding dimension seems to be 3. 5.2

Attractor reconstruction

The trajectory matrix has been projected in the singular vectors base (cj, c2, C3, C4) with a delay equal to 7 calculation steps. Figure 12 shows the projection in the space (cj, c2, c3). The structure of the attractor appears clearly. Calculation of Lyapunov exponents and dimension are going to be performed.

-15

Figure 11 : Singular values

6

-^

Figure 12 : Attractor reconstruction

Conclusion

Theory of chaos is not widely used for the study of electromechanical systems. We have applied it to a step motor. It is supposed to operate at a rotation speed proportional to the supply frequency but, when the frequency is increased, different behaviors occurs. Bifurcations from periodic steady states to subharmonic and quasiperiodic steady state have been shown using Feigenbaum diagram and Poincare sections. Some aperiodic steady states seem to be chaotic, there study need to be carried on. 7

References

1. F. BETIN et al., 2nd EPE chapter symposium on Electric Drive Design and Applications, Nancy, France, 1996, 187-192 2. M. ABIGNOLI and al., Techniques de I'Ingenieur, D3 III, 1991, D3690-1-D3690-21 3. M. KANT, Les actionneurs electriques pas a pas, Hermes, Paris, 1989 4. B. ROBERT et al., ELECTRIMACS'99, Lisboa, 1999, to be published 5. A. NAYFEH., Applied non linear dynamics, John Wiley & Sons, 1995, 233-238 6. D. BOSCQ, Statistical Estimation of the embedding dimension of a dynamic system, to appear in International Journal of Bifurcation and Chaos 7. P. GRASSBERGER, Physica 9D, 1983, 189-208 8. D.S. BROMHEAD and al., Physica 20D, 1986, 217-236 9. F. TAKENS, Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981

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SPHERES ON A VIBRATING PLATE: CLUSTERING AND COLLAPSE

JEFFREY S. URBACH and JEFFREY S. OLAFSEN Department of Physics, Georgetown University, Washington, DC 20057 A large number (-10,000) of uniform stainless steel balls comprising less than one layer coverage on a vertically shaken plate provides a simple but striking demonstration of some of the unusual phenomena associated with excited granular media. When the oscillation amplitude is large enough, the spheres are in constant motion, colliding randomly, like a two dimensional gas in equilibrium. Inter-particle collisions couple the horizontal motion of the gas to the essentially chaotic vertical motion of the spheres driven by the plate vibration. Viewed from above, the horizontal motion in the layer demonstrates interesting collective behavior as a result of inelastic particle-particle collisions. Clusters appear as localized fluctuations from purely random density distributions, as demonstrated by increased particle correlations. The clusters grow as the medium is "cooled" by reducing the amplitude of the plate vibration. The increase in local dissipation leads to the nucleation of a collapse: a close-packed crystal of particles that come to rest on the shaking plate and with respect to one another. The collapse is a motionless condensate surrounded by the remaining particles that continue in motion as a less dense gas. The two phase co-existence is hysteretic, and can be related to the chaotic trajectories of an individual inelastic sphere on an oscillating plate. 1

Introduction and Background

Granular media, systems comprised of large numbers of macroscopic particles interacting only through contact forces, display a variety of interesting behavior not found in ordinary solids, liquids, and gases. In a static granular system, e.g. a pile of sand, the unusual properties are a consequence of the disorder generated by gravitational barriers to relaxation: the energy required to move one particle over another is many orders of magnitude greater than thermal energies. When sufficient kinetic energy is added to the granular system, the grains are no longer in constant contact with one another and the medium becomes 'fluidized'. The peculiar properties of the static sandpile give way to a state characterized by rapidly fluctuating particle velocities, much like the motion of molecules in a liquid or gas. Because of the dissipation generated by colliding grains, energy must be continually added to the system, for example by external gas flow or mechanical vibration, and the results of equilibrium thermodynamics and statistical mechanics are not directly applicable. The unusual properties of granular gases have been investigated primarily through theoretical analyses and numerical simulation of freely cooling granular media.2 A collection of grains is initialized with a random velocity distribution (usually a Boltzmann distribution), and random initial positions. As energy leaves the system through dissipative inter-particle collisions, the homogenous state becomes unstable to the formation of large scale inhomogeneities in density and velocity. This instability is a consequence of the dissipation: if the density of some portion of the gas increases due to a random fluctuation, the collision rate will increase, resulting in an increase in the dissipation rate. This local 'cooling' of the granular gas will decrease the effective pressure, allowing for a further influx of particles from the surrounding gas.

365

366

Under some circumstances this instability can lead to inelastic collapse, where the particles essentially come to rest next to one another in the center of mass frame. Many researchers in the nonlinear dynamics community have become interested in fluidized granular media because of the close relationship between granular dynamics and the dynamics of other nonlinear, dissipative systems. Granular fluids display a variety of instabilities similar to those seen in fluids,1 but the underlying equations of motion are unknown. In addition, the methods used to input energy into the system are themselves interesting dynamical systems. In gas-fluidized granular media, energy is input by flowing a gas through the grains. The instabilities associated with the flow past an isolated sphere, together with the interactions between the spheres, combine to produce the complex dynamics of gas-fluidized beds. For mechanical vibration, the energy is input through collisions between the grains and the vibrating boundary. The interesting dynamics of this simple system is discussed in section 2.1. We have been investigating the behavior of a single layer composed of a large number of spherical particles on a vibrating plate. Some of our results, reviewed below, have been previously reported.3'4 2

Experimental apparatus

A schematic of the experimental apparatus is shown in Fig. 1. A 20 cm diameter horizontal aluminum plate with a smooth surface is vibrated vertically by an electromagnetic shaker. The particles placed on the surface are typically 1 mm stainless steel balls with a uniformity of better than 0.5%, and are contained by a wall around the

ccd camera

•

Diffuse Light Source

Q O Q O Q Q Q Q Q Q Q O Q Q O Q Q QOQQ

z(t) = Asin(cot)

Driving Electronics

Fig 1. A schematic of the experimental apparatus.

H—

Accelerometer

367

outside of the plate. The coverage is always less than one full layer. The acceleration of the plate is monitored with a fast-response accelerometer mounted on the bottom surface of the plate. The particles are illuminated by low angle diffuse light. This illumination produces a small bright spot at the top of each particle when viewed through a video camera mounted directly above the plate. Two digital video cameras are used for data acquisition, a high-resolution camera for studying spatial correlations (Pulnix-1040, 1024 x 1024 pixels, Pulnix America, Inc., Sunnyvale, CA), and a high-speed camera for measuring velocity distributions (Dalsa CADI, 128 x 128 pixels, 838 frames/second, Dalsa Inc., Waterloo, ON; Canada). 2.1

Relationship to a single particle on a vibrating plate

In the experimental setup we have easy access to the horizontal positions and velocities of the particles, but cannot directly measure what is happening in the vertical direction. It is nonetheless clear that the vibration of the plate inputs energy into the vertical kinetic energy of the particles, and this energy is transferred to the horizontal components through inter-particle collisions. Neither process is quantitatively understood. In thicker layers, the kinetic energy of the granular media is dissipated rapidly, and the interaction of the layer with the plate can be reasonably well modeled by a completely inelastic ball.5 For sinusoidal oscillations, the ball remains at rest on the plate if the peak plate acceleration is less than \g, the acceleration due to gravity. If the amplitude of the acceleration is increased a little above 1 g, the ball will separate from the plate at that point in the cycle where the acceleration reaches - g, and then recollide with plate and lose all relative velocity at some point after the plate acceleration has increased above - g again. At higher accelerations, period doubling bifurcations and other interesting dynamical phenomena are observed.6 The coefficient of restitution between the stainless steel balls and the aluminum plate in our experiment is approximately 0.9, so the individual particle-plate collisions are closer to completely elastic than completely inelastic, and because our coverage is always less than one full layer, inter-particle collisions do not dissipate energy very rapidly. As a consequence the particles collide with the plate randomly throughout the oscillation cycle. This can be seen directly in Fig. 2, which shows the signal recorded by the accelerometer attached to the plate. The dominant low frequency response is due to the sinusoidal forcing, but each time a particle collides with the plate, a high frequency impulse is generated as well. Measuring the rms amplitude of the high frequency component of the signal provides a measure of the relative probability of plate collisions through the cycle. As expected, collisions occur more frequently when the plate is moving upward, but no more specific correlation is observed. The dynamics of a partially elastic sphere on a vibrating plate have been extensively studied , and displays many features typically associated with chaotic dynamical systems. One relatively unusual feature is the appearance of locking, where the ball will come to rest on the plate by 'chattering' down during a period where the plate acceleration is above -g. When the plate acceleration drops below -g again, the ball will separate from the plate. Since most chaotic trajectories will eventually diffuse into a region of phase

368

Fig. 2 Signal from accelerometer for a plate oscillation of 70 Hz and r = l , showing sinusoidal forcing as well as the high frequency pulses generated by particle-plate collisions. The grid spacing is 5 milliseconds on the horizontal axis, and approximately 1/2 g on the vertical axis.

space that leads to chattering, the generic asymptotic trajectories are in fact periodic, although the periods of the trajectories can be extremely long. If the peak plate acceleration is less than g, a ball that chatters down will never again leave the plate, remaining in a trivial 'ground state' attractor. Thus initial conditions will typically lead either to relatively simple periodic trajectories or to the ground state attractor. Figure 3 shows two example trajectories for a point particle calculated from the equations of motion, using a coefficient of restitution of 0.9, and a peak acceleration of 0.95g. The initial conditions were chosen to demonstrate two characteristic behaviors: the ball in the top panel quickly chatters down to rest on the plate, while the ball in the bottom panel, dropped from a slightly lower initial height, is approaching a period 1 orbit.

10

20

30

40

50

60

70

Figure 3. Two trajectories for a point particle on an oscillating plate, for r=0.95

369

A more complete picture of the dynamics can be seen by taking a phase portrait of a number of trajectories. Figure 4 shows the result for 200 particles, followed for 100 collisions with the plate. At each collision, the phase of the plate and the kinetic energy of the particle are recorded. The dense clouds of point around an energy of 5.5 come from trajectories approaching the period-one orbit such as the bottom panel of Fig. 3, and the cluster of points close to the x-axis comes from trajectories chattering down to rest on the plate. Losert, et al9 have noted that there are apparently no periodic orbits with energies below that required for a period-one orbit (where the velocity must be such that the flight time equals one oscillation period). This 'energy gap' has important consequences for the granular layer. '

201

'

'

1

'

'

'

1

'

'

'

1

'

'

'

I

'

'

'

15 - •

10 - ~ ! "

0.0

v

'..

0.2

0.4

0.6

0.8

1.0

Figure 4. Phase portrait of 200 trajectories followed for 100 collisions with the plate. At each collision, the phase of the plate oscillation divided by 2JI (x-axis) and the kinetic energy of the ball (y-axis) are recorded.

In principle, a collection of perfectly spherical particles on a flat plate vibrating vertically with no initial horizontal velocities would not interact. Experimentally, this is not observed for any reasonable plate coverage. If the plate acceleration is increased from zero, rapid, random horizontal motion sets in almost immediately after the particles pick up significant vertical kinetic energy. This instability has been studied experimentally in a system similar to that shown in Fig. I.9 3

Results

When the plate oscillation amplitude is not too large, the spheres never hop over one another; thus the system essentially two-dimensional. Nonetheless, there is sufficient energy in the horizontal velocity component to generate fascinating dynamic phenomena.

370

At moderately large sinusoidal vibration amplitudes (maximum plate acceleration slightly larger than g), a fully fluidized state is observed. The spheres are constantly in motion and there is no large-scale spatial ordering. Figure 5(a) shows an instantaneous image of part of a cell containing 8000 particles (just under half a monolayer) in this regime. This phase is characterized by an apparently random distribution of particle positions and velocities. A sense of the dynamics can be gained from an average of 15 frames taken over a period of 1 s (Fig. 5(b)), which shows the lack of any stable structure. As the amplitude of the acceleration is slowly decreased, the average kinetic energy of the particles decreases, and localized transient clusters of low velocity particles appear. An instantaneous image in this regime (Fig. 5(c)) does not look very different from the one taken at higher acceleration, but in the time-averaged image (Fig. 5(d)) bright peaks are clearly evident, corresponding to low-velocity particles that have remained relatively close to each other over the time interval. In this regime, the clusters typically survive for 1-20 seconds. In the low density regions outside of the clusters, there are

Figure 5. Instantaneous (top row) and 1 second time-averaged (bottom row) photographs detailing the different phases of the granular monolayer, (a), (b), uniform particle distributions typical of the gas phase, (c) The clustered phase. The higher intensity points in a time-averaged image, (d), denote slower, densely packed particles. (e) A portion of a collapse, (f) The time-averaged image shows that the particles in the collapse are stationary while the surrounding gas particles continue to move. (g),(h) In a more dense system at higher frequency, there is an ordered phase where all of the particles remain in motion. (After ref. 3)

particles with anomalously high velocities, and these appear to be responsible for the breakup of the clusters. There are no attractive interactions between these particles; the cluster formation is a uniquely non-equilibrium phenomenon, resulting from the dissipation during inter-particle collisions. When the amplitude of the vibration is decreased somewhat below that of Fig. 5(c,d), the typical cluster size increases to 12-15 particles. Within a few minutes at this acceleration, one of these large clusters will become a nucleation point for a "solid' phase,

371

similar to what is referred to as 'inelastic collapse'.2 The particles in the collapse are in contact with all of their neighbors, and form a perfect hexagonal lattice (Fig. 5(d)). The collapse is surrounded by a gas of the remaining particles. The sharp interface between the coexisting phases can be seen in the time-averaged image (left panel of Fig. 5(e)) . The two-phase co-existence persists essentially unchanged for as long as the driving is maintained. The collapse is presumably nucleated by density fluctuations generated by the random motion of the particles in the gas. The behavior is analogous to first order transitions in equilibrium systems, when the high temperature phase is quenched to some temperature below the critical temperature for nucleation of the low temperature phase. The deeper the quench, the quicker the nucleation. The granular gas can be rapidly 'quenched' by abruptly decreasing the amplitude of the acceleration. The granular temperature drops almost immediately, because the inter-particle collisions rapidly dissipate the excess kinetic energy. Figure 6 shows the time required for the collapse to formed and grow to about its steady-state size after the plate acceleration is abruptly reduced from 1.15g. If the final acceleration is below about 0.88g, several nucleation sites typically form almost immediately, and the resulting collapse is polycrystalline. For quenches closer to the critical acceleration, a single nucleation site typically forms, but the time required for nucleation grows rapidly as the depth of the quench is reduced. As expected for a process dominated byfluctuations,the variation in the nucleation time

10*

10J

(sec)

1(f Domains!

10'

0.86

0.87

0.88

Homogeneous

0.89

r

0.90

0.91

0.92

Figure 6 Time until the formation of a collapse after an abrupt decrease in the acceleration from r=1.15. In the homogeneous regime, a single collapse typically nucleates, while for lower accelerations there are several nucleation points, and the resulting collapse is pollycrystalline.

372

from one run to the next can be quite large, and a more careful investigation is required before the critical behavior can be quantitatively investigated. At higher densities, instead of a transition directly from the clustering behavior to collapse, there is an intermediate phase with apparent long range order. Figure 5(g) shows a monolayer in this ordered state, where the spheres are arranged in a hexagonal lattice but are not at rest or in contact with one another. The disorder in the image is a consequence of the fluctuations induced by inter-particle collisions. When the particle positions in the ordered phase are averaged over a short time (Fig. 5(h)), the resulting image displays a nearly perfect lattice. Measurements of the correlation functions for positional and orientational order parameters in this phase suggest that the transition to this ordered phase is quantitatively similar to the solid-liquid transitions observed in a variety of equilibrium systems. The ordered phase shows long range orientational order, but algebraically decaying positional order. At slightly higher accelerations, the liquid phase shows algebraically decaying orientational order, but only short range positional correlations; in other words, it appears to be a hexatic liquid predicted by the work of Kosterlitz, Thouless, Halperin, Nelson, and Young.10 It is remarkable that this strongly dissipative, far from equilibrium system apparently displays the subtle dynamics of equilibrium two-dimensional melting.

