Elsevier

Physics Reports

Volume 329, Issue 3, May 2000, Pages 103-197
Physics Reports

The control of chaos: theory and applications

https://doi.org/10.1016/S0370-1573(99)00096-4Get rights and content

Abstract

Control of chaos refers to a process wherein a tiny perturbation is applied to a chaotic system, in order to realize a desirable (chaotic, periodic, or stationary) behavior. We review the major ideas involved in the control of chaos, and present in detail two methods: the Ott–Grebogi–Yorke (OGY) method and the adaptive method. We also discuss a series of relevant issues connected with chaos control, such as the targeting problem, i.e., how to bring a trajectory to a small neighborhood of a desired location in the chaotic attractor in both low and high dimensions, and point out applications for controlling fractal basin boundaries. In short, we describe procedures for stabilizing desired chaotic orbits embedded in a chaotic attractor and discuss the issues of communicating with chaos by controlling symbolic sequences and of synchronizing chaotic systems. Finally, we give a review of relevant experimental applications of these ideas and techniques.

Introduction

A deterministic system is said to be chaotic whenever its evolution sensitively depends on the initial conditions. This property implies that two trajectories emerging from two different closeby initial conditions separate exponentially in the course of time. The necessary requirements for a deterministic system to be chaotic are that the system must be nonlinear, and be at least three dimensional.

The fact that some dynamical model systems showing the above necessary conditions possess such a critical dependence on the initial conditions was known since the end of the last century. However, only in the last thirty years, experimental observations have pointed out that, in fact, chaotic systems are common in nature. They can be found, for example, in Chemistry (Belouzov–Zhabotinski reaction), in Nonlinear Optics (lasers), in Electronics (Chua–Matsumoto circuit), in Fluid Dynamics (Rayleigh–Bénard convection), etc. Many natural phenomena can also be characterized as being chaotic. They can be found in meteorology, solar system, heart and brain of living organisms and so on.

Due to their critical dependence on the initial conditions, and due to the fact that, in general, experimental initial conditions are never known perfectly, these systems are instrinsically unpredictable. Indeed, the prediction trajectory emerging from a bonafide initial condition and the real trajectory emerging from the real initial condition diverge exponentially in course of time, so that the error in the prediction (the distance between prediction and real trajectories) grows exponentially in time, until making the system's real trajectory completely different from the predicted one at long times.

For many years, this feature made chaos undesirable, and most experimentalists considered such characteristic as something to be strongly avoided. Besides their critical sensitivity to initial conditions, chaotic systems exhibit two other important properties. Firstly, there is an infinite number of unstable periodic orbits embedded in the underlying chaotic set. In other words, the skeleton of a chaotic attractor is a collection of an infinite number of periodic orbits, each one being unstable. Secondly, the dynamics in the chaotic attractor is ergodic, which implies that during its temporal evolution the system ergodically visits small neighborhood of every point in each one of the unstable periodic orbits embedded within the chaotic attractor.

A relevant consequence of these properties is that a chaotic dynamics can be seen as shadowing some periodic behavior at a given time, and erratically jumping from one to another periodic orbit. The idea of controlling chaos is then when a trajectory approaches ergodically a desired periodic orbit embedded in the attractor, one applies small perturbations to stabilize such an orbit. If one switches on the stabilizing perturbations, the trajectory moves to the neighborhood of the desired periodic orbit that can now be stabilized. This fact has suggested the idea that the critical sensitivity of a chaotic system to changes (perturbations) in its initial conditions may be, in fact, very desirable in practical experimental situations. Indeed, if it is true that a small perturbation can give rise to a very large response in the course of time, it is also true that a judicious choice of such a perturbation can direct the trajectory to wherever one wants in the attractor, and to produce a series of desired dynamical states. This is exactly the idea of targeting.

The important point here is that, because of chaos, one is able to produce an infinite number of desired dynamical behaviors (either periodic and not periodic) using the same chaotic system, with the only help of tiny perturbations chosen properly. We stress that this is not the case for a nonchaotic dynamics, wherein the perturbations to be done for producing a desired behavior must, in general, be of the same order of magnitude as the unperturbed evolution of the dynamical variables.

