A central problem in nonlinear dynamics is that of discovering how the qualitative dynamical properties of orbits change and evolve as a dynamical system is continuously changed. More specifically, consider a dynamical system which depends on a single scalar parameter. We ask, what happens to the orbits of the system if we examine them at different values of the parameter? We have already met this question and substantially answered it for the case of the logistic map, xn=1 = rxn(1 − xn). In particular, we found in Chapter 2 that as the parameter r is increased there is a period doubling cascade, terminating in an accumulation of an infinite number of period doublings, followed by a parameter domain in which chaos and periodic ‘windows’ are finely intermixed. Another example of a context in which we have addressed this question is our discussion in Chapter 6 of Arnold tongues and the transition from quasiperiodicity to chaos. Still another aspect of this question is the types of generic bifurcations of periodic orbits which can occur as a parameter is varied. In this regard recall our discussions of the generic bifurcations of periodic orbits of one-dimensional maps (Section 2.3) and of the Hopf bifurcation (Chapter 6).

In this chapter we shall be interested in transitions of the system behavior with variation of a parameter such that the transitions involve chaotic orbits.

Hamiltonian systems are a class of dynamical systems that occur in a wide variety of circumstances. The special properties of Hamilton's equations endow these systems with attributes that differ qualitatively and fundamentally from other sytems. (For example, Hamilton's equations do not possess attractors.)

Examples of Hamiltonian dynamics include not only the well-known case of mechanical systems in the absence of friction, but also a variety of other problems such as the paths followed by magnetic field lines in a plasma, the mixing of fluids, and the ray equations describing the trajectories of propagating waves. In all of these situations chaos can be an important issue. Furthermore, chaos in Hamiltonian systems is at the heart of such fundamental questions as the foundations of statistical mechanics and the stability of the solar system. In addition, Hamiltonian mechanics and its structure are reflected in quantum mechanics. Thus, in Chapter 11 we shall treat the connection between chaos in Hamiltonian systems and related quantum phenomena. The present chapter will be devoted to a discussion of Hamiltonian dynamics and the role that chaos plays in these systems. We begin by presenting a summary of some basic concepts in Hamiltonian mechanics.

Hamiltonian systems

The dynamics of a Hamiltonian system is completely specified by a single function, the Hamiltonian, H(p, q, t). The state of the system is specified by its ‘momentum’ p and ‘position’ q.

This second edition updates and expands the first edition. The most important change is a new chapter on control and synchronization of chaos (Chapter 10). Further additions have been made throughout the book, including new material on riddled basins of attraction, phase locking of globally coupled oscillators, fractal aspects of fluid advection by Lagrangian chaotic flows, magnetic dynamos and strange nonchaotic attractors. Also, twenty-eight new homework problems for students have been added.

The description of physical systems via classical mechanics as embodied by Hamilton's equations (Chapter 7) may be viewed as an approximation to the more exact description of quantum mechanics. Depending on the relevant time, length and energy scales appropriate to a given situation, one or the other of these descriptions may be the one that is most efficacious. In particular, if the typical wavelength in the quantum problem is very small compared to all length scales of the system, then one suspects that the classical description should be good. There is a region of crossover from the quantum regime to the classical regime where the wavelengths are ‘small’ but not extremely small. This crossover region is called the ‘semiclassical’ regime. In the semiclassical regime, we may expect quantum effects to be important, and we may also expect that the classical description is relevant as well. According to the ‘correspondence principle,’ quantum mechanics must go over into classical mechanics in the ‘classical limit,’ which is defined by letting the quantum wavelength approach zero. In a formal mathematical sense we can equivalently take the ‘classical limit’ by letting Planck's constant approach zero, ħ → 0, with other parameters of the system held fixed. This limit is quite singular, and its properties are revealed by an investigation of the semiclassical regime. Particular interest attaches to the case where the classical description yields chaotic dynamics.

