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Dec 1997

Volume 7, Issue 4, pp. 509-826

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Introduction: Control and synchronization of chaos

William L. Ditto and Kenneth Showalter

Chaos 7, 509 (1997); http://dx.doi.org/10.1063/1.166276 (3 pages) | Cited 15 times

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05.45.-a Nonlinear dynamics and chaos

Targeting in Hamiltonian systems that have mixed regular/chaotic phase spaces

Christian G. Schroer and Edward Ott

Chaos 7, 512 (1997); http://dx.doi.org/10.1063/1.166277 (8 pages) | Cited 8 times

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The problem of directing a trajectory of a chaotic dynamical system to a target has been previously considered, and it has been shown that chaos allows targeting using only small controls. In this paper we consider targeting in a Hamiltonian system, whose phase space contains a mixture of regular quasi-periodic and chaotic regions. A multistep forward–backward method targeting strategic intermediate points is found to be efficient and robust. It takes full advantage of the phase space structure and is believed to yield optimal transport times. It is robust under the influence of small noise and small modeling errors and recovers from temporary loss of control. Two illustrative examples, the standard map and the restricted circular three body problem, are presented. (The latter corresponds to motion of a space probe in the presence of the earth and the moon.) Comparisons are made of our method to other targeting strategies. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
45.05.+x General theory of classical mechanics of discrete systems

Fundamentals of synchronization in chaotic systems, concepts, and applications

Louis M. Pecora, Thomas L. Carroll, Gregg A. Johnson, Douglas J. Mar, and James F. Heagy

Chaos 7, 520 (1997); http://dx.doi.org/10.1063/1.166278 (24 pages) | Cited 213 times

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The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and “cottage industries” have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution—a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (systems with more than one positive Lyapunov exponent) to be synchronized. Several proposals for “secure” communication schemes have been advanced; major ones are reviewed and their strengths and weaknesses are touched upon. Arrays of coupled chaotic systems have received a great deal of attention lately and have spawned a host of interesting and, in some cases, counterintuitive phenomena including bursting above synchronization thresholds, destabilizing transitions as coupling increases (short-wavelength bifurcations), and riddled basins. In addition, a general mathematical framework for analyzing the stability of arrays with arbitrary coupling configurations is outlined. Finally, the topic of generalized synchronization is discussed, along with data analysis techniques that can be used to decide whether two systems satisfy the mathematical requirements of generalized synchronization. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.40.-k Geometry, differential geometry, and topology
84.40.Ua Telecommunications: signal transmission and processing; communication satellites

Control of chaos in excitable physiological systems: A geometric analysis

David J. Christini and James J. Collins

Chaos 7, 544 (1997); http://dx.doi.org/10.1063/1.166279 (6 pages) | Cited 9 times

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Model-independent chaos control techniques are inherently well-suited for the control of physiological systems for which quantitative system models are unavailable. The proportional perturbation feedback (PPF) control paradigm, which uses electrical stimulation to perturb directly the controlled system variable (e.g., the interbeat or interspike interval), was developed for excitable physiological systems that do not have an easily accessible system parameter. We develop the stable manifold placement (SMP) technique, a PPF-type technique which is simpler and more robust than the original PPF control algorithm. We use the SMP technique to control a simple geometric model of a chaotic system in the neighborhood of an unstable periodic orbit (UPO). We show that while the SMP technique can control a chaotic system that has UPO dynamics which are characterized by one stable manifold and one unstable manifold, the success of the SMP technique is sensitive to UPO parameter estimation errors. © 1997 American Institute of Physics.
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87.10.-e General theory and mathematical aspects
05.45.-a Nonlinear dynamics and chaos
02.40.-k Geometry, differential geometry, and topology

The OPCL control method for entrainment, model-resonance, and migration actions on multiple-attractor systems

E. Atlee Jackson

Chaos 7, 550 (1997); http://dx.doi.org/10.1063/1.166283 (10 pages) | Cited 10 times

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A survey is given of three different control objectives that can be achieved with the use of the Open-Plus-Closed-Loop (OPCL) control method, developed by Jackson and Grosu. For a system that can be characterized by N first-order ordinary differential equations, these objectives are: (1) the asymptotic entrainment of the system’s dynamics to a prescribed “goal” dynamics, g(t); (2) an experimental-search method to determine an approximate dynamic model; (3) the transferal of the system from one attractor to any “target” attractor. For one class of systems, this may be accomplished without a model, by using only a short-duration record of the natural dynamics in the target attractor, as demonstrated experimentally using the Chua system. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.30.Hq Ordinary differential equations