0.95

r

6~^(a) N = 8k

Nucleation O-O Evaporation

0.85 0.75

collapse + gas H

h

H

\—i—i—i—h

h

(b) N = 14.5k

r ordered

30

40

50

60

70

80

90

v(Hz) Figure 7. The phase diagram for (a) N=8000 and (b) N=14,500 particles. The filled circles indicate the acceleration where the collapse nucleates (see text). The open circles in (a) indicate the point where the collapse disappears when the acceleration is increased. The diamonds in (b) indicate approximately where the transition to the ordered state occurs. (After ref. 3)

100

373

Figure 7 shows phase diagrams of the system with two different densities. The hysteresis is omitted from the lower diagram for clarity. The data for the nucleation points were taken by decreasing the plate acceleration in steps of about 0.003g, and waiting 5 minutes at each step to see if the collapse nucleates. From Fig. 6, it is clear that the precise location of the nucleation line will depend on the waiting time, but only very weakly. It is not immediately clear what causes the frequency dependence in the phase diagram. For an ideal spherical particle on an oscillating plate with a velocityindependent coefficient of restitution, the dynamical behavior depends only on T, and the frequency sets the timescale for the motion, and the length scale through g/a , which is proportional to the distance a ball falls during one oscillation cycle. Thus as the frequency is reduced for fixed Y, balls will bounce higher. Because the balls in this system interact with their neighbors, it is possible that the frequency dependence enters through the ratio of this length scale to the ball diameter. In fact the rapid increase in the acceleration where collapse forms for frequencies below about 50 Hz (see Fig. 7) occurs when the particles begin to bounce high enough to hop over one another, resulting gradual transition from primarily 2D to 3D dynamics. This suggests that the frequency dependence comes from a characteristic frequency vc=(g/d)1/2, where d is the sphere diameter. Figure 8 shows the phase diagram measured for two different sets of particles, one with a diameter of 1.2 mm, and one with a diameter of 1.6 mm. The upturn at low frequencies and the appearance of the ordered phase occur at lower frequencies for the larger particles, but the scaled phase diagrams lie right on top of one another.

IT

0* o

1.2

gas

J*
1.0 -

C^cftt^

ordered

+

0.8

gas + collapse 0.6

0.0

0.5

1.0

1.5

2.0

v/v„ Figure 8. The phase diagrams for 1.2 mm diameter particles (o, 0) and 1.6 mm (+,x). The frequency is scaled by vc=(g/d)"2.

374 3.1

Non-Maxwellian velocity distributions in the gas phase

As described above, the clustering is characterized by a decrease in the relative velocities of particles in the cluster. Dissipative particle systems will not necessarily show a Maxwellian velocity distribution, although models often assume that form for lack of anything better to use, and by arguing that it should be appropriate in the limit of low dissipation. The velocity distribution has been experimentally measured in a few cases, but not with sufficient precision to resolve the functional form, in part because these systems were inhomogeneously forced, with the energy input coming from the sides or from the bottom of a thick granular layer. The translational invariance of the system we are investigating facilitates precise measurements of statistical quantities because the entire system (sufficiently far from the walls) can be included in the ensemble.

10°

10"1

P(v/v0) 10" 2

10" 3 -3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

v/v Figure 9. Probability distribution function for a single component of the horizontal velocity. The data represents several different experimental parameters for values of T from 0.76 to 1.0. Each distribution is scaled by v0=(2v2m2)I/2, where v2m is the second moment of the distribution. (After ref. 3)

With the use of a high speed digital video camera, the particle velocities between collisions can be determined in the granular monolayer. Our measurements of the velocity distributions in the plane of the granular gas clearly show non-Gaussian behavior. In Fig. 9 the experimentally measured velocity probability distributions for several different parameters are shown on a semi-log plot, and compared to a Gaussian distribution (solid line). The distributions are scaled only with a characteristic velocity proportional to the second moment of the distribution. As no asymmetry is observed, the distributions contain both vx and vy data. The straight tails in Fig. 9 suggest an exponential velocity distribution.

375

The deviations from the Maxwell velocity distribution can be understood as clustering in momentum space: when the clusters form in real space they generate a population of low-velocity particles in a the high-density region. The corresponding areas of low density result in regions of decreased dissipation that allow for high-velocity particles. Thus, as seen in Fig. 9, the populations of low- and high-speed particles are higher than that expected from a Maxwell distribution, and the intermediate velocity population is smaller. The behavior of the velocity distributions at higher accelerations and an investigation of the cross-correlations between density and energy fluctuations is reported in reference 4, and non-Gaussian velocity distributions have been reported in a similar system.11 4

Discussion

An understanding of the interesting dynamics displayed in both the gas phase and the two-phase coexistence regions will require a better understanding of the flow of energy from the plate to the granular layer. Energy flow into thicker layers from a vibrating surface have been extensively studied,12 but the dynamics of the monolayer system are quite different. The net energy transferred to the vertical motion by the plate must balance the energy dissipated by inter-particle collisions. It is this balance that determines the steady-state horizontal granular temperature. The energy dissipated by a two-dimensional inelastic gas can be calculated using kinetic theory.13 There are several assumptions that go into the theory that have not been tested experimentally, but numerical simulations suggest that the equations provide a reasonably good model, provided the density is not too large.14 In order to generate an equation of state from kinetic theory, however, an expression for the energy input by the plate is required. The motion of a single ball on a plate displays the characteristics of low-dimensional chaos, but when coupled with a large number of similar systems through inter-particle collisions, the result is apparently a very regular rate of energy flow, producing a system that looks in many ways very much like an equilibrium system of a large number of interacting particles. The interaction of the spheres with the plate, and in particular the apparent lack of any periodic or chaotic orbits with average energies less than the period-one orbit, may partially explain the two-phase coexistence and hysteresis observed in this system.9 If the kinetic energy of the spheres drops too low, they will fall into the 'ground state', where they remain at rest on the plate, and the energy input drops to zero. Collisions from neighboring spheres may keep a sphere from falling into the ground state, but also dissipate energy. Quantitative predictions of the conditions under which density fluctuations can nucleate a collapse, as well as the stability of the collapse-gas interface, may be derivable from considerations of the sphere-plate dynamics, coupled with the kinetic theory of dense, inelastic gases. 5

Acknowledgements

This work was supported by an award from the Research Corporation, a grant from the Petroleum Research Fund, and by the National Science Foundation under grant DMR-9875529.

376 6 1

References

H. M. Jaeger, S. R. Nagel, and R. P. Behringer, Rev. Mod. Phys. 68, 1259 (1996) I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619 (1993); S. McNamara and W. R. Young, Phys. Rev. E 53, 5089 (1996). 3 J. S. Olafsen and J. S. Urbach, Phys. Rev. Lett. 81, 4369 (1998). J. S. Olafsen and J. S. Urbach, Phys. Rev. E, in press. See also http://www.physics.georgetown.edu/~granular 5 F. Melo, P. B. Umbanhowar, H. L. Swinney, Phys. Rev. Lett. 75, 3838 (1995) 6 A. Mehta and J. M. Luck, Phys.Rev. Lett. 65, 393 (1990). 7 For a review, see An Experimental Approach to Nonlinear Dynamics and Chaos, N. B. Tufillaro, T. Abbott, and J. Reilly (Addison-Wesley, New York, 1992). 8 J. M. Luck and A. Mehta, Phys. Rev. £.48, 3988 (1993). 9 W. Losert, D.G.W. Cooper, and J. P. Gollub, Phys. Rev. E 59, 5855 (1999). 10 For a review, see K.J. Strandburg, Rev. Mod. Phys. 60, 161 (1988) 1 ' W. Losert, et al, to appear in Chaos. 12 S. McNamara and S. Luding, Phys. Rev. £ 58, 813 (1998), and references therein. 13 J. T. Jenkins and M. W. Richman, Phys. Fluids 28, 3485 (1985). 14 C. Bizon, et al, cond-mat/9904132, submitted to Phys. Rev. E. 2

XII. Hydrodynamics

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Dynamics, Statistics and Vortex Crystals in the Relaxation of 2D Turbulence C.F. Driscoll, D.Z. Jin, D.A. Schecter, E.J. Moreau and D.H.E. Dubin Physics Dept., Univ. of California, San Diego, La Jolla, CA 92093

Abstract Magnetically confined electron columns evolve in (r, 6) as essentially inviscid, incompressible 2D fluids with a single sign of vorticity. Turbulent initial states with 50-100 vortices relax due to vortex merger and filamentation, in general agreement with a recent dynamical scaling theory. However, this relaxation is sometimes halted when 3-20 vortices "anneal" into a fixed pattern, or "vortex crystal." A new "regional maximum fluid entropy" theory predicts the crystal patterns and background vorticity distribution, by assuming conservation of the robust flow invariants and preservation of the intense vortices. Other simulations show that the character of the relaxed state depends strongly on initial conditions and dynamics.

Electron Plasmas and Euler Flows Magnetically confined pure electron columns are excellent systems for quantitative observations of 2D fluid vortices, turbulence and self-organization [1]. A "generic" experimental apparatus is shown schematically in Fig. 1. The electrons are contained within a grounded conducting cylinder. A uniform axial magnetic field (B < IT) provides radial confinement, and negative voltages (V < 50 Volts) applied to end cylinders provide confinement at the ends. The apparatus is operated in an inject/manipulate/dump-and-measure cycle. For injection, the leftmost cylinder is briefly grounded, allowing electrons to enter from the negatively biased tungsten filament. Then the confined plasma is diagnosed and manipulated by antennas on the wall. Finally, after a confinement time which may be as long as hundreds of seconds, the rightmost cylinder is grounded, and the zintegrated electron density n(r, 9, t) is measured by accelerating the electrons onto a phosphor screen and imaging the resulting light with a CCD camera. The (r, 9) flow of the electrons across the magnetic field occurs due to the strong electric field E(r, 9, z) = —V(/>(r, 9, z) from the unneutralized electron plasma. For the experiments discussed here, the electron gyrofrequency fi9 is a few GHz, which is much higher than other frequencies in the system. The gyroradius rg is a few /jm, 379

380

±L B

CCD Camera

-/h~~~ Electrons * - - - 4 — Filter

T

-50V

-50V

Phosphor

H>

Figure 1. The cylindrical experimental apparatus with phosphor screen/CCD camera diagnostic.

which is much smaller than the diameter of the plasma column (2RV « 2 cm) or of the conducting wall (2RW = 7 cm). In addition, the electron density n is far below the Brillouin limit (n/ns ~ -01), so electrons move slowly enough in (r,9) that electron inertia can be ignored. In this regime the electron velocity is well-described by E x B drifts as v(r,6,t) = E x B / B 2 , giving a bulk plasma rotation //?(r) = vg(r)/2nr « 10 kHz. Finally, the individual electrons move rapidly along the magnetic field lines, bouncing between the ends of the plasma at a rate /{, ~ 10 6 /sec. Since /& 3> fn, the electrons behave as rigid "bounce-averaged rods" of charge. In this approximation, the (r, 9) flow of the electrons is described by the 2D drift-Poisson equations [1], which can be written in terms of the vorticity ((r,0,t) = (4n\e\c/B)n and the scaled electrostatic potential ij;(r,0,t) = (c/B)(j>(r,6,t) as

dt

+

v

• V( = 0

-Vip x z ,

vV = C • These drift-Poisson equations are isomorphic to the Euler Equations. The flow vorticity ( is proportional to the electron density n, which is directly measured. A column of electrons in vacuum surrounded by a conductor thus evolves as would a 2D vortex in an incompressible inviscid fluid surrounded by a circular free-slip boundary. We emphasize that here, there is only one sign of vorticity (taken to be positive), because the density of electrons can only be positive, and there are no charges of opposite sign. There are also small unwanted diffusive effects due to the end confinement fields [2], and weak "viscous" effects on small spatial scales due to electron-electron collisions [3], but these are not modelled by the Euler or the Navier-Stokes equation. However,

381 we believe the effects described in this paper do not depend on the details of the fine-scale dissipation. Euler flows are strongly constrained by integral invariants. The total circulation (number of electrons), scaled angular momentum, and scaled electrostatic energy, given by

Jd2r(,

T tot

=

Pe

=

^2/d2r(l-r2)C/Co,

H

=

-(l/2)C2/d2r(C/Co)(^o),

are well conserved. Here, ( 0 = Ttot/Ri invariants such as enstrophy, given by

and ip0 = r tot /47r. However, less robust

Z2 = (l/2)RZ2Jd2r(C/Cof, vary significiantly, due to measurement coarse-graining or dissipation of small spatial scales. The simplest stable flow is a centered, symmetric vortex with monotonically decreasing vorticity profile £(r) and azimuthal flow velocity v#(r). Small shape distortions of this nominally symmetric equilibrium can be analyzed as a spectrum of waves with azimuthal and radial mode numbers (m, k), varying as gk(r) exp(im9 — iu^kt)- These waves are generalizations of the surface distortions on vortex patches referred to as Kelvin waves [4]. Recent analyses have elucidated the process of inviscid damping due to wave-fluid interactions at critical radii rc where uik/Sn = m fR(rc)[5: 6], and this damping is routinely observed in electron plasma experiments [7]. These modes have recently been analyzed in terms of "discrete" and "continuum" eigenfunctions [8, 9], with application to atmospheric circulations [9]. For even moderate wave amplitudes, this observed damping is typically nonlinear, and the damping may decrease [6, 7] or cease when the resulting "cat's-eye" flows generate fine-scale filaments inside the vortex. For "sharp-edged" vorticity profiles, the resonant radii rc can be completely outside the vortex, in which case no direct resonance damping occurs. Also, experiments have shown the importance of nonlinear wave-wave couplings: even otherwise stable modes may exhibit "beat-wave damping" [10], whereby energy is observed to flow to longer azimuthal wavelengths. If the vorticity profile ((r) is "hollow" rather than monotonically decreasing, some of these modes may be unstable, giving the Kelvin-Hemlholtz (shear-flow) instability [11, 12]. Both the frequencies and growth rates of these unstable modes are reasonably well characterized by computational solution of the eigenvalue equation using the measured density profiles [13]. One exception is m = 1, where we observe a robust exponential instability [14] where cold fluid theory predicts only algebraic growth;

382 here, finite length effects may cause the instability [15, 16].

Vortex Merger and the Relaxation of Turbulence The merger of like-sign vortices is fundamental to the relaxation of 2D turbulence at high Reynolds numbers. When two columns of electrons are injected symmetrically on either side of the trap center, they either orbit around the trap center, or merge together at the center, as shown in Fig. 2. Experimentally [17], the vortices are observed to merge within a few orbit times when the spacing between vortex centers D is less than 1.6 times the individual vortex diameter 2RV; and to orbit without merger for more than 104 orbits when Dj2Rv > 1.7. The merger after 104 orbits apparently results from weak non-ideal effects causing Rv to increase, thereby satisfying D/2RV < 1.6. However, the 104 : 1 ratio attests to the weakness of "viscous" effects, and suggests an effective Reynolds number Re « 104 — 105. Some of the circulation originally trapped in the two individual vortices is lost to filamentation during the process of merger. These filaments eventually get stretched to such fine spatial scales that they can not be imaged, and they form a weak background of vorticity. To study the relaxation of fully developed turbulence, we start with highly filamented initial conditions, which rapidly form 50-100 vortices, and then freely relax toward a 2D meta-equilibrium [18]. The 2D meta-equilibrium state then persists until viscous or other dissipative processes cause further relaxation to a thermal equilibrium state [19]. In the initial inviscid relaxation, chaotic mutual advection and vortex merger are clearly important dynamical processes. The final meta-equilibrium is typically strongly peaked on center, reflecting the single intense vortex resulting from repeated mergers, superimposed on a weaker background vorticity resulting from filamentation and mixing. Surprisingly, this relaxation is sometimes halted when individual vortices settle into a stable, rotating "vortex crystal" pattern which persists for thousands of rotation times. Figure 3 shows the measured ?-averaged electron density n(r, 6, t) at five times for two slightly different initial conditions: the upper sequence forms vortex crystals, whereas the lower sequence relaxes rapidly to a monotonically-decreasing profile [18]. The observed vortex crystal states consists of 5-20 individual vortices each 4-6 times the background vorticity, arranged in a lattice pattern which rotates with the background. That is, rods of enhanced electron density (n v ~ 7 x 106cm~3) are maintaining self-coherence and positions relative to each other for several seconds, while E X B drifting with a diffuse background (ubkg ~ 2 x 10 6 cm~ 3 ). Vortex crystal states are repeatably observed over a range of filament bias voltages, but the characteristics of the initial n(r, 9) required for these states are not fully understood. Figure 4 shows the number of distinct vortices Nv, and the enstrophy Z2 for the two sequences. In each sequence, the unstable filamentary initial condition forms Nv = 50 — 100 vortices of roughly equal circulation, after which Nv initially decreases

383

Vortex Merger E^\

Vorticity (arb. units) -100.