The idea of chaos control was enunciated at the beginning of this decade at the University of Maryland [1]. In Ref. [1], the ideas for controlling chaos were outlined and a method for stabilizing an unstable periodic orbit was suggested, as a proof of principle. The main idea consisted in waiting for a natural passage of the chaotic orbit close to the desired periodic behavior, and then applying a small judiciously chosen perturbation, in order to stabilize such periodic dynamics (which would be, in fact, unstable for the unperturbed system). Through this mechanism, one can use a given laboratory system for producing an infinite number of different periodic behavior (the infinite number of its unstable periodic orbits), with a great flexibility in switching from one to another behavior. Much more, by constructing appropriate goal dynamics, compatible with the chaotic attractor, an operator may apply small perturbations to produce any kind of desired dynamics, even not periodic, with practical application in the coding process of signals.

It is reasonable to assume that one does not have complete knowledge about the system dynamics since our system is typically complicated and has experimental imperfections. It is better, then, to work in the space of solutions since the equations, even if available, are not too useful due to the sensitivity of the dynamics to perturbations. One gets solutions by obtaining a time series of one dynamically relevant variable. The right perturbation, therefore, to be applied to the system is selected after a learning time, wherein the dependence of the dynamics on some external control is tested experimentally. Such perturbation can affect either a control parameter of the system, or a state variable. In the former case, a perturbation on some available control parameter is applied, in the latter case a feedback loop is designed on some state variable of the system.

The first example of the former case is reported in Ref. [1]. Let us draw the attention on a chaotic dynamics developing onto an attractor in a D-dimensional phase space. One can construct a section of the dynamics such that it is perpendicular to the chaotic flow (it is called Poincaré section). This (D−1)-dimensional section retains all the relevant information of the dynamics, which now is seen as a mapping from the present to the next intersection of the flow with the Poincare’ section. Any periodic behavior is seen here as a periodic cycling among a discrete number of points (the number of points determines the periodicity of the periodic orbit). Since all periodic orbits in the unperturbed dynamics are unstable, also the periodic cycling in the map will be unstable. Furthermore, since, by ergodicity, the chaotic flow visits closely all the unstable periodic orbits, this implies that also the mapping in the section will visit closely all possible cycles of points corresponding to a periodic behavior of the system. Let us then consider a given periodic cycle of the map, such as period one. A period one cycle corresponds to a single point in the Poincaré section, which repeats itself indefinitely. Now, because of the instability of the corresponding orbit, this point in fact possesses a stable manifold and an unstable manifold. For stable (unstable) manifold we mean the collection of directions in phase space through which the trajectory approaches (diverges away from) the point geometrically. The control of chaos idea consists in perturbing a control parameter when the natural trajectory is in a small neighborhood of the desired point, such that the next intersection with the Poincaré section puts the trajectory on the stable manifold. In this case, all divergences are cured, and the successive natural evolution of the dynamics, except for nonlinearities and noise, converges to the desired point (that is, it stabilizes the desired periodic behavior). Selection of the perturbation is done by means of a reconstruction from experimental data of the local linear properties of the dynamics around the desired point.

In some practical situations, however, it may be desirable to perform perturbations on a state variable accessible to the operator. This suggests the development of some alternative approaches. The first was introduced in Ref. [2]. It consists in designing a proper feedback line through which a state variable is directly perturbed such as to control a periodic orbit. This second method requires the availability of a state variable for experimental observation and for the perturbations. In such a case, a negative feedback line can be designed which is proportional to the difference between the actual value of the state variable, and the value delayed of a time lag T. The idea is that, when T coincides with the period of one unstable periodic orbit of the unperturbed system, the negative feedback pushes to zero the difference between the present and the delayed dynamics, and the periodic orbit is stabilized. Furthermore, as soon as the control becomes effective, this difference goes effectively to zero, so that the feedback perturbation vanishes. Moreover, as before, a preliminary learning time is needed, for learning the periods of the unstable periodic orbits. In the above mechanism, the proportionality constant entering in the feedback loop is given in Ref. [3] where an adaptive technique has been introduced which automatically selects this constant by adaptively exploiting the local dynamics of the system.