Controlling chaos in a fast diode resonator using extended time-delay autosynchronization: Experimental observations and theoretical analysis

David W. Sukow, Michael E. Bleich, Daniel J. Gauthier, and Joshua E. S. Socolar

Chaos 7, 560 (1997); http://dx.doi.org/10.1063/1.166256 (17 pages) | Cited 35 times

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We stabilize unstable periodic orbits of a fast diode resonator driven at 10.1 MHz (corresponding to a drive period under 100 ns) using extended time-delay autosynchronization. Stabilization is achieved by feedback of an error signal that is proportional to the difference between the value of a state variable and an infinite series of values of the state variable delayed in time by integral multiples of the period of the orbit. The technique is easy to implement electronically and it has an all-optical counterpart that may be useful for stabilizing the dynamics of fast chaotic lasers. We show that increasing the weights given to temporally distant states enlarges the domain of control and reduces the sensitivity of the domain of control on the propagation delays in the feedback loop. We determine the average time to obtain control as a function of the feedback gain and identify the mechanisms that destabilize the system at the boundaries of the domain of control. A theoretical stability analysis of a model of the diode resonator in the presence of time-delay feedback is in good agreement with the experimental results for the size and shape of the domain of control. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
84.30.Bv Circuit theory
84.30.-r Electronic circuits

Sharp diffraction peaks from chaotic structures

Alfred Hübler, Ulrich Kuhl, Rolf Wittmann, and Takashi Nagata

Chaos 7, 577 (1997); http://dx.doi.org/10.1063/1.166257 (13 pages) | Cited 1 time

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Recently various models for spatially chaotic structures have been proposed. We study the diffraction patterns produced by plane chaotic waves incident on one-dimensional chaotic point scatterers. The spacing between the scatterers and the dynamics of the incident wave are given by a logistic map or standard map. We find a sharp diffraction peak when the incident dynamics is produced by the same map as the structure of the spatial configuration. The diffraction pattern is symmetric about the incident direction only if the map dynamics is invertible. Diffraction patterns with chaotic incident waves have a large signal-to-noise ratio and are well suited for pattern identification. We discuss possible applications to the scattering of microwaves from aperiodic structures. © 1997 American Institute of Physics.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
05.45.-a Nonlinear dynamics and chaos

Stability analysis of fixed points via chaos control

M. Löcher, G. A. Johnson, and E. R. Hunt

Chaos 7, 590 (1997); http://dx.doi.org/10.1063/1.166258 (7 pages)

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This paper reviews recent advances in the application of chaos control techniques to the stability analysis of two-dimensional dynamical systems. We demonstrate how the system’s response to one or multiple feedback controllers can be utilized to calculate the characteristic multipliers associated with an unstable periodic orbit. The experimental results, obtained for a single and two coupled diode resonators, agree well with the presented theory. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
84.30.Bv Circuit theory

Multistability and the control of complexity

Ulrike Feudel and Celso Grebogi

Chaos 7, 597 (1997); http://dx.doi.org/10.1063/1.166259 (8 pages) | Cited 28 times

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We show how multistability arises in nonlinear dynamics and discuss the properties of such a behavior. In particular, we show that most attractors are periodic in multistable systems, meaning that chaotic attractors are rare in such systems. After arguing that multistable systems have the general traits expected from a complex system, we pass to control them. Our controlling complexity ideas allow for both the stabilization and destabilization of any one of the coexisting states. The control of complexity differs from the standard control of chaos approach, an approach that makes use of the unstable periodic orbits embedded in an extended chaotic attractor. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

Maintenance of chaos in a computational model of a thermal pulse combustor

Visarath In, Mark L. Spano, Joseph D. Neff, William L. Ditto, C. Stuart Daw, K. Dean Edwards, and Ke Nguyen

Chaos 7, 605 (1997); http://dx.doi.org/10.1063/1.166260 (9 pages) | Cited 18 times

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The dynamics of a thermal pulse combustor model are examined. It is found that, as a parameter related to the fuel flow rate is varied, the combustor will undergo a transition from periodic pulsing to chaotic pulsing to a chaotic transient leading to flameout. Results from the numerical model are compared to those obtained from a laboratory-scale thermal pulse combustor. Finally the technique of maintenance (or anticontrol) of chaos is successfully applied to the model, with the result that the operation of the combustor can be continued well into the flameout regime. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
82.33.Vx Reactions in flames, combustion, and explosions
44.25.+f Natural convection

Nonlinear prediction, filtering, and control of chemical systems from time series