• 10. 1

1

1. 40 uS

80 uS

200 uS

Figure 2. Experimental images of the vorticity C(r> times during the merger of 2 identical vortices, showing fine-scale filamentation which eventually leads to a weak background of vorticity.

Figure 3. Images of vorticity a t five times for two sequences from similar initial conditions.

384

10* ? Nv 10 1

o Crystal **•

_

*+SS8oe<,, o • • „o *+ + + +

oo

+ +++++++++++++++++••-

o

2Erv/rtot

10"

+^ * ^ | N * + t 1 * # | + 10"

J

10" 10"

I

J

1

I

10*

J

I ••••

10*

t/T„ Figure 4. Number of surviving vortices Nv, their circulation ^ r v , and their average radius < r v > for the two evolutions of Fig. 2. as Nv ~ t~t, with £ « 1. This relaxation is generally consistent with a dynamical scaling based on conserved quantities in repeated vortex merger [20]. The observed f range from 0.2 to 1.1 as the initial conditions are varied, with 0.8 being commonly observed. In the evolution of the top sequence in Fig. 3, the relaxation is arrested by the "cooling" of the chaotic vortex motions, with formation of vortex crystals by WrR. (Here, TB = 1 / / R ( 0 ) = 170/^s.) The diamonds in Fig. 4 show that 8 to 10 distinct vortices survive for about 104TR. Since the surviving vortices all have about the same circulation, the patterns are quite regular, as seen at 600TH in Fig. 3. After 10 4 TR, JVV decreases to 1 as the individual vortices decay away in place due to nonideal "viscous" effects. Other experimental images show that as Nv decreases, the remaining vortices re-adjust to a new rigidly rotating, symmetrically spaced pattern. The measured integral quantities for both sequences are consistent with 2D inviscid motion on large scales and dissipation on fine scales. Experimentally, the circulation, angular momentum, and energy are robust invariants. In contrast, the enstrophy Z2 is a "fragile" invariant, and initially decays a factor of about 4 in both sequences. For the crystals sequence, Z2 is constant from IOTR until 10 4 TR, after which time the individual vortices decay in place due to non-ideal effects. Reduction of the chaotic advective motions of the individual vortices is required to form the vortex crystal states. Figure 5 shows the average magnitude of the random velocities of the individual vortices, with respect to their common rotating frame \SV\. The random velocities decrease a factor of 6 between 2 TR and 100 TR for the

385

10 - 1

i 11 linn—rrrrnnj—i iniiiii—ITMIIIII—i i MINI—i i niiiij—r iiinu

V 00 X)

a> >

10 - 2

o Crystal + Monotonic

>

Mllllll

"

O

'"

I

10 - 2

10*

t / T

o

o •

W O © 11 mill

'"'

10<

c

Figure 5. Decrease in the average random vortex velocity \SV\ as the crystal sequence "cools."

crystals sequence, whereas only slight cooling is seen before relaxation to Nv = 1 for the monotonic sequence [18]. We believe this cooling and cessation of relaxation through mergers is a nearinviscid 2D fluid effect, i.e. independent of the details of the fine-scale dissipation. However, two essential characteristics of this system are the non-zero total circulation and the boundary of the vorticity patch; these effects may not be present in other systems. Here, because there is no "negative" vorticity, the diffuse background necessarily persists, and the vortex/background interactions are more pronounced. In order to show that the cooling of turbulent flow to vortex crystals is a 2D inviscid effect, we have compared the experiments directly to vortex-in-cell simulations with up to 106 point vortices [21] to approximate 2D Euler dynamics. The cooling curves are approximately the same in simulation and experiment, and they appear to follow power law decay as \5V\ oc t~a. Simulations with a different number of point vortices have verified that the vortex cooling and crystallization is not sensitive to this discreteness. The point vortex gas is equivalent to an ideal Euler fluid only in the mean field approximation. Microscopic fluctuations of the vorticity about the mean field cause distributions of point-vortices to eventually relax to global 2D maximum entropy states. We believe that cooling occurs through the chaotic mixing of background vorticity by the strong vortices, as opposed to processes such as deformations of individual vortices in the crystal pattern. To test this, we start with the simulated flow at t = \ATR, then we artificially multiply all "background" vorticity by a constant ranging from 0 to 3. Evolving this artificial system forward, we observe cooling which depends on the strength of the background vorticity. Figure 6 shows the cooling exponent a versus r b k g / r t o t - There is no cooling in the absence of the background,

386

o

r

u, 7T bkgd

i tot

Figure 6. Cooling exponent a versus relative strength of the background vorticity. and there is no cooling if almost all of the circulation is in the background. When the background is too strong, cooling is apparently countered by shears and fluctuations in the background vorticity.

Dynamics and Entropy In recent years, two radically different theories have had success in describing the free relaxation of 2D turbulence. One is the Punctuated Scaling Theory (PST) referred to above [20], which states that the turbulent flow is dominated by strong vortices which generally follow Hamiltonian dynamics of point vortices, punctuated by the occasional merger of like-sign vortices. The relaxed state is then one single vortex of each sign, or just one vortex in our case. The PST agrees with the observed power law decrease in the number of strong vortices [18, 20], but the theory can not explain why several strong vortices remain and anneal into an equilibrium pattern in the final state of the turbulent relaxation. A diametrically opposite approach is incorporated in the global maximum fluid entropy (GMFE) theory [22]. This approximates the turbulent flow as a collection of non-overlapping, incompressible microscopic vorticity elements that become ergodically mixed in the relaxed state. Clearly, the GMFE theory can not explain the vortex crystals, since the theory predicts a smooth vorticity distribution, whereas it is clear from experiments and simulation that vorticity elements in the strong vortices do not mix with the background. A new Regional Maximum Fluid Entropy (RMFE) theory approach [23, 24] characterizes the vortex crystal states by maximizing the fluid entropy S [22] in the presence of persistent strong vortices. The key idea is that some regions of the flow are well-mixed, while other regions are not. The strong vortices ergodically mix the

387

FIGURES

a

b

c

d

e

Figure 7. A selection of experimentally observed vortex crystal patterns (top), and the patterns predicted by RMFE theory (bottom). background, driving it into a state of maximum fluid entropy. This mixing, in return, affects the punctuated dynamics of the strong vortices, "cooling" their chaotic motion, and driving them into an equilibrium pattern. However, the vorticity in the strong vortices is trapped and remains unmixed. The quantities that determine the RMFE state include the conserved quantities of the measured flow that survive coarse graining: the total circulation r i o l , the angular momentum Pg, and the energy H. The diffuse background vorticity is assumed to consist of incompressible microscopic vorticity elements of fixed strength (,"/, with Q taken to be the maximum observed vorticity. Coarse-graining over these randomized vorticity elements gives the observed background vorticity Ct(r)In addition to the above quantities, we need to know the number M of surviving strong vortices and their vorticity distributions {£•'(»"), i = 1,2...M}. These properties of the strong vortices depend on the details of the early dynamical evolution of the flow, and are beyond the scope of the statistical theory. The statistical theory treats the flow only after the mergers of the strong vortices have ceased. Given these inputs, two properties of the relaxed vortex crystal state can be predicted: the coarse-grained vorticity distribution of the background Ct(r), a l l ( l 'he equilibrium positions {R,} of the strong vortices [23]. The equations that characterize the RMFE states are obtained by maximizing the fluid entropy S'[£(,(r)] associated with the background distribution Q{r). This entropy can be calculated by counting the number of ways of arranging microscopic vorticity elements, each with vorticity £/, to obtain the given coarse-grained vorticity £k(r). The maximization of S while keeping H, L and T constant is done by finding the extrema of 5 = S — (3 (H — QPg + /iV) with respect to the independent variables {R.;} and &(r). Here /?, 0 and n are Lagrange multipliers that can be interpreted as inverse "temperature," rotation frequency and "chemical potential." Maximization

388

Figure 8. Measured and predicted theta-averaged vorticity distributions for the 5 vortex crystals of Fig. 7.

with respect to £& gives

aW = C//(e"c'* + i), where $ = ip + ^Q,r2+/j, is the stream function in the rotating frame. This is analogous to the Fermi distribution in quantum statistics, which is to be expected, since the microscopic vorticity elements are incompressible. Maximization with respect to {R;} yields d[H — QPe] / <9R; = 0. This equation states that the strong vortices are stationary in a frame rotating with frequency Si, implying that they form a vortex crystal equilibrium. The parameters j3, Si, fi, {Ri} are then varied numerically until the solution has the proper r t o t , P$ and Hj,. The RMFE solutions reproduce the observed vortex crystal patterns, as shown in Fig. 7. Also, the observed background vorticity is close to the theory, as can be seen in the #—averaged vorticity profiles shown in Fig. 8. The edge of the background distribution falls off gradually, since the vorticity elements near the edge can fluctuate in energy by an amount of order 1//3. Also, near a strong vortex the background vorticity is slightly depressed, since tp tends to increase due to the influence of the strong vortex, as can be observed around the large central vortex in Fig. 7(d). Thus, the following physical picture of vortex crystal formation emerges: the strong vortices undergo chaotic mergers described by punctuated scaling theory, but they also ergodically mix the low vorticity background. The mixing of the background, in return, cools the chaotic motion of the vortices, and drives the vortices into a vortex crystal equilibrium. The interaction between the strong vortices and the background, a process neglected in the PST, may be important in understanding the relaxation of 2D turbulence in other situations as well. Interestingly, recent theory [24] has established estimates for the number of vortices which survive to form the vortex crystal state, by equating the time to merge to

389 the time to cool. Here, the estimates are based on the dynamical scaling exponents £ and rj, which determine the number of surviving vortices Ny(t) and their total circulation Ty(t) as

N,{t) = Nv(t0) ( ! )

rv(t) = rv(t0) ( i )

v

Note that the assumptions of PST imply rj = 1/2, but somewhat different values (0.2 < r) < 0.8) are observed in experiments and simulations. The time to merge is given by l/rm(t) = (d/dt)Nv, and the cooling time is estimated from mixing arguments as rc(t) = A/aNyTv, where A is the area of the vorticity patch, and a PH 0.03. Surprisingly, these simple estimates predict Nc to within about a factor of two.

=0

1.5

7 *\

vorticity Figure 9. Gradient driven radial separation of a clump and hole in a circular shear low. Other recent theory work has analyzed the dynamics of intense positive vortices (clumps) or negative vortices (holes) on a non-uniform background of (positive) vorticity [25]. The analysis clearly shows that clumps move up the vorticity gradient, and holes move down the gradient. Figure 9 shows this effect from a numerical simulation used to. check the analysis. The full analysis necessarily treats the flow shear dfR/dr separately from the vorticity gradient d(/dr, since they are related only by a spatial integral. The analysis also shows that there can also be stationary self-trapped"states lor small ratios of{dCfdrj/(dfR/dr).

390

Minimum Enstrophy State Theorists often suggest that relaxed states may be determined from the robust invariants T, Pe, H by either maximization of entropy S [22], or by minimization of the enstrophy Z2 [26, 27, 28, 29], including generalizations thereof [30]. Obviously, the occurrence of vortex crystal states shows that neither principle holds universally! ^ Nevertheless, early electron plasma experiments found that a range of unstable initial conditions relaxed to near the minimum enstrophy state. Specifically, minimization of enstrophy accurately predicts the meta-equilibrium profiles for hollow initial conditions of moderate energy [31]. These profiles are significantly different from the predictions of maximum entropy. The minimization is subject to constant N, Pe, and H^, and subject to the physical constraint that n > 0. The explicit Pe dependence of the solution can be removed by rescaling the enstrophy as Z2 = 47r(l — Pg)Z2, and considering the excess energy

Here, Hgm = 1/4 - 1/2 ln(2 - 2P9) is the minimum energy possible for given N and Pe, i.e. the energy of a uniform density column. 1.09

n t=0 1.08 0.92 I

20

H^^H^-H^

1 1

30

( 1 0-"3°v)

Figure 10. Predicted (curve) and measured (points) enstrophy Z2 vs excess energy H% Figure 10 shows the measured Z2 (circles) and the predicted minimum Z2 (curve) for a range of excess energies. Also shown is the value of Z2 which would be obtained from a maximum entropy for one evolution. Here, the experimental measurements are typically 2-3 times closer to the minimum enstrophy predictions than to the maximum entropy predictions, both in the measured Z2 and in the measured profiles n{r) For

391 higher excess energies, theory shows that this class of symmetric, monotonic minimum entropy solutions do not exist. These experiments suggested that the relaxed states could the robust invariants. However, recent computer simulations annuli of vorticity show that different dynamics and therefore can be obtained from the same invariants [32]. Specifically, the simulations were started from two annuli

I

The six parameters {au

ai

for

indeed be predicted by starting from unstable different relaxed states of vorticity given by

Oi < r < 61


D T I

1 " 1 '1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

IJ

X r

X

0 Generic s e q u e n c e —

|

X Vortex c r y s t a l s

01

•

0

\

+ Min. e n s t r o p h y

+\

+

1

° 0 X ^

0

•

0

-

1 •

x*

1 1 1 1 1 1 1 1 1 1 1 1 i_.i 1 1

2

3

4

5

6

CT

1

Figure 11. Types of relaxed states obtained for different initial conditions characterized by (<Ti,<72). Extreme initial vorticity values lead to persistent strong vortices, giving vortex crystals or generic (single centered vortex) profiles. Less extreme initial conditions lead to vorticity holes, filamentation, and near-minimum-entropy states. leaving 3 parameters which were chosen to be {<7i,<72,62} for comparison to the experiments. {V,Pe,H} were chosen to give H exc = 6 x 10 - 3 , and b2 was generally near 0.5 Rw. The initial annuli also had a small m = 1 or m = 2 seed asymmetry. Figure 11 shows that completely different relaxed states can be obtained, depending on the details of these initial conditions. The relaxed states were characterized as (1) generic, meaing peaked on center due to complete vortex merger and centerization; (2) vortex crystals, where the "cooling" process prevented complete merger from occurring; and (3) minimum enstrophy, where no strong vortices persist, because filamentation dynamics dominated the evolution. We note that the "minimum enstrophy" profiles are always closer to the minimum enstrophy prediction than to

392

the maximum entropy prediction, suggesting that there may be some validity to Batchelor's inverse cascade hypothesis [29], due to the formation of arbitrarily finescale filaments. Thus, at present, we can only conclude that both dynamics and statistics contribute to the inviscid relaxation of 2D turbulence with a single sign of vorticity. This work was supported by National Science Foundation grants PHY94-21318 and PHY98-76999, and Office of Naval Research grant N00014-96-1-0239.

References [1] C.F. Driscoll and K.S. Fine, "Experiments on Vortex Dynamics in Pure Electron Plasmas," Phys. Fluids 2, 1359 (1990). [2] A.J. Peurrung and J. Fajans, "A Limitation to the Analogy Between Pure Electron Plasmas and 2D Inviscid Fluids," Phys. Fluids B 5, 4295 (1993). [3] D.H.E. Dubin, "Collisional Transport in Nonneutral Plasmas," Phys. Plasmas 5, 1688 (1998). [4] W. Kelvin, "On a Disturbing Infinity in Lord Rayleigh's Solution for Waves in a Plane Vortex Stratum," Nature 23, 45 (1880). [5] R.J. Briggs et al., "Landau Damping in Crossed-Field Electron Beams and Inviscid Shear Flow," Phys. Fluids 13, 421 (1970). [6] N.S. Pillai and R.W. Gould, "Damping and Trapping in 2D Inviscid Fluids," Phys. Rev. Lett. 73, 2849 (1994). [7] A.C. Cass, "Experiments on Vortex Symmetrization in Magnetized Electron Plasma Columns," Ph.D. dissertation, UCSD (1998). [8] D.A. Schecter, Ph.D. dissertation, UCSD (1999). [9] M.T. Montgomery and C. Lu, "Free Waves in Barotropic Vortices. Part I: Eigenmode Structure," J. Atmos. Sci. 54, 1868 (1997). [10] T.B. Mitchell, C.F. Driscoll and K.S. Fine, "Symmetrization of 2D Vortices by Beat-Wave Damping," Phys. Rev. Lett. 73, 2196 (1994). [11] R.H. Levy, "Diocotron Instability in Cylindrical Geometry," Phys. Fluids 8, 1288 (1965). [12] R.C. Davidson, Physics of Nonneutral Plasmas (New York: Addison-Wesley, 1990).