Many other techniques have been introduced with the aim of establishing control over chaos that will be referred to and described along this Report. Among the many available reviews, books, and monographies on this matter, here we address the reader the most recent ones, contained in [4], [5], [6], [7], [8]. In face of this huge number of theoretical studies, experimental realizations of chaos control have been achieved with a magnetoelastic ribbon [9], a heart [10], [11], a thermal convection loop [12], [13], a yttrium iron garnet oscillator [14], a diode oscillator [15], an optical multimode chaotic solid-state laser [16], a Belousov–Zhabotinski reaction diffusion chemical system [17], and many other experiments.

While control of chaos has been successfully demonstrated experimentally in many situations, the control of patterns in space-extended systems is still an open question. This is the reason why most of the interest has moved actually from the control of periodic behaviors in concentrated systems, to the control of periodic patterns in space-extended systems, with the aim of controlling infinite dimensional chaos, or even space–time chaos. The applications would be enormous, ranging from the control of turbulent flows, to the parallel signal transmission and computation to the parallel coding-decoding procedure, to the control of cardiac fibrillation, and so forth.

One of the major problems in the above process is that one can switch on the control only when the system is sufficiently close to the desired behavior. This is warranted by the ergodicity of chaos regardless of the initial condition chosen for the chaotic evolution, but it may happen that the small neighborhood of a given attractor point (target) may be visited only infrequently, because of the locally small probability function. Thus the unperturbed dynamics may take a long time to approach a given target, resulting in an unacceptably large waiting time for the operator to apply the control of chaos process.

Efficient targeting methods can, instead, reduce the waiting time by orders of magnitude, and so they can be seen as a preliminary task for chaos control, independent of the particular control algorithm that one applies. In this Report, we devote Section 4 to the problem of targeting of chaos, since it is crucial for the realization of the control procedure, and summarize the different proposed methods for directing chaotic trajectories to target points in the attractor.

Another section of this Report is devoted to the problem of the control of desired chaotic behaviors and its major applications. The critical sensitivity to initial conditions of a chaotic system can, indeed, be exploited not only to produce a large number of possible periodic behaviors (the different unstable periodic orbit), but, much more generally, any desired behavior compatible with the natural evolution of the system. Therefore, one can imagine to select suitable perturbations to slave the chaotic system toward a particular “desired” chaotic behavior.

Among the practically unlimited possible applications of the control of chaotic behavior, herewith we concentrate on two applications, which have attracted considerable attention in the scientific community over the past few years; namely the control of chaotic behavior for communicating with chaos and for the synchronization of chaotic systems for various communication schemes. In the first case, a chaotic system is conveniently perturbed, in order to give rise to a particular chaotic trajectory carrying a given message. In the second case, the process of chaos synchronization is applied to a communication line between a message sender and a message receiver, allowing the synchronization between them.

There is a simple connection between chaos and communication theory. Chaotic systems can be viewed, indeed, as information sources that naturally produce digital communication signals. The formal connection between chaotic dynamics and information theory began with the introduction of the concept of measure-theoretic entropy in ergodic theory [18], [19], [20]. Chaotic systems are, indeed, characterized by having positive entropies and thus they are information sources. By assigning a discrete alphabet to the system state space using the formalism of symbolic dynamics, the chaotic system becomes a symbol source, and because it is a continuous-time waveform source, it is also a digital signal source. A chaotic system is, therefore, a natural source of digital communication signals. This concept has been recently shown to be more than formal [24]. Controlling the output of an oscillator via small guiding current pulses allows for the transmission of a desired message without effectively altering the time-evolution equations for the system. As an example, a very simple chaotic electrical oscillator can produce a seemingly random sequence of positive and negative (bipolar) voltage peaks [21]. If these bipolar peaks are assigned binary symbols 0 and 1, respectively, then the signal can be viewed as a binary communication waveform. We can furthermore encode any desired message into the waveform by using small perturbing pulses to control the sequence of peaks representing the symbols 0 and 1. More sophisticated waveforms and encodings are possible, but this example suffices to convey the basic concept. In this Report, we also summarize the most relevant achievements in communicating with chaos, and we suggest some problems still unsolved.