Valery Petrov and Kenneth Showalter

Chaos 7, 614 (1997); http://dx.doi.org/10.1063/1.166261 (7 pages) | Cited 1 time

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Prediction, filtering and control of nonlinear systems is formulated in terms of corresponding nonlinear surfaces in the phase space of delayed system readings and control parameters. The construction of these surfaces from time series and their use is demonstrated with a simple chemical model in the chaotic regime. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.50.-r Probability theory, stochastic processes, and statistics
82.30.-b Specific chemical reactions; reaction mechanisms

Adaptive strategies for recognition, noise filtering, control, synchronization and targeting of chaos

F. T. Arecchi and S. Boccaletti

Chaos 7, 621 (1997); http://dx.doi.org/10.1063/1.166262 (14 pages) | Cited 10 times

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Combining knowledge of the local variation rates with some information on the long time trends of a dynamical system, we introduce an adaptive recognition technique consisting in a sequence of variable resolution observation intervals at which the geometrical positions are sampled. The sampling times are chosen so that the sequence of observed points forms a regularized set, in the sense that the separation of adjacent points is almost uniform. We show how this adaptive technique is able to recognize the unstable periodic orbits embedded within a chaotic attractor and stabilize anyone of them even in the presence of noise, through small additive corrections to the dynamics. These techniques have been applied to the synchronization of three chaotic systems, assuring secure communication between a message sender and a message receiver; furthermore they provide robust solutions to the problems of targeting of chaos and of filtering the noise out of an experimental chaotic data set. Implementation of adaptive methods to chaotic Lorenz, three and four dimensional Roessler models and Mackey-Glass delayed system are reported.© 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.10.Ab Logic and set theory
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

Synchronizing spatiotemporal chaos

Ljupčo Kocarev, Žarko Tasev, Toni Stojanovski, and Ulrich Parlitz

Chaos 7, 635 (1997); http://dx.doi.org/10.1063/1.166263 (9 pages) | Cited 12 times

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We show analytically and numerically that a pair of uni-directionally coupled spatially extended systems can synchronize. For the case of partial differential equations the synchronization can be achieved by applying the scalar driving signals only at finite number of space points. Our approach is very general and can be useful for practical applications since the synchronization is achieved via feeding in the response system only the information from certain (discrete) spatial locations of the drive system. We also stress some open problems in the field of synchronization of spatiotemporal chaos. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.30.Jr Partial differential equations

Control and synchronization of chaos in high dimensional systems: Review of some recent results

Mingzhou Ding, E-Jiang Ding, William L. Ditto, Bruce Gluckman, Visarath In, Jian-Hua Peng, Mark L. Spano, and Weiming Yang

Chaos 7, 644 (1997); http://dx.doi.org/10.1063/1.166284 (9 pages) | Cited 18 times

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Controlling chaos and synchronization of chaos have evolved for a number of years as essentially two separate areas of research. Only recently it has been realized that both subjects share a common root in control theory. In addition, as limitations of low dimensional chaotic systems in modeling real world phenomena become increasingly apparent, investigations into the control and synchronization of high dimensional chaotic systems are beginning to attract more interest. We review some recent advances in control and synchronization of chaos in high dimensional systems. Efforts will be made to stress the common origins of the two subjects. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

On a simple recursive control algorithm automated and applied to an electrochemical experiment

M. A. Rhode, R. W. Rollins, and H. D. Dewald

Chaos 7, 653 (1997); http://dx.doi.org/10.1063/1.166264 (11 pages) | Cited 1 time

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We review a simple recursive proportional feedback (RPF) control strategy for stabilizing unstable periodic orbits found in chaotic attractors. The method is generally applicable to high-dimensional systems and stabilizes periodic orbits even if they are completely unstable, i.e., have no stable manifolds. The goal of the control scheme is the fixed point itself rather than a stable manifold and the controlled system reaches the fixed point in d+1 steps, where d is the dimension of the state space of the Poincaré map. We provide a geometrical interpretation of the control method based on an extended phase space. Controllability conditions or special symmetries that limit the possibility of using a single control parameter to control multiply unstable periodic orbits are discussed. An automated adaptive learning algorithm is described for the application of the control method to an experimental system with no previous knowledge about its dynamics. The automated control system is used to stabilize a period-one orbit in an experimental system involving electrodissolution of copper. © 1997 American Institute of Physics.
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64.75.-g Phase equilibria
05.45.-a Nonlinear dynamics and chaos
82.45.-h Electrochemistry and electrophoresis