393 [13] C.F. Driscoll, J.H. Malmberg, K.S. Fine, R.A. Smith, X.-P. Huang and R.W. Gould, "Growth and Decay of Turbulent Vortex Structures in Pure Electron Plasmas," Plasma Physics and Controlled Nuclear Fusion Research 1988 3, 507 (Vienna: IAEA, 1989). [14] C.F. Driscoll, "Observation of an Unstable I = 1 Diocotron Mode on a Hollow Electron Column," Phys. Rev. Lett. 64, 645 (1990). [15] R.A. Smith and M.N. Rosenbluth, "Algebraic Instability of Hollow Electron Columns and Cylindrical Vortices," Phys. Rev. Lett. 64, 649 (1990); R.A. Smith, "Effects of Electrostatic Confinement Fields and Finite Gyroradius on Instability of Hollow Eoectron Columns," Phys. Fluids B 4, 287 (1992). [16] J. Finn, "Compressional Effects in Nonneutral Plasmas, a Shallow Water Analogy and t = 1 instability" (1999). [17] K.S. Fine, C.F. Driscoll, J.H. Malmberg and T.B. Mitchell, "Measurements of Symmetric Vortex Merger," Phys. Rev. Lett. 67, 588 (1991). [18] K.S. Fine, A.C. Cass, W.G. Flynn and C.F. Driscoll, "Relaxation of 2D Turbulence to Vortex Crystals," Phys. Rev. Lett. 75, 3277 (1995). [19] Daniel H.E. Dubin and T.M. O'Neil, "Trapped Nonneutral Plasmas, Liquids, and Crystals (The Thermal Equilibrium States)," Rev. Mod. Phys. 7 1 , 87 (1999). [20] G.F. Carnevale et al., Phys. Rev. Lett. 66, 2735 (1991); J.B. Weiss and J.C. McWilliams, Phys. Fluids A 5, 608 (1993). [21] D.A. Schecter, D.H.E. Dubin, K.S. Fine, and C.F. Driscoll, "Vortex Crystals from 2D Euler Flow: Experiment and Simulation," Phys. Fluids 11, 905 (1999). [22] J. Miller, Phys. Rev. Lett. 65, 2137 (1990); R. Robert and J. Sommeria, J. Fluid Mech, 229, 291 (1991); J. Miller, P.B. Weichman, and M.C. Cross, Phys. Rev. A 45, 2328 (1992), and references therein. [23] D.Z. Jin and D.H.E. Dubin, "Regional Maximum Entropy Theory of Vortex Crystal Formation," Phys. Rev. Lett. 80, 4434-4437 (1998). [24] D.Z. Jin, "Theory of Vortex Crystal Formation in Two-Dimensional Turbulence," Ph.D. dissertation, UCSD (1999). [25] D.A. Schecter and D.H.E. Dubin, "Vortex Motion Driven by a Background Vorticity Gradient" (1999). [26] W.H. Matthaeus and D. Montgomery, "Selective Decay Hypothesis at High Mechanical and Magnetic Reynolds Numbers," Ann. N.Y. Acad. Sci. 357, 203 (1980).

394 [27] C.E. Leith, "Minimum Enstrophy Vortices," Phys. Fluids 27, 1388 (1984). [28] W.H. Matthaeus, W.T. Stribling, D. Martinez, S. Oughton, and D. Montgomery, "Selective Decay and Coherent Vortices in Two-Dimensional Incompressible Turbulence," Phys. Rev. Lett. 66, 2731 (1991). [29] G.K. Batchelor, "Computation of the Energy Spectrum in Homogeneous TwoDimensional Turbulence," J. Fluid Mech. 78, 129 (1976). [30] B.M. Boghosian, "Thermodynamic Description of the Relaxation of 2D Turbulence using Tsallis Statistics," Phys. Rev. E 53, 4754 (1996). [31] X.-P. Huang and C.F. Driscoll, "Relaxation of 2D Turbulence to a MetaEquilibrium Near the Minimum Enstrophy State," Phys. Rev. Lett. 72, 2187 (1994); X.-P. Huang, "Experimental Studies of Relaxation of Two-Dimensional Turbulence in Magnetized Electron Plasma Columns," Ph.D. dissertation, UCSD (1993). [32] E.J. Moreau et ai, "The Influence of Dynamics on 2D Turbulent Relaxation: A Numerical Study" (1999).

GROWTH OF DISORDERED FEATURES IN A TWO-DIMENSIONAL CYLINDER WAKE

PETER VOROBIEFF1 and ROBERT E. ECKE Center for Nonlinear Studies and Condensed Matter and Thermal Physics Group Los Alamos National Laboratory Los Alamos, NM 87545, USA ABSTRACT We investigate the behavior of a quasi-two-dimensional wake behind a circular cylinder inserted in a gravity-driven flow of soap film. For Reynolds numbers below 200, the near wake immediately downstream of the cylinder exhibits a well-known, highlyregular behavior (Benard - von Karman vortex street). Farther downstream, the wake transforms into a nearly-parallel shear flow, from which a secondary vortex street forms. With increasing downstream distance, the latter becomes disordered, and the flow begins to acquire features normally associated with two-dimensional turbulence (e.g. coarsening of the flow structure). We present instantaneous flow visualization showing this evolution, as well as the analysis of the regularity of the wake at different downstream distances.

1. Introduction The problem of vortex shedding behind a bluff body immersed in a flow of fluid is perhaps the most fundamental problem of hydrodynamics. The discovery of a staggered pattern of counter-rotating vortices in a bluff-body wake is usually attributed to Theodore von Karman1, who presented an analysis of stability for this flow pattern in 1911. However, it would be more accurate to claim that Henri Benard2 reported the first actual observation of a vortex street in 1908. Figure 1 reproduces the sketch done by Benard and the excellent-quality 1912 flow visualization photograph by von Karman and Rubach3. These images give a good notion of the flow pattern known as the Benard-von Karman vortex street. The periodic shedding of counter-rotating vortices on alternating sides of a body moving through fluid is the characteristic of a wide range of flows. The main parameters that define the vortex-street flow are the Reynolds number Re=u dlv and Strouhal number St=fd/u . These numbers are dimensionless combinations of the system parameters, namely u - the freestream velocity, d - size of the shedding body (e.g. cylinder diameter), v - the kinematic viscosity a n d / - the frequency with which vortices are shed on one side of the body. Vortex streets and the relationship between Re and St in three spatial dimensions have been extensively studied. In a wide range of Reynolds

New address after August 16, 1999: Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 87131, USA

395

396 numbers, the basic structure -of the flow is very robust and effectively two-dimensional, although, as Re increases, both two-and three-dimensional instabilities develop in the flow, changing the vortex street behavior (see Noack4 for a concise review). To date most of the studies, however, have concentrated on the near wake, i.e. the flow immediately downstream of the cylinder. Many issues related to the far wake flow (100 and more cylinder diameters downstream), in particular, the growth of disorder with downstream distance and eventual transition to turbulence, remain unclear.

Fig. 1. Vortex streets: a) schematic of counter-rotating staggered vortices in a vortex street (Benard2, 1908); b) vortex street visualization (von Kannan and Rubach3, 1912).

In this paper, we present experimental data on far-wake behavior at distances up to 500 cylinder diameter downstream. Our results are acquired in a quasi-two-dimensional flowing soap film - the promising experimental testbed for 2D hydrodynamics described in the second section of the paper. With the growth of downstream distance, we observe vortex pairing in the near-wake vortex street. This vortex pairing leads to the formation of a nearly-parallel shear flow from which a secondary vortex street emerges. The latter begins with a regular vortex pattern but becomes progressively more disordered as the vortices move downstream. 2. Experimental Apparatus and Data Acquisition System Couder5 was the first to study two-dimensional hydrodynamics of soap films, and Couderetal.6 provide a good review of soap-film physics. The most familiar everyday manifestation of soap film is in soap bubbles. The films, typically several microns thick, form the walls of a soap bubble. If a soap film is stretched on a rigid frame, an object can be inserted into the film without rupturing it. The object can also be moved in the plane of the film, producing a wake flow. The first soap-film hydrodynamics experiments were

397 set up this way5"7 and, not surprisingly, one of these studies7 dealt with vortex streets in a cylinder wake. The flow visualization employed a sodium lamp, making it possible to see a fringe pattern of the isosurfaces of film thickness. Thickness in 2D hydrodynamics behaves as a scalar advected by the flow, making it a convenient quantity to visualize. Our experimental setup is somewhat more complicated, and it represents a twodimensional analog of a wind or water tunnel - thus the name "soap film tunnel" (Fig. 2). Goldburg et al.8 developed the basic concepts of the apparatus. Two thin nylon wires are suspended from the nozzle attached to a reservoir with a 1-2% solution of household liquid soap in water. The wires are stretched vertically, with the soap solution flowing down the wires due to gravity. Pulling the wires apart produces a flowing soap film. In our apparatus, the whole setup can be tilted at an arbitrary angle, thus providing additional control over the freestream velocity. Instead of a sodium lamp used in many earlier studies, we employ a pair of Xenon arc flashes backlighting the plane of the flow. Seeding the flow with a small (5><10"6) volume fraction of submicron-sized titanium dioxide particles makes it possible to acquire instantaneous images of the film thickness. Double-pulse exposures also allow to analyze seeding particle displacement and thus measure velocity9'10, although in this work we used velocity measurements only to establish the value of the freestream velocity - 1.32±0.02 m/s. Our earlier study10 measured instantaneous fields of velocity u and vorticity co=Vxn and found a strong correlation between vortex cores (a> peaks) and the areas of low film thickness. Thus vortices can be located in the thickness-field images as local thickness minima.

Fig. 2. Schematic of a tilted gravity-driven soap-film tunnel. 1-camera, 2-flash. Second flash not shown.

In the experiments reported here, the mean film thickness h was 7 microns. The film viscosity decreases with thickness6, and we estimate the corresponding film viscosity is 5.28 times that of bulk water11. The following section presents instantaneous images of thickness fields acquired at different downstream distances and the analysis of the geometry of the wake derived from these images. The worst-case error in distance between vortex cores estimated from a thickness image should not exceed 5%. For a detailed description of the setup and data acquisition techniques used in the experiments, as well as the study of the accuracy of the measurements, refer to our earlier work.10

398 3. Observations Earlier we investigated the range of Reynolds numbers from 23 to 1300.11 This work concentrates on the wake behavior at /te=180. At this value, we observed no secondary instabilities in the near wake altering the appearance of the von Karman vortex street. The near-wake flow was periodic and regular. With increase in the downstream distance, however, the flow pattern evolved, ending with a rather disordered flow at the extreme downstream extent (up to 500 cylinder diameters) of the area we investigated. Figure 3 shows three 'composite images made from instantaneous thickness-field snapshots taken at different downstream distances. The boundaries of the flow zones do undergo some fluctuations, but the overall spatial pattern remains fairly stable, as comparison of the three composites in Fig. 3 shows. First (von Kirman) wake

Second wake Disorder, transition to turbulence?

Near-parallel shear flow ,200

300

,400

x/d

Fig. 3. Composite images of the thickness field in the two-dimensional wake of a circular cylinder at /te=180. Scale indicates downstream distance x nondimensionalized by cylinder diameter d. The physical length of the imaged area is 18.2 cm.

The first transition from the regular vortex-street pattern occurs at a downstream distance of about 70 cylinder diameters, where the staggered structure is replaced by a paired-vortex pattern. The vortices arrange themselves in pairs aligned in the streamwise direction. Vortical structures also show considerable elongation, indicating shear in the flow. Figure 4 presents a view of this transitional region. The transitional zone extends to approximately 90 cylinder diameters downstream. Then a second vortex street emerges. Additional study is required to examine the relationship between its vortex-shedding frequency and that of the near wake. For Re=180, we observe the former to be about 50% of the latter. However, thefrequencyof the second vortex street may be unrelated to the von Karman near-wake frequency, as Cimbala et al.12 observed in a three-dimensional flow. Farther than 200 cylinder diameters downstream, the second vortex street becomes more disordered, while the width of the wake in the crossflow direction increases appreciably. It is also noteworthy that the number of vortices per frame decreases with

399 downstream distance, consistent with the notion of vortex cannibalization specific for 2D hydrodynamics.10

Fig.4. Transition from a von Karman vortex street (left) to a shear flow with vortex pairing (right). The physical horizontal extent of the image is 1.8 cm.

How to quantify the growth of disorder in the wake? A measure of irregularity in the flow can be obtained by evaluating the distances between the alternating vortices I in the freestream direction. If the flow is a regular vortex street, then these distances are fairly uniform and the standard deviation from the mean-distance value should be small. As the flow becomes disorganized, however, this standard deviation should grow (see Fig. 5 illustrating the concept).

Fig. 5. Distances between alternating vortices immediately behind the cylinder (left) and at 435 cylinder diameters downstream (right).

Assuming that the vortices are transported downstream with thefreestreamvelocity u , and the velocity of their relative motion is substantially smaller than u , one can easily estimate consider the Strouhal number knowing I: St=d/l. Thus we can measure instantaneous values of St at a fixed downstream position, average the result and find the standard deviation. This standard deviation should give us a good notion of the local regularity of the wake. Figure 6 shows plots of mean St and the standard deviation

400 normalized by the mean versus the dimensionless downstream position x/d. The horizontal line on the mean St graph represents the value of St from the empirical Re-St relationship acquired in 3D experiments.13 Not surprisingly, at x/d<70 the St value we measure is quite close to this line, and the standard deviation is only about 10%. At x/d~70 the standard deviation increases to 35%, as the staggered vortex pattern reorganizes into a vortex-paired shear flow. It drops again at the onset of the second wake (x/d~90). The Strouhal number of the second wake is about 60% of the near-wake value, indicating a longer period. The second wake is also less regular, with standard deviations as high as 30%. The region of the second wake can be identified to 200 cylinder diameters downstream. At higher x/d values, the wake can no longer be adequately characterized in terms of the Strouhal number, because the standard deviation increases to values exceeding 100%. Is this disorder in the wake indicative of transition to turbulence? To address this question directly, further analysis is required, including the study of velocity and vorticity structure functions at different downstream locations. 1.2 1.0 t-0 U-i

o OS G


II

4

S0.6 Q

H w 0.4

H#f j (

0.2 00

0

100

200

300 x/d

400

500

0

100

200

300

400

500

x/d

Fig. 6. Estimates of St vs. downstream distance (left) and the growth of the normalized standard deviation of St (right). The horizontal line indicates the 3D empirical value of St for Re=180 according to Roshko.13

4. Conclusions We use a gravity-driven soap film tunnel to study the growth of disorder in a twodimensional cylinder wake flow at a Reynolds number i?e=180. Investigation of this fundamental problem of hydrodynamics in a two-dimensional setting is motivated primarily by the interest in the influence of dimensionality on hydrodynamics instabilities driving the transition to turbulence in the far downstream region. Our experimental arrangement also allows us to acquire quantifiable data at extreme downstream distances (up to 500 cylinder diameters) where previously not much information was available from either two- or three-dimensional experiments or numerical simulations. While the wake immediately behind the cylinder exhibits the classical vortex-street behavior, the flow

401 becomes progressively more disordered with increase in downstream distance. As the downstream distance grows, the width of the wake increases, while the number of vortex structures per unit area goes down. The wake dilation in the crossflow direction is commonly observed both in two- and three-dimensional wakes. However, the coarsening of vortex structures (which apparently leads to the decrease in the number of vortices) is the unique property of two-dimensional flows. As the downstream distance increases, we observe several transitions in the cylinder wake. At a downstream distance x/d~70, the vortex street transforms into a flow with apparent strong shear and vortex pairing. From this flow, a second vortex street emerges at x/d~90. Finally, at 200<x/c?<500, visualizations and our analysis show that the flow becomes irregular. Our further studies will characterize this irregularity in terms of the statistics of the velocity and vorticity.