In the preceding chapters we have been mainly concerned with studying the properties of chaotic dynamical systems. In this chapter we adopt a different, more active, point of view. In particular, we ask, can we use our knowledge of chaotic systems to achieve some desired goal? Two general areas where this point of view has proven useful are the control of chaos and the synchronization of chaotic systems. By control we shall generally mean feedback control. That is, we have some control variable that we can vary as a function of time, and we decide how to do this variation on the basis of knowledge (perhaps limited) of the system's past history and/or current state. In the synchronization of chaos, we generally shall be considering two (or more) systems that are coupled. The evolution is chaotic, and we are interested in the conditions such that the component systems execute the same motion. Both control of chaos and synchronization of chaos have potential practical applications, and we shall indicate these as this chapter proceeds.

Control of chaos

Two complementary attributes sometimes used to define chaos are (i) exponentially sensitive dependence and (ii) complex orbit structure. A quantifier for attribute (i) is the largest Lyapunov exponent, while a quantifier for attribute (ii) is the entropy (e.g., the metric entropy or the topological entropy, Section 4.5). These attributes of chaos can be exploited to fashion chaos-control strategies.

Although chaotic dynamics had been known to exist for a long time, its importance for a broad variety of applications began to be widely appreciated only within the last decade or so. Concurrently, there has been enormous interest both within the mathematical community and among engineers and scientists. The field continues to develop rapidly in many directions, and its implications continue to grow. Naturally, such a situation calls for textbooks to serve the need of providing courses to students who will eventually utilize concepts of chaotic dynamics in their future careers. A variety of chaos texts now exists. In my teaching of several courses on chaos, however, I found that the existing texts were not altogether suitable for the type of course I was giving, with respect to both level and coverage of topics. Hence I was motivated to prepare and circulate notes for my class, and these notes led to this book. The book is intended for use in a graduate course for scientists and engineers. Accordingly, any mathematical concepts that such readers may not be familiar with (e.g., measure, Cantor sets, etc.) are introduced and informally explained as needed. While the intended readers are not mathematicians, there is a greater emphasis on basic mathematical concepts than in most other books that address the same audience. The style is pedagogical, and it is hoped that the very interesting, sometimes difficult, concepts that are the backbone for studies of chaos are made clear.

Perhaps the most basic aspect of a set is its dimension. In Figures 1.10(a) and (b) we have given two examples of attractors; one is a steady state of a flow represented by a single point in the phase space, while the other is a limit cycle, represented by a simple closed curve. While it is clear what the dimensions of these attracting sets are (zero for the point and one for the curve), it is also the case that invariant sets arising in dynamical systems (such as chaotic attractors) often have structure on arbitrarily fine scale, and the determination of the dimension of such sets is nontrivial. Also the frequency with which orbits visit different regions of a chaotic attractor can have its own arbitrarily fine scaled structure. In such cases the assignment of a dimension value gives a much needed quantitative characterization of the geometrical structure of a complicated object. Furthermore, experimental determination of a dimension value from data for an experimental dynamical process can provide information on the dimensionality of the phase space required of a mathematical dynamical system used to model the observations. These issues are the subjects of this chapter.

The box-counting dimension

The box-counting dimension (also called the ‘capacity’ of the set) provides a relatively simple and appealing way of assigning a dimension to a set in such a way that certain kinds of sets are assigned a dimension which is not an integer.

In previous chapters we considered synchronization in purely deterministic systems, neglecting all irregularities and fluctuations. Here we discuss how the latter effects can be incorporated in the picture of phase locking. We start with a discussion of the effect of noise on autonomous self-sustained oscillations. We show that noise causes phase diffusion, thus spoiling perfect time-periodicity. Next, we consider synchronization by an external periodic force in the presence of noise. Finally, we discuss mutual synchronization of two noisy oscillators.

Self-sustained oscillator in the presence of noise

No oscillator is perfectly periodic: all clocks have to be adjusted from time to time, some even rather often. There are many factors causing irregularity of self-sustained oscillators, for simplicity we will call them all noise. A detailed analysis of noisy oscillators must include a thorough mathematical description of the problem, where fluctuations of different nature (e.g., technical, thermal) should be taken into account. This has been done for different types of oscillators (see, e.g., [Malakhov 1968]); here we want to discuss the basic phenomena only.