The control of chaotic behavior has another important application, namely, the synchronization of chaotic systems. If one consider two identical chaotic systems starting from different initial conditions, then the critical sensitivity to initial conditions implies that their difference grows exponentially in time, and that they will evolve in an unsynchronized manner. The feeding of the right signal from one system to another can, however, reduce to zero such difference, and push the two systems into a synchronized manifold, wherein the chaotic motion is now developed so as the system are in step during the course of time. This proposal was intensively pushed forward at the beginning of this decade [22]. In the present Report, we simply summarize a possible application of synchronization of chaos, consisting in making secure the transmission of a signal between a message sender and a receiver along a communication line.

Finally, we devote a section to summarize the most relevant experimental applications of the above ideas and techniques. Since it would be unrealistic to cover the whole body of experimental implementations of chaos control, herewith we limit ourselves to focus on few prototypical experiments, and we suggest to the interested reader to the most relevant literature.

The OGY ideas found experimental applications in several different fields, such as mechanical oscillations (magnetoelastic ribbon), electronic circuits (diode resonator), chemical systems (Belouzov–Zhabotinski reaction), nonlinear optics (multimode laser). Different control techniques were also experimentally tested on fluid dynamical systems leading to the control of convective instabilities, and on biomechanical systems for the control of the cardiac activity in a rabbit heart, and of the neuronal activity of an hippocampal slice. In every experimental example, we point out the relevance of the achievements, the difficulties for the practical realization of the theoretical proposals, and the perspective opened by such implementations.

The present Report is organized as follows.

In Section 2, the OGY method is illustrated with applications to one-dimensional and two-dimensional mappings. The pole placement technique is then discussed for the control of higher-dimensional situations. In Section 3, we discuss alternative schemes for chaos control, and we describe in detail the adaptive strategy with application to delayed dynamical systems, since it constitutes a bridge between concentrated and spatially extended systems. Section 4 is devoted to the discussion of the targeting problem. We show how the OGY criterion and adaptivity can provide suitable tools for directing the chaotic trajectories to desired targets. Furthermore, we show a possible application for the control of fractal basin boundaries. In Section 5, we discuss the issue of stabilizing desirable chaotic trajectories, and we point out two main applications: the communication with chaos, and communication through chaos synchronization. Section 6 summarizes the main experimental work in chaos control, and points out the perspective open in different fields by this process.

Section snippets

The basic idea

Besides the occurrence of chaos in a large variety of natural processes, chaos may also occur because one may wish to design a physical, biological or chemical experiment, or to project an industrial plant to behave in a chaotic manner. The idea of Ott, Grebogi, and Yorke (OGY) is that chaos may indeed be desirable since it can be controlled by using small perturbation to some accessible parameter [1], [23] or to some dynamical variable of the system [24].

The major key ingredient for the

The basic idea

Many alternative approaches to the OGY method have been proposed for the stabilization of the unstable periodic orbits (UPO) [47] of a chaotic dynamics. In general the strategies for the control of chaos can be classified into two main classes, namely: closed loop or feedback methods and open loop or non feedback methods.

The first class includes those methods which select the perturbation based upon a knowledge of the state of the system, and oriented to control a prescribed dynamics. Among

Overview

We consider the following situation: suppose there is a nonlinear dynamical system whose trajectories lie on a chaotic attractor. Suppose further that one of the uncountably infinite number of chaotic orbits embedded in the chaotic attractor corresponds to a desirable operational state of the system. Our goal is to apply only small feedback control to keep trajectories originating from random initial conditions in the vicinity of the desirable chaotic orbit. In what follows, we present a

Acknowledgements

The authors are grateful to F.T. Arecchi, E. Barreto, G. Basti, E. Bollt, A. Farini, R. Genesio, A. Giaquinta, S. Hayes, E. Kostelich, A.L. Perrone, F. Romeiras and T. Tél for many fruitful discussions. SB acknowledges financial support from the EEC Contract no. ERBFMBICT983466. CG was supported by DOE and by a joint Brasil-USA grant (CNPq/NSF-INT). YCL was supported by AFOSR under Grant No. F49620-98-1-0400 and by NSF under Grant No. PHY-9722156. HM and DM acknowledge financial support from

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