5. Acknowledgements We gratefully acknowledge the useful suggestions of Michael Rivera (University of Pittsburgh, Pennsylvania). This research is funded by the U.S. Department of Energy.

6. References 1. Th. von Karman, Gottingen Nachrichten, mathematisch-physikalische Klasse (1911), 509; (1912), 547. 2. H. Benard, Comptes Rendus 147 (1908), 839. 3. Th. von Karman and H. Rubach, Physik. Zeitsch. 13 (1912), 49. 4. B.R. Noack, Z. Angew. Math. Mech. 79 (1999), S223; S227. 5. Y. Couder, J. Phys. Lett. 45 (1984), 353. 6. Y. Couder, J.M. Chomaz, and A. Rabaud, Physica D 37 (1989), 384. 7. Y. Couder and S. Basdevant, J. Fluid Mech. 173 (1986), 225. 8. W.I. Goldburg, M.A. Rutgers, and X.L. Wu, Physica A 239 (1997), 340. 9. M. Rivera, P. Vorobieff, and R.E. Ecke, Phys. Rev. Lett. 81 (1998), 1417. 10. P.Vorobieff, M. Rivera, and R.E. Ecke, Phys. Fluids 11 (1999), 2167. 11. P. Vorobieff and R.E. Ecke, Phys. Rev. E 60 (1999), 2953. 12. J.M. Cimbala, H.M. Nagib, and A. Roshko, J. Fluid Mech. 190 (1988), 265. 13. A. Roshko, NACA TN 2913 (1953).

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DOUBLE SCREEN TRANSITION EFFECTS IN NEAR EARTH PLASMA TURBULENCE STUDYING. N.V.MURAVIEVA, YU.V.TOKAREV, G.N.BOIKO, E.YU.RYNDYK, Radiophysical Research Institute, B.Pecherskaya St., 25, Nizhny Novgorod 603600, Russia and M.L.KAISER. NASA/Goddard Space Flight Center, Grinbelt, USA

ABSTRACT Solar wind observations by means of bi-static decametric radar formed by ground based facility and spacecraft allow studying a fine structure of solar wind plasma irregularities with resolution up to 40 km. Both spectral analysis and Hurst exponent methods were applied to the time series resulting from scintillation of radio signal transmitted through double screen of turbulent layers of ionosphere and solar wind plasma. That was used to separate the signal components, which reflect influence of either Earth ionosphere or Solar wind. As a result of the analysis it has been resumed that the main contribution into interplanetary scintillation was supplied by a region of enhanced solar wind turbulence in the site of near Earth bow shock.

1. Introduction The paper is devoted to the analysis of scintillation of radio signal transmitted through turbulent layers of ionosphere and solar wind plasma. Experiments were performed by means of radio scintillation technique 1 . This method is based on sounding media by monochromatic radio waves. Radio signal radiated from ground based SURA facility 2 in Russia propagates through ionosphere and solar wind turbulence and is detected by the receiver placed at NASA WIND spacecraft 3 . Electron density irregularities of ionosphere and solar wind plasma scatter electromagnetic waves that causes low frequency scintillation of detected radio signal. Statistical analysis of this radio scintillation can provide information about scattering media such as characteristic scales of turbulence, power low spectral index p, S(f)oc f'p'2, scintillation index ^ 4 = ( < / >- 2 ) / < I >2, where / - intensity of scintillation . In specific conditions of our experiments the signal statistics is defined by both ionosphere and solar wind plasma, and at this point we come to the problem of separation parameters associated with influence of either ionosphere or solar wind turbulence. 403

404 2. Radio scintillation technique. According the theory of week scattering' the maximum on the scintillation spectra is determined by both characteristic internal spectral scale of turbulence and the size of first Fresnel zone JAZ e / , where distance of scattering media from transmitter, Zm r - distance of scattering media from receiver and Z,_r- distance of receiver from transmitter. The size of first Fresnel zone varies with distance of spacecraft from Earth. The spacecraft orbits for the period, discussed in the paper, are shown at Fig.l. Since 1997 WIND was located at elliptic orbits near Earth 150 100 ecliptic plane with apogee 80 + 230 DistancefromEarth Fig. 1 Ecliptic cut of spacecraft orbits for periodfromNov Earth's radii and perigee 10+15 14, 1996 to Nov 12, 1997. Abscissa is directed to Sun. Earth radii. Thus, the size of first Hyperbolic curves border area confined by Earth bow Fresnel zone differs between 40 and shock and inagnetopause. http://sscweb.gsfc.nasa.gov 100 km for Earth bow shock area and stays about 3 km for ionosphere. For the first time we used such a technique, which allows observing so fine structure of solar wind. At the same time the trace geometry in the sunward direction kept less that 0.1 astronomical unit. It means that we study near Earth solar wind itself without huge averaging with heliocentric distances Results of more than 30 sounding sessions at 9 MHz 0.1 1.0 10. frequency were collected and Frequency (Hz) analyzed. Almost every sample of Fig. 2 Scintillation spectra, top - Dec 29, 1996, WIND scintillation spectra calculated for distance from Earth Z = 33.4R M , elongation angle SURA signals has a high frequency e = 82.5', V^rK_ = 320km/sec; middle - Apr 28,1997, = 6-5', V^„ =331/tm/sec; bottom -Jul 7, singularity located at 0.4+6 Hz 1997 Z = 27«„ l , £ = 4 4 . 5 \ VMrWM =368368/sec km/sec.range, that might be associated with 50 +800 km solar wind scales or

405

O.Ol-s-0.25 km ionosphere turbulence scales. Scintillation spectra obtained for positions of spacecraft with different distances and sun relative directions are shown on Fig.2. Low frequency part of the plot is associated with influence of ionosphere turbulence which is known to have a power law spectra with power law index about p<* 2.5*3. High frequency spectral peculiarity exists all time when spacecraft stays outside of bow shock area and disappears after spacecraft crosses magnetopause toward Earth (see Fig 3). So we can conclude that we do observe radio waves scattering by solar wind clouds. 0.1 1.0 In weak scattering approximation the ray Frequency (Hz) propagation through turbulent media and chaotic Fig. 3 Power spectra shifted vertically, 1996, distribution of resulting electromagnetic field may 7-Dec.,29, Z = 33AR^„ (WIND is outsite be explained by a fluctuation of medium refractive bowshock), £ = 82.5- , V^ ^ =320km/sec; r index. This may be modeled as a series of periodic 2 -Dec.,30,Z = lAlR^ (WIND is insite phase screens, which provides phase modulation bowshock), e = 127.5' of electromagnetic waves . Numeric simulation performed by Ryndyk5 has shown that wave propagation for this model can be described by analogy with Brownian motion and can be analyzed with rescaled range analysis. 3. R/S analysis. Detailed description of Hurst's rescaled range analysis (R/S analysis) can be found in many places. This is a statistical method devised by Hurst6 and later discussed extensively by Mandelbrot and Wallis 7'8. In our investigation we have obtained a discrete record in time of radio signal scintillation that reflects scatter of electromagnetic waves by ionosphere and solar wind turbulence. By choosing a time period (lag) shorter than the total record we calculated the Range R(.T) = maxX(t,T)-mmX(t,r), where X(t,r)= £{£-<£> }- accumulated departure of random time series %(t)frommean value <|> r =X#(0 The range depends on the lag and we expect it to increase with increasing of t. By dividing R with standard deviation S = (— I{£(t)~<£> } ) r '••

we obtain a dimensionless number R/S for• the lag in

r

question. Hurst, and later Mandelbrot, found that the following empirical relation very well describes the observed rescaled range, R/S, for many records in time

406 R/S = (T/2)H, where H is referred to as Hurst exponent. Hurst and Feller have shown that in the absence of long-run statistical dependence R/S should become asymptotically proportional to R/S = (JIT/2) ' for records generated by statistically independent processes with finite variances. Statistical results collected by Hurst shows that for many natural phenomena H > 'A. Mandelbrot has described data when H > 'A as persistent and when H < 'A the data are said to be antipersistent. As Gouyet shows 8 that the following relationship between fractal dimension D, dimension of Brownian motion d, spectral power index/) and Hurst exponent H exists D = d + l-H

=d +@

^ .

That leads to P = 2H + \.

4. R/S results.

s*.s

If wAv^ AiVWvy^ y 3-14

a.i*

3

Fig. 4 Zoom of time record shows similarity of curves. estimated about 1000 km. Fig 6 demonstrates the same calculations performed for data obtained when spacecraft was located inside magnetopause. As it is seen, for all scales data shows persistent character with Hurst exponent about 0.7-0.8

Fig 4. represents the time records for various recorded time periods. The similarity in the curves lets to suggest that we have data be of fractal nature. Hurst curves calculated on Fig 5 for the case when spacecraft was positioned outside bow shock area show antipersistent behavior for small lag while for larger scale of averaging Hurst exponent became close to 0.7 that is in well agreement with value obtained by Hurst for natural phenomena. The critical scale when behavior of data changes can be ^

v

l

*

i^WhJk n

ar

5. Conclusions In our experiments new "inverse" geometry of observations has been Fig.5 Hurst curves calculated for spacecraft demonstrated. For the first time a lowest position inside magnetopause area. most effectively scattered frequencies near ionosphere cut-off have been used.

407

Location of spacecraft in near Earth (< 0.1 a.u.) space and high time resolution of its receiver allows studying a fine (40100 km) structure of solar wind without averaging along a broad range of ioyu 100U iMXl tuu «>t1 heliocentric distances. i •r-S Of o% S Double scattering of decameter >--r .*"' 7 signal has been observed as ionosphere and 0 2. 1 0 } solar wind spectral components. -. Characteristic scales of the region of enhanced plasma inhomogeneity have been 9 f 0.6 s \r detected to be near internal boundary of 0I solar wind turbulence. ? '' ' L1L. Rescaled range analysis of time series shows that scintillation of radio signal Fig. 6 Hurst curves calculated for distant spacetransmitted through turbulent media is of craft position fractal nature and the influence of solar wind may be recognized through antipersistent behavior of signal scattered by Earth bow shock turbulence.. •*00

*r^~l

L

1

6. Acknowledgments This material is based upon work made in frame of project of U.S. Civilian Research and Development Foundation (award No RPl-260) with partial support by INTAS-CNES (grant No 97-1450). 7. References 1. I.Yakovlev, Space Radiophysics, (Moscow, RFFI, 1998). 2. B.F.Belov, V.V.Bychkov, G.G.Getmantsev, et al, Preprint 167 (NIRFI, Gorky, 1983). 3. J.-L.Bougeret, M.L.Kaiser, P.J.Kellogg, et al., Space Sci. Rev. 71 (1995) 231. 4. L.M.Erukhimov, V.P.Uryadov, Radiophysics and Quantum electronics, 11 (1968) 2. 5. E.Yu.Ryndyk, V.V.Chugurin, L.M.Erukhimov, Waves in Random media 4:(1) (1994), 21-28. 6. J.Feder, Fractals, (Plenum Press, New York, 1988). 7. B.B.Mandelbrot and J.R.Wallis, Water Resource Res. 4 (1969) 321. 8. J.F.Gouyet, Physics and Fractal Structures (Masson, Paris, 1996).

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XIII. General

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Experimental Evidence for Microscopic Chaos M. E. Briggs University of Utah, Salt Lake City, Utah 84112, USA

P. Gaspard Universite Libre de Bruxelles, B-1050 Brussels, Belgium

M. K. Francis, R. V. Calabrese, R. W. Gammon, J. V. Sengers, J. R. Dorfman University of Maryland, College Park, Maryland 20742, USA

Abstract Irreversibility and the transport behavior of physical systems is likely created by chaos at the microscopic level. We discuss experimental evidence for the existence of this microscopic chaos. The experiment provides measurements of the dynamic epsilon-entropy for a colloid particle undergoing Brownian motion in water. The results show a positive epsilon entropy, which is consistent with positive Lyapunov exponents and thus microscopic chaos. We point out the general utility of the epsilon entropy for characterizing random systems, and mention a follow-up experiment. I. INTRODUCTION Theorists can now use chaos theory to make a quantitative connection between the deterministic microscopic-dynamics and the irreversibility of many-body systems [1-7]. This is a fundamental advance, since for the first time transport properties can be calculated directly from the underlying deterministic equations of motion without any stochastic assumption about correlations in the fluid. Such asssumptions amount to creating irreversibility externally, whereas using the theory of microscopic chaos, stochasticity is self-generated by the chaotic dynamics of the system itself. Two questions that arise are: 1.) Is this indeed how randomness arises? 2.) Is there any experimental evidence to support the presence of microscopic chaos?

412

The large number of degrees of freedom of many-body systems renders a measurement of the full chaos in such systems impossible. Gaspard has shown that, nevertheless, evidence for the existence of microscopic chaos could be found experimentally through an analysis of the Dynamic (e, r)— entropy per unit time of the Brownian motion of a colloidal particle [8]. We performed this experiment, and found that the dynamic epsilon entropy is indeed > 0 [9], which is true for systems which are chaotic. This is strong evidence for the presence of microscopic chaos; the answer to question two above is yes. We will describe this experiment below, focusing on the technique, rather than the the analysis, which has already been described in reference [9]. The answer to question one requires further experimentation in which the length scale probed would become small enough to reveal the presence of deterministic behavior. We mention briefly in the conclusion a candidate experiment. As several workers have pointed out, there are theoretical systems which are not chaotic, but which still show positive dynamical entropies [10]. However, these systems are not realistic models of fluids. All of these introduce the randomness externally, e.g, through explicit calls to random number generators in the placement of scatterers, and all fail to capture an essential feature of Brownian motion, that the diffusion constant is inversely proportional to viscosity of the surrounding fluid. The experimental objectives are laid out clearly by Gaspard [8]: obtain as large a catalog as possible of the different trajectories followed by the Brownian particle over time scales and spatial scales as broad as possible. We were aware of video particletracking work in which the motion of the particle is imaged by a microscope into a video camera [11]. In this technique, a computer digitizes a sequence of images from the video camera, and fits each image for the location of the center of the particle. The data is thus a time series of two-dimensional particle-locations, with a time step controlled by the time between video frames. The time resolution of this technique is thus limited for standard U.S. video-equipment to an unremarkable 1/60 of a second. The spatial resolution, however, is a very remarkably 25 nm. This is possible because of the ability of the fitting routine to interpolate a center for the image of the particles, which have diameters of 1-2.5 x 10~6m (1-2.5/mi). This interpolation in turn relies on the 1-2% uniformity in the radii of commercially available colloid particles, and the quality of the optical and video systems. Using this technique, we were able to produce trajectories of sufficient spatial and dynamic range to give two decades of scaling behavior in the e-entropy analysis of the data reported in [9]. We present in the rest of this paper the hardware and software details, and performance and error analysis.