Tracking controlled chaos: Theoretical foundations and applications

Ira B. Schwartz, Thomas W. Carr, and Ioana Triandaf

Chaos 7, 664 (1997); http://dx.doi.org/10.1063/1.166285 (16 pages) | Cited 9 times

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Tracking controlled states over a large range of accessible parameters is a process which allows for the experimental continuation of unstable states in both chaotic and non-chaotic parameter regions of interest. In algorithmic form, tracking allows experimentalists to examine many of the unstable states responsible for much of the observed nonlinear dynamic phenomena. Here we present a theoretical foundation for tracking controlled states from both dynamical systems as well as control theoretic viewpoints. The theory is constructive and shows explicitly how to track a curve of unstable states as a parameter is changed. Applications of the theory to various forms of control currently used in dynamical system experiments are discussed. Examples from both numerical and physical experiments are given to illustrate the wide range of tracking applications. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

Phase synchronization of chaotic oscillations in terms of periodic orbits

Arkady Pikovsky, Michael Zaks, Michael Rosenblum, Grigory Osipov, and Jürgen Kurths

Chaos 7, 680 (1997); http://dx.doi.org/10.1063/1.166265 (8 pages) | Cited 55 times

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We consider phase synchronization of chaotic continuous-time oscillator by periodic external force. Phase-locking regions are defined for unstable periodic cycles embedded in chaos, and synchronization is described in terms of these regions. A special flow construction is used to derive a simple discrete-time model of the phenomenon. It allows to describe quantitatively the intermittency at the transition to phase synchronization. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

Sensitive dependence on initial conditions for cellular automata

Jesús Urías, Raúl Rechtman, and Agustín Enciso

Chaos 7, 688 (1997); http://dx.doi.org/10.1063/1.166266 (6 pages) | Cited 6 times

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The property of sensitive dependence on intial conditions is the basis of a rigorous mathematical construction of local maximum Lyapunov exponents for cellular automata. The maximum Lyapunov exponent is given by the fastest average velocity of either the left or right propagating damage fronts. Deviations from the long term behavior of the finite time Lyapunov exponents due to generation of information are quantified and could be used for the characterization of the space time complexity of cellular automata. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Generalized entropies of chaotic maps and flows: A unified approach

R. Badii

Chaos 7, 694 (1997); http://dx.doi.org/10.1063/1.166267 (7 pages) | Cited 3 times

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A thermodynamic study of nonlinear dynamical systems, based on the orbits’ return times to the elements of a generating partition, is proposed. Its grand canonical nature makes it suitable for application to both maps and flows, including autonomous ones. When specialized to the evaluation of the generalized entropies Kq, this technique reproduces a well-known formula for the metric entropy K1 and clarifies the relationship between a flow and the associated Poincaré maps, beyond the straightforward case of periodically forced nonautonomous systems. Numerical estimates of the topological and metric entropy are presented for the Lorenz and Rössler systems. The analysis has been carried out exclusively by embedding scalar time series, ignoring any further knowledge about the systems, in order to illustrate its usefulness for experimental signals as well. Approximations to the generating partitions have been constructed by locating the unstable periodic orbits of the systems up to order 9. The results agree with independent estimates obtained from suitable averages of the local expansion rates along the unstable manifolds. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
05.70.Ce Thermodynamic functions and equations of state
02.50.Cw Probability theory
02.40.Pc General topology

Entropy potential and Lyapunov exponents

Stefano Lepri, Antonio Politi, and Alessandro Torcini

Chaos 7, 701 (1997); http://dx.doi.org/10.1063/1.166268 (9 pages) | Cited 1 time

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According to a previous conjecture, spatial and temporal Lyapunov exponents of chaotic extended systems can be obtained from derivatives of a suitable function, the entropy potential. The validity and the consequences of this hypothesis are explored in detail. The numerical investigation of a continuous-time model provides a further confirmation to the existence of the entropy potential. Furthermore, it is shown that the knowledge of the entropy potential allows determining also Lyapunov spectra in general reference frames where the time-like and space-like axes point along generic directions in the space-time plane. Finally, the existence of an entropy potential implies that the integrated density of positive exponents (Kolmogorov-Sinai entropy) is independent of the chosen reference frame. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
05.70.Ce Thermodynamic functions and equations of state
02.60.-x Numerical approximation and analysis

On chaotic dynamics in “pseudobilliard” Hamiltonian systems with two degrees of freedom

V. M. Eleonsky, V. G. Korolev, and N. E. Kulagin

Chaos 7, 710 (1997); http://dx.doi.org/10.1063/1.166269 (21 pages)