413 II. CELL DESIGN A. Construction The cell is a nice example of the benefits of making things as simple as possible, but not simpler. The design presented in reference [11] was for an experiment which required precise control of the thickness of the layer of fluid in which the particles move. We only had to provide sufficient thickness to ensure bulk diffusion. We found that the original workers in this area, including Brown himself, would simply put a ring of grease on a microscope slide, put in a drop of the sample, and push a cover slip onto the grease. The resulting cell was sealed against evaporation, adjustable in height (you could just squish the cover slip down further), and cheap! We tried this approach, making an upgrade to modern materials and using a reflection microscope (Fig. 1). We used modern vacuum grease, which is very clean, and replaced the microscope slide with a polished silicon-wafer. These wafers are available atomically flat and completely dust free, right out of the box, as required by the semiconductor industry. This means that the optical quality of these wafers is excellent—they look like perfect mirrors. In fact, if you place one of these under even the best microscope, it is virtually impossible to see the surface until you have gotten it dirty to provide you with a bit of scattered light! So, the wafers provide us with an exceedingly uniform background against which to image tiny particles. For comparison, the microscope slides must be carefully cleaned, have micron-scale deviations from flatness across their length, and have polish marks on them which reduce the contrast and uniformity of the background in the image, and will affect the motion of the particle near the surface. We begin constructing the cell by carefully cleaning and inspecting some cover slips and glass vials with lids, to make sure they are free of dust and residue. For the cleaning we boil the glassware in purified water and Fisher lab detergent, and immediately rinse with a large volume of purified deionized water. We obtained the water from a "Milli-Q" laboratory water system, which produced water with a resistance reported by the device of 18 Mft cm. We followed the water rinse immediately with a flush of filtered nitrogen, which would push the liquid out and off of the glassware before it could dry and leave behind a residue. We would then quickly carry the glassware to a clean-room area maintained under a laminar-flow hood, which is a device that produces a column of dust-free air that falls on your work area. We had in this area silicon wafers, vacuum grease, more of the deionized water, syringes, an inspection microscope, and the vials of colloid particles. We had already greased the bottom edges of the cover slip with a very thin ridge of the vacuum grease for use when the colloid solution was ready. We needed the final concentration of particles to be low enough that we could

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follow a single particle for at least 10 minutes without encountering another particle, yet high enough that we did not have to search forever to find a particle to start tracking. After a few trips back and forth between preparation and the microscope, we learned to judge the correct concentration by looking at the amount of light scattered from the vial in which we were preparing the solution. Typically, for a 5 ml vial, we would put in one drop of the concentrated colloid solution as shipped from the manufacturer [12], fill the vial with the deionized water, empty nearly all the contents, fill again, empty nearly all again, and fill a final time. We would then place a single, very small drop of the final solution on the silicon wafer using a single-use plastic syringe with a cleaned stainless steel 22 gauge needle. Here again, a short period of experimentation allowed us to find the correct size drop. We would fill most of the volume of the sample, leaving a small vapor bubble to avoid sealing problems caused by excess liquid forcing its way out through the grease ring. Finally, we placed the cover slip down over the drop, and we pushed down around the edges of the cover slip until the grease sealed the cell completely. A complete seal was evident from the change in appearance when the grease had wetted both the cover slip and wafer. Once the sample had stabilized for an hour or so, we saw no evidence of convection or any other flow along the edge of the vapor bubble or elsewhere in the cell. Note that flow was easily detected visually because of the effectiveness of the particles as markers. It is of course vital that no grease be on top of the cover slip, because of the very short working distances of high-magnification objectives. B. Quality One sensitive indication of the quality of a colloidal sample is how long the particles persist in showing Brownian motion: The colloids we use [12] are "charge stabilized," meaning that electrostatic repulsion due to negative charge-clusters anchored around the surface prevent the particles from sticking to each other or to the walls of the container (the surfaces of which generally acquire negative polarity during cleaving and polishing). With time, impurities neutralize these charges and the particles coalesce or adsorb on the surroundings. Other reports in the literature typically give one week as a survival time. Brownian motion in our samples persisted for several weeks, with one sample lasting more than two months. We did not use any form of deionizer within the cell. The durability of the cell is undoubtedly due to good quality water and the simplicity of the cell, which allows for speedy, clean assembly.

III. HARDWARE, PROCEDURE, AND PARTICLE TRACKING SOFTWARE A. Hardware The images we obtained were created by a Nikkon microscope using brightfield illumination and CF-N-DIC plan achromat 40X and 100X dry objectives (i.e.. differential-interference objective). We first simply measured the trajectory of the particle by tracing it on a TV monitor screen, to check that the particle was not being held by the illumination or associated heat. We found that we had good diffusive behavior (x2 oc t) for several minutes. The microscope focused the images onto a Cohu 640X480 monochrome camera (mounted on the microscope), and the camera sent the images in standard US split-field 30 frames/second analog signal to be recorded. Initially, this was done on VHS tape by a Panasonic computer-controllable VCR, model AGC 6300 VTR. The data from the tapes were digitized by a DATA Translation DT3851 frame-grabber board in the EISA slot on a Pentium machine, using a 3rd party VCR controller card BCD-1000 to step the Panasonic recorder from frame to frame. The digitized images were stored on a hard drive for the particle-tracking analysis. The final spatial calibration was 0.098 pm/pixel for the 100X objective (a pixel is the spatial unit reported by the frame-grabber), which we found by digitizing the image of a calibrated 10 ^m ruling. In order to prevent vibrations, we mounted the microscope on a air-mounted honeycomb optical bench. We obtained a measure of the stability of the system by doing power-spectral analysis on particles stuck to the bottom of specially prepared slides. The resulting power spectrum was over two decades below the power spectrum of moving particles, as we will see later. We did not control temperature beyond the lab room-temperature control. If temperature effects were present, they would be changes to the diffusion constant for the particles, and convection from non-uniformity. Because the particles make excellent visual flow-trackers, we were convinced that there was no convection, and the room was controlled to ±1°C, so that variations from D = kT/(Girqr) would be ~ .3%. Our expectations are born out by the pure (Ax)2 oc t behavior we observe. B. Batch capture The retrieval of images from the VCR suffered from two significant problems. First, the signal quality was reduced by the loss of resolution from the video format. Second, the video signal from the VCR was less stable than that from the camera, 415

416

causing lost images from the VCR during digitizing—the frame grabber was often unsuccessful at detecting required synchronization signals at the proper time. We solved both problems to produce the data reported in [9] by using some additional features of the frame grabber to acquire a small portion of the full image directly to memory on board the video card at the full video rate (30 frames per second). The process is complicated by two limitations placed on the system by the EISA expansion slot in which the frame grabber resides. The EISA slot limits the transfer of memory from the frame grabber to the host in a single block to 64 kilobytes. The frame grabber is capable of acquiring a small rectangle located at the users direction within the full image. The card also has a megabyte of memory that can be used for storage. Initially, a small square is selected with a width W in pixels to enclose the desired particle. Then, two sets of buffers are allocated on the video card, each set containing N buffers. We required for speed that the sum of the data in N buffers occupy less than 64 kilobytes, so that the transfer of the data from the video card to the computer could occur in a single block dump. The tracking process starts by acquiring N consecutive image areas at the location which was initially centered on the particle, and storing them in the N buffers of the first set. When this is finished the process continues by acquiring N consecutive images into the second set of buffers. During this second acquisition, the data in the first set are combined into a single 64 kilobyte buffer and this buffer is transferred to the computer. The computer then calculates a new x, y position for the particle. We do this by finding the center of gravity of the grayscale levels in the last image acquired in the transferred data. This new video region is then sent to the frame grabber card, which then moves the small box it is capturing to the new location. This process is repeated with the two sets of buffers alternating between being used for acquiring images or being used for writing the data to the computer. The process is stopped either at the request of the user or when the random walk of the particle takes it outside of the field of view of the full image. C. Particle Tracking The one-micron diameter particles appear on the image as a dark spot, and the 2.5 micron diameter particles appear as a dark ring. These are the diffraction-limited result of the scatter from the particles seen against the bright background from the polished silicon substrate. We approximated the intensity distribution across the spot by a 2-D Gaussian of half-width d peaked at the location of the center of the particle RQ. We described the ring-shaped profile of the 2.5 micron particle by displacing the center of a one dimensional Gaussian a distance w from RQ and rotating the Gaussian 360° around the tip of RQ. RQ is thus still the center of the particle. The routine to

417 track the particles proceeds by using the parameters Ro, w, etc., found image to fit each point f in the two-dimensional square around the new image for the new value of RQ, etc. There is a start-up mode to parameters. Thus, the function we use to fit the measured intensity 8-bit number, is I = B + Ae-^r'w^'d2,

in the previous particle in the find the initial / , which is an

(1)

where B is the background intensity, A is the amplitude of the Gaussian, and Ar = | f — RQ I gives the distance between the point in the image r and the center of the particle RQ. Notice that for the spot particle, w is 0 and d is a measure of the particle radius, while for the ring particle, w is this measure. The last procedure in the data acquisition is thus to go through all of the images stored on the hard drive, which nearly filled the drive for the longest series reported here, and fit each image for the optimum values of the parameters in Eq. (1). In fact, although the time between the raw images is 1/30 second, the raw images are composed of two interlocking images (called fields) separated by only 1/60 second. Although these fields contain only 1/2 the spatial resolution of the complete image, there is still more than enough position information to give the 25 nm resolution reported for example in Ref. [11]. Therefore, we split the raw image up into its two constituent fields, so that the final data output is a time series of fit parameters with a time step of 1/60 second, with the value of Ro being the location of the particle at that time. We wanted to track as small a particle as we could resolve, since these would move the most, and therefore give us the best ratio of signal to noise. It is just fortuitous that these happen to stay in the field of view at our magnifications long enough to give us the long time series we want. Usually, it would take a few tries before a particle stayed in view for the required 10 minutes for the long file. If the particle strayed close to the edge of the field of view, the urge to persuade the particle to reverse course was very strong, so that we were often seen talking to the particle via the TV monitor in the course of this work. IV. ANALYSIS A. Diffusion We know that low concentrations of spherical particles diffuse in accordance with Einstein's model [13], in which sufficiently low concentrations of particles behave as an ideal gas, obeying the standard diffusion-equation

418

WM

= DV*f(x,t),

(2)

where /(£, t) is the number of particles per unit volume at location x and time t, and D = kT/(6irr)r), k = Boltzmann's constant, T = absolute temperature, 77 = solvent viscosity (water here), and r = the particle radius. This behavior has been demonstrated numerous times in the literature for particles freely diffusing at low concentrations [14]. Therefore, our first task was to look for this behavior in our own time series. We show in Fig. 2 results for the one-micron diameter particle moving freely in bulk solution. The ruling we used to calibrate the conversion from pixels to distance had a nominal accuracy of 10%. The measured values for D are Dx = 0.482 fj,m2/s and Dy = 0.474 /^m2/s, agreeing within 5% with the value of D = 0.462 /im2/s predicted from the standard model discussed in the previous paragraph; we used the nominal radius for the batch of particles we used, f = 0.5 /zm, a viscosity of rj = .0095 dyne/cm2 for water at 22 °C. For further discussion of uncertainties, see Sec. IV B. We are therefore confident that we have purely diffusive motion in our particles. In order to obtain this data, we constructed a cell with thickness of about 50 ^im, determined by measuring the vertical displacement between focusing on the bottom of the cell and focusing on the bottom of the cover slip. We then set the focus half-way between these two limits, so that we were imaging particles in the middle of the cell, away from the surfaces. Then we just had to wait until by chance a particle stayed in focus long enough to provide good statistics over a few decades of xTms(t) before it happened to move up or down too much to be imaged. Gaspard analyzed this data and found good agreement with his expectations, noting the expected exponential decrease in the trajectory probability with length of the trajectory, and therefore Lyapunov exponents > 0. However, the exponential range was an unsatisfying one decade. We could not go to shorter times, so the only way to increase the range of scaling was to increase the duration of our time series. However, this clearly would not be possible for particles moving in bulk, because no particle was going to stay in the required vertical focusing-band for long enough unless we sacrificed spatial resolution by going to lower magnification. We therefore began tracking particles which were heavy enough to sediment, that is, execute Brownian motion on or near the bottom of the cell. Such particles would stay in focus for as long as they remained in the horizontal field of view, which is much larger than the depth of field. The results of tracking 2.5 /im particles are shown in the plot of mean-squared distance vs. time presented in Fig. 3. The apparent diffusion-constant is reduced by .75 from the expected value. We attribute this to the well known wall-drag effect, which arises from an effective increase in the viscosity of the solvent due to hydrodynamic interactions near the surface [15]. Note that the datum at the shortest time does not follow the trend of the rest of the data. This

419 is consistent with our signal/noise estimate of 1 for this time, as we discuss in Sec. IV B. Clearly, Brownian motion is present, with only an adjustment to the value of D required to analyze the average behavior of the particle. Further, this method did allow us to track the particle for > 10 minutes, which was sufficient to give us the length of time series we required. However, we questioned whether sedimented motion would represent a good test of Gaspard's ideas. Our guess was that the presence of wall-drag would surely need to be included in Gaspard's analysis in order to describe correctly the crossover from stochastic to deterministic behavior expected for the system at sufficiently small time and length scales. However, that regime is the scattering time and mean-free-path of the colloid in water, which is 10's of decades smaller in time, and several in space, below the regime accessible to this apparatus [8]. At the large spatial-scale and slow time-scale of this experiment, we expect the behavior of the particle to be purely stochastic. Therefore, the surface-restricted diffusion should serve as a good test (without modifying Gaspard's description) of the positivity of the Lyapunov exponents for diffusion, given the measured value of D. Indeed, the results of Gaspard's analysis show pure stochastic behavior with the expected behavior of the e — T entropy [9].

B. Noise and Spatial Resolution We think of the data as having two important types of uncertainties. The first type is the accuracy of the change in spatial and time positions of the particle, and the diffusion constant D, which we take from measurements of (Ax)2 vs. t. The second type is the noise baseline, where the apparent random-motion of the particle changes from being actual particle motion into being an artifact due to some extraneous source, such as vibration or changes in the background in the image. Dealing first with the issue of accuracy, an effort to scrutinize the accuracy of x, t and D separately beyond the nominal values implied by the accuracies of the calibrated ruling, camera, and particle radius is not useful here. The comparison with theory hinges on the value of D, which has built into it the uncertainty of the radius r of the individual particle we are tracking, through D = kT/Qin^r. We had no way to measure the radius of the individual particle we are tracking, other than to calculate the value from D itself. Thus, we had to use for r the nominal value for the batch of colloidal particles we used. Therefore, we are not be able to distiguish an inaccuracy in x or t from a difference between the true radius r of the individual particle we tracked, and the nominal radius f we use to find D. The root-mean-squared deviation of particle radius from the lot mean is reported by the manufacturer to be 2%. This is already comparable to the agreement we have between our measured and expected

420

values for D, and so the issue is not worth pressing. It would however be interesting as an experimental issue to push these things further...say, using a strobed light-source to calibrate the timing, measuring the motion of the particle as calculated from the tracking program against a calibrated spatial ruling present in the same image in the background, and starting the experiment by trapping the particle in a laser beam and using the scattered light to make an independent determination of r. Moving on to the determination of the noise baseline, we used two methods to characterize the limits of our experiment. First of all, a power-spectrum analysis of the trajectories shows a uniform exponential decrease down to the shortest time of the experiment. Second of all, by letting a sample dry, and then re-wetting the sample, we are able to image stuck particles in the same optical condition as the moving particles. The stuck particles show a flat power spectrum a decade below the lowest value present in the moving particle's spectrum (Fig. 4). We expect the actual extraneous (i.e., not due to Brownian motion) noise in the moving particle to be higher than that suggested by the stuck particle because of changes in the background. Perhaps this accounts for the small rise in the power spectrum at high frequencies, which is about twice that expected from aliasing [16]. The possibility of vibrations as the origin of the excess noise seems ruled out by the fact that the power spectrum from the stuck particle is insufficient in strength to account for the increase. Our best estimate for the actual spatial noise-level for a moving particle comes from inspecting this power spectrum to find the time at which the spectrum deviates from the stochastic l / / 2 behavior associated with the Brownian motion. This suggests that we treat the rms displacement we measure at the shortest time as the uncertainty level. In this way we arrive at a value of ±25 nm for the 2.5 nm particle. The plots of rms displacement vs. time suggest that this is a worst case estimate, since they are still obeying the expected (Ax)2 oc t behavior. Such a scheme for estimating the noise also suggests different baselines for different particles, but this is reasonable, as the larger particles provide better imaging and more information for the interpolation. V. CONCLUSION We have produced trajectories of individual colloidal particles undergoing Brownian motion in water, with a time resolution of 1/60 of a second and a spatial resolution of 25 nm in a time series 146 thousand elements long. The resulting behavior is purely stochastic, and shows a positive dynamic epsilon-entropy consistent with microscopic chaos, as reported in [9]. An experiment which is able to span the stochastic and deterministic regions is the logical next work to do. One candidate is to suspend a tiny mirror in a closed cell and study the trajectories as a function of the gas density

421

in the cell. As the gas is evacuated, the mean free-path would quickly become larger than the size of the cell, and correlations should appear in the motion of the mirror. This would reduce the rate at which new trajectories appear, and thus reduce the dynamical entropy. Such behavior would be the first sign of underlying deterministic behavior. Gaspard and Wang have argued that in general the e—entropy and in particular the multiple-time correlation function used in the analysis are a powerful tool as a quantitative measure to characterize the spatial and time-scale dependence of different random systems [17]. The analysis in reference [9] of the experiment discussed here shows how this technique reveals the scaling of the randomness with space and time. The purpose of the measurements was to provide a large catalogue of particle trajectories for analysis of the e-entropy associated with Brownian motion, and the data met this purpose well. The video particle-tracking technique we use gives noise less then 10% for the mean-squared displacement over four decades of time for the motion of a sedimented particle. The diffusive behavior for the motion in bulk fluid is well described (within a few percent) by the usual Stokes-drag model, and that of the sedimented particle shows a reduced rate of diffusion consistent with other workers' measurements of the wall-drag effect. ACKNOWLEDGMENTS M.E. Briggs would like to thank the Physics Department at the University of Utah and the organizers of the 5th Experimental Chaos Conference for support to attend this conference.

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REFERENCES [1] Gaspard, P., Chaos, Scattering Theory, and Statistical Mechanics (Cambridge University Press, 1998). [2] Bunimovich, L.A. & Sinai, Ya G., "Statistical Properties of Lorentz gas with periodic configuration of scatterers," Commun. Math. Phys. 78, 479-497 (1981). [3] Evans, D.J., Cohen, E.G.D. and Morris, G.P., "Viscosity of a simple fluid from its maximal Lyapunov exponents," PRA 42, 5990-5997 (1990). [4] Gaspard, P. and Nicolis, G., "Transport properties, Lyapunov exponents, and entropy per unit time," PRL 65, 1693-1696 (1990). [5] Dorfman, J.R. and Gaspard, P., "Chaotic scattering theory of transport and reaction-rate coefficients," PRE 51, 28-35 (1995). [6] Ruelle, D., "Positivity of entropy production in nonequilibrium statistical mechanics," J. Stat. Phys. 85, 1-23 (1996). [7] Chernov, N.I., Eyink, G.L., Lebowitz, J.L. and Sinai, Y., "Derivation of Ohm's law in a deterministic mechanical model," PRL 70, 2209-2212 (1993). [8] Gaspard, P., "Can we observe microscopic chaos in the laboratory?" Adv. Chem. Phys. XCIX, 369-392 (1997). [9] Gaspard, P., et al., "Experimental Evidence for Microscopic Chaos," accepted for publication in Nature. [10] Private communications with several workers, including E.G.D. Cohen, C. Dettman, P. Grassberger and H. van Beijeren, concerning models such as the wind-tree model, the Rayleigh flight of a tracer particle in an ideal non-interacting gas, or the motion of an impurity in an infinite harmonic crystal. These discussions are currently submitted to Nature for the comment and response section. [11] Grier, D. G. & Murray, C. A., "Video Microscopy of Strongly Interacting Colloids and Supramolecular Aggregates in Solution" in NATO ASI Series C, 369, Chen, S. I., Huang, J. S. & Tartiglia, P., Eds., Boston: Kluwer (1991). [12] Interfacial Dynamics Corporation, Portland, Oregon, USA, IDC Spheres (tm), Polystyrene Latex, Surfactant Free, Group: Sulfate, sizes in microns 0.99 ± 2.4% (8.4% solids, batch 10-110-13.169), and 2.5 ± 2.8% (8.3% solids, batch 10-265-37.) [13] A. Einstein, Investigations on the theory of the Brownian Movement, Dover, New York, 1956. [14] See for example W.B. Russel, D.A. Saville and W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, New York (1989), Chapter 3. [15] Schaertl, W. & Sillescu, H., "Dynamics of colloidal hard spheres in thin aqueous suspension layers," J. Coll. Int. Sci., 155, 313-318 (1993). [16] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, 2 ^ Ed., Cambridge, Cambridge (1992). [17] Gaspard, P. & Wang, X.-J., Physics Reports 235, 321 (1993).

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Magnetic Resonance Imaging of Structure and Coarsening in Three-Dimensional Foams Burkhard A. Prause and James A. Glazier Department of Physics, University of Notre Dame, Notre Dame, IN 46556 We used Magnetic Resonance Imaging (MRI) to study nondestructive^ the interior of slowly coarsening disordered foams. We developed two methods for data analysis that can independently provide detailed structural information, including the distributions of bubble sizes, faces per bubble and edges per face to study foams with varying liquid fractions and polydispersity. We analyzed several hundred bubbles in detail, and found neither Kelvin nor complete Weaire-Phelan polyhedral structures. Our analysis provided values for the average number of faces, {/), the averaged zero growth value, /o, and the volume rate of change for individual bubbles with constant topology. While the average volume rate of change depends on the initial disorder and wetness, the growth rate for each foam is compatible with Glazier's linear three-dimensional growth law. Neither foam coarsened self-similarly during an average volume increase of a factor of 3.5 agreeing with simulation results that equilibration is very slow in three-dimensional foams.

I. INTRODUCTION Early this century D'Arcy Thompson [1] attributed the formation of many regular biological patterns, such as the bee's honeycomb, sponges and cucumber skin, to the action of purely physical forces, stressing the analogy between the development of foam structure and the formation of other natural patterns. Smith noted the similar nature of coarsening in bubbles and growth in metallic grains [2], proposing foams as a convenient model for the universal dynamics of grain growth. Today, foam experiments model cellular materials in two and three dimensions, in which simple physical constraints, such as diffusion and surface energy minimization drive structure and dynamics. Industries including brewing, printing, fire-fighting and oil exploration all use foams. Recent simulations have led to a better theoretical understanding of threedimensional growth laws and the scaling properties of grains [3-6]. Analogous to von Neumann's law for two-dimensional grain growth [7], based on computer simulations, Glazier proposed an averaged growth law for three-dimensional grains [3]: 427

428

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The averaged volume rate of change for a group of bubbles with / faces depends only on its number of faces. On average, bubbles with / greater than /o will grow, while bubbles with a smaller number of faces will shrink, k is a diffusion constant. For the relation between / and /o , Weaire and Glazier deduced that [4]: fo = ^ ( 1

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where n% = (J2) — (/) 2 measures the disorder of the foam. Eqn. 2 is exact provided that the average volume of a bubble with /faces scales as (V/) oc / 3 , which was true in Glazier's Potts model simulations. A large amount of work addresses the minimal surface area partition of three-dimensional space into polyhedral domains of uniform volume. Weaire and Phelan recently showed that the /^-Tungsten lattice partition with a unit cell composed of a regular pentagonal dodecahedron and six tetrakaidecahedral barrels (with two hexagonal faces and twelve pentagonal faces) has a lower surface area than a partition into regular Kelvin tetrakaidecahedra (six square faces and eight hexagonal faces) [8-10]. Experimental verification of the existence of any of these structures in bulk three-dimensional foams has proven difficult, in part due to the difficulty of visualizing the foam interior. Both structures can occur in foams confined to tubes narrow compared to the typical bubble radius. Durian et al. used diffusing wave spectroscopy (DWS) to measure the rates of rearrangement and the averaged grain volume growth exponents in bulk shaving cream [11-13]. They found that the average radius grew according to a power law (r) oc t@, with /? = 0.5, consistent with "self-similar" growth, in which the normalized distribution of bubble volumes does not change over time. DWS cannot provide direct information on bubble size distributions or grain shapes and topologies which are needed to verify growth laws, but it can examine very large numbers of bubbles (hundreds of thousands) over many decades of growth in length scale. Confocal optical tomography (COT) can determine the growth and shapes of individual bubbles in dry foams [14,15], but only for foams with a few interior bubbles (Monnereau et al. studied a total of 28). COT is anisotropic with limited spatial resolution, and requires black film boundaries, which restricts it to extremely dry foams. The botanist Matzke made the most careful study of three dimensional bubble shapes, using a simple binocular microscope [16,17] and near endless patience. He studied 1000 individual bubbles, noting their numbers of faces and edges. While providing valuable data on the shapes of such grains, he could

429 not provide information on volumes, lacking the ability to determine exact vertex locations, or to track a bubble's evolution over time. Magnetic Resonance Imaging provides detailed information on the structure and evolution of individual bubbles in a large variety of foams. II. MRI EXPERIMENTS MRI has evolved into a widely used and irreplaceable tool in diagnostic medicine since its discovery in 1973 [18]. It detects the precessing magnetic moments of excited nuclei in the presence of magnetic field gradients, making it ideal to study biological tissues which contain a high natural abundance of excitable water [19]. Non-medical MRI has developed comparatively slowly, due to the varying requirements of different physical, chemical and material science applications, which require custom imaging techniques and high performance imagers and computers [20,21]. German et al. made the first MRI foam images [22], investigating foam drainage by measuring the water content inside a column of liquid foam over time. The two-dimensional images could only distinguish a few very large bubbles. Gonatas et al. succeeded in imaging two-dimensional slices, measuring the bubble size distributions over long periods of time [23]. Their foam did not reach self-similar growth (the scaling state), having a growth exponent, 0 = 0.3. Our experiments used a gelatin based liquid with SDS as surfactant and trace amounts of CUSO4 and DyCl3 (TTHA) as relaxation agents. We used a 7 Tesla SQUID magnetometer to measure the magnetization of the liquid with varying amounts of relaxation agents. Matching the magnetic susceptibility of the liquid to that of air minimized susceptibility artifacts in the foam during imaging, as well as increasing the susceptibility induced transverse relaxation time T2 [20]. We produced foams by two methods, resulting in different degrees of initial wetness and polydispersity. Blowing air through a millipore (fish-tank) filter produced very dry (volume liquid fraction $ = 3%) foam, with an initial average bubble diameter of 2.0mm ± 0.3mm. By whipping the liquid with an electric mixer, we obtained an initially wet (4> = 50%) foam, which drained to about $ = 10% after 90 minutes, resulting in a moderately dry polyhedral foam, with bubble diameters ranging from 30/xm to 300^m. Due to the non-zero liquid fraction, gravitational drainage of liquid through Plateau borders and edges affects coarsening in the foam, as well as gas diffusion across the films, vertex coalescence and wall breakage. Holding the sealed glass cells containing the foam at a constant temperature of 280°K, limited the rate of drainage and kept air from drying the foam walls. Samples prepared in this manner coarsened without breaking down for as long as six days, at which time the gelatin in the faces started polymerizing, inhibiting growth.

430

FIG. 1. Maximum intensity projections of three-dimensional MRI reconstructions of a foam at three stages of development, (a) = 24 hrs. (b) = 36 hrs. (c) = 48 hrs.

The stability and magnetic homogeneity of the samples allowed us to image the foams in three dimensions, using our own Bruker 300 MHz (7 Tesla) imaging spectrometer and a set of water-cooled high field gradients (96 G/cm), to achieve sufficient spatial resolution in this strongly relaxing material. The samples were placed inside 15mm or 25mm rf-coils and kept at constant 280°K±1°K. A customized 3D spin-echo pulse sequence with very high repetition rates (TE = 1.9ms, TR = 50ms) provided 256 x 128 x 128 real data points, at isotropic resolutions of 101/j,m in the 15mm coil, and 140/xm in the 25 mm coil. A single average of the entire sample took 14 minutes. To improve the signal to noise ratio, initial scans employed two averages, extended to four and later eight, as the sample drained. Thus a "snapshot" of the foam took 30 minutes initially (for about 24 hours) and 120 minutes at the end of the run. We took data for both relatively homogeneous (first method) and polydisperse (second method) foams in both rf-coils. The larger volume of the 25mm coil allowed the foam to coarsen to a longer length scale but reduced the signal intensity, which is twice as great in the 15mm coil. Figure 1 shows three-dimensional, maximum intensity reconstructions of a dry, initially homogeneous foam at different stages of development.

III. ANALYSIS We have developed two methods to analyze the three-dimensional data sets. The more labor intensive method concentrates on exact reconstruction of each

431 individual bubble. We cut the data run into slices one pixel deep. The images show signal along all intersections of bubble faces (edges). Scanning through the slices, we identify any point at which four edges converge as a vertex and record its location. We process the lists of vertices that belong to each bubble using the qhull algorithm [24] (qhull is available for free from the University of Minnesota Geometry Center), creating a list of simplicial (triangular) facets that constitute the smallest convex hull around the set of vertices. Prom this list we calculate the bubble volumes. We then merge the facets to create non-simplicial faces that have the same number of edges as the corresponding polygonal walls of the bubbles. This method allows us to investigate the detailed shapes of any number of interior bubbles, limited only by the cumbersome procedure of tracing all vertices in the sample. We analyzed over 300 bubbles at different stages of coarsening between 24 and 48 hours. We found no Kelvin tetrakaidecahedra, or full Weaire-Phelan structures, but did find six pentagonal dodecahedra. Only two dodecahedra were present in the sample at the same time and they did not adjoin tetrakaidecahedral barrels. At 36 hours, the mean number of edges per face was (n) = 5.06, and the mean number of faces (/) = 12.32 ± 0.56. Fig. 2(a) shows the averaged volume for /-faced bubbles as a function of the number of faces for a 36 hour foam. Fig. 2 shows the distributions of (b) volumes, (c) numbers of edges and (d) numbers of faces. For (Vf) oc fa, we found a = 2.7 ± 0.36, consistent with the value for Potts model simulations, a = 3 [4]. Using Eqn. 2 we found a zero growth value /o = 16.3±0.74, consistent with /o = 15.8 for the Potts model [3]. The disorder was ^{f) = 54. Manual reconstruction allows us to trace even the smallest polygonal shapes in the samples, with volumes as low as 36 voxels, providing a high degree of accuracy for very polydisperse foams. We developed a second method to eliminate the manual identification of vertices. We first processed the reconstructed three-dimensional images to remove all imaging artifacts and random noise. Artifact correction employed an appropriate neighborhood ranking filter [25]. To avoid eliminating vertex or edge signal, while removing all disconnected noise ("buckshot") we created a binary image by thresholding and used a closing operator to fill small gaps. A scanning program then recorded the locations of all pixels neighboring at least one voxel with signal. Fig. 3 shows the result of this process on a slice from a three-dimensional data set. We used the list of connected voxels to create a Euclidean distance map of the lattice as shown in Fig. 4, in which the intensity at each voxel is the Euclidean distance to the nearest voxel containing signal.

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Next we group localized clusters of local maxima into single centers using a third nearest neighbor scan. We eliminate spurious maxima that can occur at the centers of bubble faces by requiring a zero overlap between the zones of influence of all pairs of maxima, imposing the geometric constraint that the diameters of adjacent bubbles are larger than that of their joining face. We use a three-dimensional Delauney triangulation [26] from the center locations to compute neighbors for each center and estimate the averaged radius of the equivalent sphere for each center as the mean of the distances to all its neighbors, weighted by the ratio of the distances between the center and each neighbor. This approximation is the chief source of error in our volume determination. Finally, we discard bubbles in contact with an outside wall or the edge of the image. We could thus track individual bubbles over time, identifying bubbles that did not change their number of faces between consecutive images. For each time step we compared the centers to those in the previous image and de-

433

termined the most probable pairings according to center location, volume and number of faces. Between consecutive time steps, only 10 to 15% of all bubbles did not change their number of sides. Of those, we could uniquely map the center locations of over 90%. This information provides /o in Eqn. 1 directly. Integrating Eqn. 1 for constant topology yields, for the volume rate of change: dV

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Fig. 5 shows the volume rate of change as a function of / for two different foam samples. The first sample, foam-10-16 was an initially ordered, very dry foam with # < 3%, while the second sample, foam-2-02 was very polydisperse and relatively wet ($o ~ 10%). The zero growth values were /o = 11.6 ± 3.8 and /o = 12.1±3.3, with (/) = 12.32±0.69 and
434

The large errors in / 0 are due to the small number of bubbles that maintained a constant / between consecutive time steps. During a series of 36 runs the number of interior bubbles decreased from over 200 to about 40. The values are in very good agreement with each other. They are lower but within error of the values for (/) and /o found by optical tomography [14] and in simulations [3]. The linearity of the computed derivative in /strongly supports the linear dependence on topology of Glazier's growth law, Eqn. 3. While the large error in our volume determination makes it difficult to determine whether the scatter in growth rate is intrinsic or due to measurement error, the fact that the linearity is better than would be expected from a random error suggests that the volume scatter is intrinsic, i.e. that the law holds only on average as in the Potts model.