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A new class of Hamiltonian dynamical systems with two degrees of freedom is studied, for which the Hamiltonian function is a linear form with respect to moduli of both momenta. For different potentials such systems can be either completely integrable or behave just as normal nonintegrable Hamiltonian systems with two degrees of freedom: one observes many of the phenomena characteristic of the latter ones, such as a breakdown of invariant tori as soon as the integrability is violated; a formation of stochastic layers around destroyed separatrices; bifurcations of periodic orbits, etc. At the same time, the equations of motion are simply integrated on subsequent adjacent time intervals, as in billiard systems; i.e., all the trajectories can be calculated explicitly: Given an initial data, the state of the system is uniquely determined for any moment. This feature of systems in interest makes them very attractive models for a study of nonlinear phenomena in finite-dimensional Hamiltonian systems. A simple representative model of this class (a model with quadratic potential), whose dynamics is typical, is studied in detail. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

Waves of excitation on nonuniform membrane rings, caustics, and reverse involutes

Attila Lázár, Horst-Dieter Försterling, Henrik Farkas, Péter Simon, András Volford, and Zoltán Noszticzius

Chaos 7, 731 (1997); http://dx.doi.org/10.1063/1.166270 (7 pages) | Cited 5 times

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Chemical wave experiments on concentric nonuniform membrane rings are presented together with their theoretical description. A new technique is applied to create a slow inner and a fast outer zone in an annular membrane. An abrupt qualitative change of the wave profile was observed while decreasing the wave velocity in the inner zone. This phenomenon and all the experimental wave profiles can be adequately described by assuming that waves are involutes of a relevant caustic. A possible connection with recent models of atrial flutter is also set forth. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
41.20.Jb Electromagnetic wave propagation; radiowave propagation
82.39.Wj Ion exchange, dialysis, osmosis, electro-osmosis, membrane processes

Synchronizing Moore and Spiegel

N. J. Balmforth and R. V. Craster

Chaos 7, 738 (1997); http://dx.doi.org/10.1063/1.166271 (15 pages) | Cited 3 times

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This paper presents a study of bifurcations and synchronization {in the sense of Pecora and Carroll [Phys. Rev. Lett. 64, 821–824 (1990)]} in the Moore–Spiegel oscillator equations. Complicated patterns of period-doubling, saddle-node, and homoclinic bifurcations are found and analyzed. Synchronization is demonstrated by numerical experiment, periodic orbit expansion, and by using coordinate transformations. Synchronization via the resetting of a coordinate after a fixed interval is also successful in some cases. The Moore–Spiegel system is one of a general class of dynamical systems and synchronization is considered in this more general context. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

Fractional kinetic equations: solutions and applications

Alexander I. Saichev and George M. Zaslavsky

Chaos 7, 753 (1997); http://dx.doi.org/10.1063/1.166272 (12 pages) | Cited 88 times

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Fractional generalization of the diffusion equation includes fractional derivatives with respect to time and coordinate. It had been introduced to describe anomalous kinetics of simple dynamical systems with chaotic motion. We consider a symmetrized fractional diffusion equation with a source and find different asymptotic solutions applying a method which is similar to the method of separation of variables. The method has a clear physical interpretation presenting the solution in a form of decomposition of the process of fractal Brownian motion and Lévy-type process. Fractional generalization of the Kolmogorov–Feller equation is introduced and its solutions are analyzed. © 1997 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.60.-k Transport processes
05.20.Dd Kinetic theory

Ring wave solutions of a n+1-dimensional Sine–Gordon model

A. Di Garbo, L. Fronzoni, and S. Chillemi

Chaos 7, 765 (1997); http://dx.doi.org/10.1063/1.166273 (4 pages)

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The dynamical properties of the ring wave solutions of the model ψtt−∇n2ψ+sin ψ+ϵ sin (ψ/2)+αψt = 0 (0 ⩽ ϵ≪1,0 ⩽ α≪1) are studied analytically and numerically for spatial dimension (n = 2,3). The model is obtained as a continuum approximation of a multidimensional Frenkel–Kontorowa lattice. We will show that in the case ϵ>0, α = 0 (or α>0) the return effect of the ring wave does not occur only for well defined values of ϵ. It will be shown numerically that the dissipative perturbation αψt (α>0) stabilizes both the velocity and the wave profile of the ring wave when the return effect does not occur. © 1997 American Institute of Physics. 
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11.10.Lm Nonlinear or nonlocal theories and models
05.45.-a Nonlinear dynamics and chaos
02.60.-x Numerical approximation and analysis
11.10.Cd Axiomatic approach
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