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For comparison we used Eqn. 2 to compute the zero growth value for the two foams at each time step. Averaged over time we calculated /o = 14.07 ±0.25 for foam-01-16 and / 0 = 14.88 ±0.48 for foam-2-02. The values for f0 found by the

435

two methods for both samples are consistent with each other, due to the large error of the computation using Eqn. 3. The slope K = 4.31 X 1 0 - 2 ± 1.43 x 1 0 - 2 mm 2 /hr for foam-10-16 was nearly twice as large as the K = 2.24 x 10 _ 2 ±6.0 x 10~ 3 mm 2 /hr for foam-2-02. K is proportional to the diffusion constant, k, in Eqn. 1, which we expected to be lower than for foam-2-02, due to foam-2-02's larger liquid fraction. Fig. 5 suggests that neither foam reached a scaling state, where (V(t)) oc ta, with a = 1.5. During coarsening over an average volume increase by a factor of 3.5 in both samples, we found a = 0.834 ± 0.055 in foam-10-16, and a — 0.936 ± 0.029 in foam-2-02. This result is consistent with earlier findings in three-dimensional foams [23,14], and predictions from computer simulations, that disordered three-dimensional foams need several orders of magnitude of length scale coarsening to attain self-similar growth [3]. IV. CONCLUSIONS We have used M M to analyze the detailed interior structure of a cellular material, and experimentally verified its dynamics. The data presented are consistent with Glazier's growth law and predictions concerning scaling behavior in three-dimensional foams. They also agree with earlier optical studies on disordered foam structures by Matzke and Monnereau [16,17,15]. We did not find Kelvin tetrakaidecahedra or Weaire-Phelan structures. Both of our methods for analyzing three-dimensional foam structure have advantages. Hand tracing distinguishes very small bubbles, with volumes near the MRI resolution limit. It also allows faithful reconstruction of bubble shapes. Values for (/), /o, and growth exponents V(f) and V(i), are highly accurate. The method is too labor intensive to determine the time evolution of large numbers of bubbles. The automated analysis using Euclidean distance maps and Delauney triangulations is computation intensive. Its current implementation loses small bubbles because it requires absolutely noise free data. In future experiments we can improve the large error bars for (/) and /o by slowing coarsening rates and further reducing imaging times (with minimal loss of signal to noise ratio). Faster imaging will allow us to track more bubbles between rearrangements, improving statistics. To improve our volume estimates, we are currently incorporating a threedimensional Voronoi tessellation [26] into our automated analysis, to reconstruct exact hulls around individual bubbles as in our manual method. ACKNOWLEDGMENTS This research was supported by NSF/DMR92-57011, NSF/CTS-9601691

436 a n d NSF/INT-96-03035-0C, by t h e American Chemical Society/Petroleum Research Fund, NASA grant UGA99-0083 a n d D O E grant DE-FG0299ER-45785. We t h a n k Dr. Dieter Gross a t Bruker Analytik in G e r m a n y for his vital help with t h e imaging experiments.

[1] D. W. Thompson, On Growth and Form (Cambridge University Press, Cambridge, 1942). [2] C. S. Smith, in Metal Interfaces (American Society for Metals, Cleveland, 1952), pp. 65-108. [3] J. A. Glazier, Phys. Rev. Lett. 70, 2170 (1993). [4] D. Weaire and J. A. Glazier, Phil. Mag. Lett. 68, 363 (1993). [5] C. Sire, Phys. Rev. Lett. 72, 420 (1994). [6] R. M. C. de Almeida and J. C. M. Mombach, Physica A 236, 268 (1997). [7] J. von Neumann, in Metal Interfaces (American Society for Metals, Cleveland, 1952), pp. 108-110. [8] D. Weaire and R. Phelan, Phil. Mag. Lett. 69, 107 (1994). [9] D. Weaire and R. Phelan, Phil. Trans. R. Soc. Lond. A 354, 1989 (1996). [10] D. Weaire, The Kelvin Problem (Taylor& Francis, London, 1996). [11] D. J. Durian, D. A. Weitz, and D. J. Pine, J. Phys. Condens. Matt. 2, SA433 (1990). [12] D. J. Durian, D. A. Weitz, and D. J. Pine, Science 252, 686 (1991). [13] D. J. Durian, D. A. Weitz, and D. J. Pine, Phys. Rev. A 44, R7902 (1991). [14] C. Monnereau and M. Vignes-Adler, Phys. Rev. Lett. 80, 5228 (1998). [15] C. Monnereau and M. Vignes-Adler, J. Colloid Interface Sci. 202, 45 (1998). [16] E. B. Matzke, Am. J. Bot. 33, 58 (1946). [17] E. B. Matzke, Am. J. Bot. 33, 130 (1946). [18] P. C. Lauterbur, Nature 242, 190 (1973). [19] P. Mansfield and P. Morris, NMR Imaging in Biomedicine (Academic Press, New York, 1982). [20] P. T. Callaghan, Principles of Nuclear Magnetic Resonance Microscopy (Clarendon Press, Oxford, 1991). [21] R. A. Komoroski, Analytical Chemistry 65, 1068A (1993). [22] J. B. German and M. J. McCarthy, J. Agric. Food Chem. 37, 1321 (1989). [23] C. P. Gonatas et al., Phys. Rev. Lett. 75, 573 (1995). [24] C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, ACM Transactions on Mathematical Software 22, 469 (1996). [25] J. C. Russ, The Image Processing Handbook (CRC Press, Boca Raton, 1995). [26] F. Aurenhammer, ACM Computing Surveys 23, 345 (1991).

Using Unstable Periodic Orbits to Approximate Noisy Chaotic Time Series T. L. CARROLL Code 6345, Naval Research Lab, Washington, DC 20375 ABSTRACT Many noise reduction techniques have been proposed for chaotic signals, but most techniques only work when the added noise is small. I show in this work how to approximate a chaotic signal by building sequences based on unstable periodic orbits (UPO's). The UPO approximation may be used to recognize attractors even in the presence of large noise, including noise that is deterministic.

1. Introduction At one point, several research groups suggested that the deterministic nature of chaotic signals might make it easy to separate these signals from background noise " . Most of these methods only worked when the noise was smaller than the chaotic signal. In this work, I report a technique that approximates a chaotic signal based on the unstable periodic orbits (UPO's) of the attractor that generated the chaotic signal. The UPO approximation can be made to work for any noise level, depending on the amount of computation one is willing to do. The UPO approximation is essentially an attractor recognition technique; it recognizes the presence of a signal from a desired attractor and can reject signals from other attractors. My approach is this: I first extract a set of low period UPO's from one or more time series generated by a chaotic system. There are several techniques available for this ' . I then assemble sequences of these UPO's. I seek to construct all possible sequences of a given length. There may be many possible sequences for a particular length, but most sequences never show up in the actual attractor, so a large number of possible sequences can be discarded. I then compare each of these sequences to a fixed length segment of a time series from the chaotic attractor. Different chaotic systems become uncorrelated with each other, so I compare each unstable periodic orbit sequence with the time series segment by taking the cross-correlation between them. I take the UPO sequence with the highest cross-correlation as the best approximation to the segment-from the chaotic time series. The fit is never exact because noise (including roundoff error) causes the chaotic attractor never to be exactly on a UPO, and the chaotic attractor may move to a different UPO at some unpredictable time. 2. Constructing UPO Sequences The process of constructing UPO sequences begins with isolating the UPO's. Typically, I isolate UPO's up to period 16, although isolating more or fewer may not Q

greatly affect the approximation. Reference shows some typical UPO's for the Lorenz equations. All possible sequences for this set of UPO's are then constructed up to some maximum sequence length. It is necessary in this construction to decide which UPO's can follow which other UPO's. For the work in this paper, the distance from the end of 1 UPO to the beginning of the next determined which orbits could go in series. The maximum 437

438 distance allowed between each component of the end of 1 orbit and the beginning of the next was orbits was 2% of the RMS amplitude of that component of the chaotic attractor. This distance could also be adjusted to change the approximation. All possible phases of each UPO were used. For a first pass, I created sequences about 2 cycles long (64 points at the sampling rate I used). These 2 cycle UPO sequences were the basic building blocks of the approximation. To create longer UPO sequences, rather than go through the prediction step again, these 2 cycle (64 point) sequences were simply combined sequentially to yield 4 cycle (128 point) sequences. There were a large number of 4 cycle sequences, and many of these sequences contained large discontinuities or did not correspond to physically realizable sequences. To reduce the number of sequences, I used several long time series from my original chaotic circuit, and measured the cross correlation between each 4 cycle UPO sequence and a 4 cycle segment from the chaotic time series. All signals were normalized and DC offsets were removed before cross correlating. The 4 cycle UPO sequence with the largest cross correlation was chosen as the best fit to the time series segment. This process was repeated for a large number of different time series segments, on the order of 10,000 to 100,000 total segments. From the data on which 4 cycle UPO sequence was the best fit to each time series segment, a histogram was made. Some UPO sequences showed up more often than others, and some never showed up. All UPO sequences that occurred fewer than some threshold number of times were discarded, so that the total number of UPO sequences was reduced by a factor of 4 to 10. Excluding too many UPO sequences degraded the cross-correlations noticeably, so the threshold was set so that the cross-correlations were not degraded by too much. This truncation of the set of UPO sequences worked because many of the UPO sequences never actually occurred in the chaotic attractor (there has been some work on creating "grammars" ' which describe what sequences of UPO's can occur in a chaotic attractor). In addition, some sequences were very similar to each other, so substituting one sequence for another only slightly reduced the cross correlation. I repeated the procedures above to generate UPO sequences of length 256 points. Longer sequences could be generated if necessary. 3. Application to a Chaotic Circuit In order to extract the UPO's, I digitized the x, y, and z signals from a piecewise linear Rossler (PLR) chaotic circuit described in n . I digitized the signals at the rate of 80,000 points/sec. Figure 1 is the attractor from this circuit. The x, y, and z signals from the PLR circuit were used to reconstruct the attractor, and the method of close approaches 6 was used to extract UPO's up to 565 points long, which was approximately period 10. The UPO's could be extracted from a single time series embedded by the method of delays, but the additional signals were used because they were available. The UPO's were all downsampled by a factor of 2 (equivalent to digitizing at a rate of 40,000 points/sec) and were combined to generate UPO sequences with a length of 64 points by the method described above. There were 10,446 UPO sequences of length 64 (the exact number will vary depending on how the combination is done). These sequences were compared to a set of time series of the x signal from the PLR circuit

439 digitized at 40,000 points/sec. From this comparison, it was determined that a set of the 244 U P O sequences that occurred most often were all that was necessary to give an adequate approximation to the x time series (the definition of adequate is somewhat arbitrary, and may depend on how much computation one is willing to do). The U P O sequences of length 64 were then combined to give sequences of length 128, and after again eliminating most of the length 128 sequences, length 256 sequences were constructed in the same way. Eventually, it was found that a set of 436 U P O sequences of length 256 gave an adequate approximation to the x signal from the P L R circuit.

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4. Attractor Recognition The UPO sequences may be used to indicate when a chaotic time series comes from a particular chaotic circuit, even when noise is present. The UPO sequences are chosen so that the maximum of the cross-correlation between each UPO sequence and a time series signal from the same chaotic system is large (usually greater than 0.9), while the cross correlation between each UPO sequence and a time series from a different chaotic circuit will be smaller. Figure 2 shows histograms made by comparing each UPO sequence to a time series signal. The histogram is made by recording the largest cross correlation between the set of UPO sequences (time lag is also varied for each sequence) and the signal, using UPO sequences 256 points long. Figure 2(a) shows the cross correlations when the UPO sequences are compared to a time series from the x signal from the PLR circuit. The cross correlation is large, usually just over 0.9. Larger cross correlations could be obtained by using a larger set of UPO sequences, but the computational burden would also be greater. Figure 2(b) shows the cross correlations when the UPO sequences are compared to a sine wave (generated by a function generator) whose frequency is the same as the peak frequency for the PLR x

440 signal. It can be seen that the cross correlation is almost always lower than the cross correlation with the PLR x signal, so the UPO approximation is able to distinguish between the PLR signal and a similar sine wave. Figure 2(c) shows the cross correlations between the UPO sequences and a signal from a hysteretic circuit described in . The hysteretic circuit has been modified so that its peak frequency is the same as the PLR circuit. Once again, the cross correlations are almost always lower than when the PLR x signal is used, so the UPO approximation technique is able to distinguish between these two chaotic circuits.

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441

5. Approximations in the Presence of Noise The attractor recognition described above also works when noise is added to the time series signals. To test the attractor recognition in the presence of noise, white noise from a noise generator was electronically added to the time series signals. The RMS amplitude of the noise was the same as the RMS amplitude of the signals, so the signal to noise ratio was 1.

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442 When the signal to noise ratio is 1, is difficult to distinguish the different time series by eyeball. Figure 3 shows the cross correlations between the set of UPO sequences for the PLR circuit and the three noisy signals. While the histograms in Fig. 3 are not as well separated as the histograms in Fig. 2 (no noise), it is still possible to distinguish the three signals based on the UPO approximation. It would be easier to distinguish between the different time series signals if longer UPO sequences were used.

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6. Communications The attractor recognition method also makes the UPO approximation technique useful for communications. For a very simple communications method, it is possible to take a chaotic signal that is not symmetric about 0 and multiply by ±1. In the receiver the largest magnitude of the cross correlation maximum will be greater than 0 if a +1 was sent and less than 0 if a -1 was sent. In order to show how noise affects the detection process, histograms were made for the PLR x signal multiplied by +1 when noise was added. These histograms are similar to those presented before, but the largest magnitude of the

443

cross correlation is measured and plotted with the proper sign, so that negative cross correlations are possible. Positive cross correlations mean that a +1 was received when a +1 was sent, while negative cross correlations mean that a -1 was received when a +1 was sent. Figure 4 shows the histogram of cross correlations when white noise is added. Figure 4(a) is when the noise amplitude is 0. In Fig. 4(b), the noise is as large as the signal, and it can be seen that the cross correlation with the greatest magnitude is sometimes negative, leading to an error is signal detection. In Fig. 4(c), the noise is twice the size of the signal, so more errors occur. Figure 5 shows what happens when other types of interfering signals are added to the chaotic carrier. Figure 5(a) shows the result of adding a signal from the hysteretic circuit with the same amplitude as the signal from the PLR circuit. While some errors are present, they are not common, so the receiver is able to reject interference from another transmitter that uses a different chaotic carrier. Figure 5(b) shows the histograms when the interfering signal comes from a sine wave with the same RMS amplitude as the PLR x signal. Some errors are present, but the receiver is able to reject some of this periodic interference. Standard methods of communications characterization are presented in .

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correlation Figure 5. Histograms of cross correlations when interfering signals with the same RMS amplitude as the PLR x signal are added to the PLR x signal. In (a), the interfering signal is from a hysteretic circuit, while in (b) the interfering signal is a sine wave from a sine wave generator.

7. Conclusions It is possible to approximate a chaotic signal by building sequences from the UPO's of the chaotic attractor that generated the signal, even when a large amount of

444 noise is added to the signal. The noise level that may be overcome is proportional to the length of the UPO sequences. There are some computational issues that have not yet been resolved. There may be many possible UPO sequences for a given length, and the number of possible sequences increases as the sequence length increases. It is possible to eliminate some of these sequences because they do not occur in the actual attractor, while other sequences may be dropped at the expense of the accuracy of the approximation. One may still be left with a large number of sequences to compare to the received time series. A better search algorithm could improve the efficiency of the UPO approximation. 8. References 1. R. Cawley and G. H. Hsu, Phys. Rev. A 46 (1992) 3057. 2. J. D. Farmer and J. J. Sidorowich, Physica D 47 (1991)373. 3. E. J. Kostelich and J. A. Yorke, Phys. Rev. A 38 (1988) 1649. 4. E. Rosa, S. Hayes and C. Grebogi, Phys. Rev. Lett. 78 (1997) 1247. 5. T. Sauer, Physica D 58 (1992) 193. 6. D. P. Lathrop and E. J. Kostelich, Phys. Rev. A 40 (1989) 4028. 7. P. So, E. Ott, S. J. Schiff, D. T. Kaplan, T. Sauer and C. Grebogi, Phys. Rev. Lett. 76 (1996) 4705. 8. T. L. Carroll, Phys. Rev. E 59 (1998) 1615. 9. D. Auerbach and I. Procaccia, Phys. Rev. A 41 (1990)6602. 10. G. B. Mindlin, X. J. Hou, H. G. Solari, R. Gilmore and N. B. Tufillaro, Phys. Rev. Lett. 64 (1990)2350. 11. T. L. Carroll, Am. J. Phys. 63 (1995) 377. 12. J. Neff and T. L. Carroll, Scientific American 269 (1993) 120. 13. T. L. Carroll, unpublished (1998)