As the first model, we consider a self-sustained oscillator subject to a noisy external force. In revising the basic equations of forced oscillators of Chapter 7, one can see that only the approximation of phase dynamics is valid in the case of a fluctuating force as well, since we do not assume any regularity of the force in the derivation of Eq. (7.15).

In this chapter we consider the effects of synchronization due to the interaction of two oscillating systems. This situation is intermediate between that of Chapter 7, where one oscillator was subject to a periodic external force, and the case of many interacting oscillators discussed later in Chapters 11 and 12. Indeed, the case of periodic forcing can be considered as a special case of two interacting oscillators when the coupling is unidirectional. However, two oscillators form an elementary building block for the case of many (more than two) mutually coupled systems. The problem can be formulated as follows: there are two nonlinear systems each exhibiting self-sustained periodic oscillations, generally with different amplitudes and frequencies. These two systems can interact, and the strength of the interaction is the main parameter. We are interested in the dynamics of the coupled system, with the main emphasis on the entrainment of phases and frequencies.

In Section 8.1 we develop a phase dynamics approach that is valid if the coupling is small – the problem here reduces to coupled equations for the phases only. Another approximation is used in Section 8.2, where the dynamics of weakly nonlinear oscillations are discussed. Finally, in Section 8.3 we describe synchronization of relaxation “integrate-and-fire” oscillators. No special attention is devoted to coupled rotators: their properties are very similar to those of oscillators.

Phase dynamics

If the coupling between two self-oscillating systems is small, one can derive, following Malkin [1956] and Kuramoto [1984], closed equations for the phases. The approach here is essentially the same as in Section 7.1. It is advisable to read that section first: below we use many ideas introduced there.

The effect of mutual synchronization of two coupled oscillators, described in Chapter 8, can be generalized to more complex situations. One way to do this was described in Chapter 11, where we considered lattices of oscillators with nearest-neighbor coupling. However, often oscillators do not form a regular lattice, and, moreover, interact not only with neighbors but also with many other oscillators. Studies of three and four oscillators with all-to-all coupling give a rather complex and inexhaustible picture (see, e.g., [Tass and Haken 1996; Tass 1997]). The situation becomes simpler if the interaction is homogeneous, i.e., all the pairs of oscillators are equally coupled. Moreover, if the number of interacting oscillators is very large, one can consider the thermodynamic limit where the number of elements in the ensemble tends to infinity. Now, when the oscillators are not ordered in space, one usually speaks of an ensemble or a population of coupled oscillators. An analogy to statistical mechanics is extensively exploited, so it is not surprising that the synchronization transition appears as a nonequilibrium phase transition in the ensemble.

We start with the description of the Kuramoto transition in a population of phase oscillators. An ensemble of noisy oscillators is then discussed. Different generalizations of these basic models (e.g., populations of chaotic oscillators) are described in Section 12.3.

The word “synchronous” is often encountered in both scientific and everyday language. Originating from the Greek words χρόνος (chronos, meaning time) and σύν (syn, meaning the same, common), in a direct translation “synchronous” means “sharing the common time”, “occurring in the same time”. This term, as well as the related words “synchronization” and “synchronized”, refers to a variety of phenomena in almost all branches of natural sciences, engineering and social life, phenomena that appear to be rather different but nevertheless often obey universal laws.

A search in any scientific data base for publication titles containing the words with the root “synchro” produces many hundreds (if not thousands) of entries. Initially, this effect was found and investigated in different man-made devices, from pendulum clocks to musical instruments, electronic generators, electric power systems, and lasers. It has found numerous practical applications in electrical and mechanical engineering. Nowadays the “center of gravity” of the research has moved towards biological systems, where synchronization is encountered on different levels. Synchronous variation of cell nuclei, synchronous firing of neurons, adjustment of heart rate with respiration and/or locomotory rhythms, different forms of cooperative behavior of insects, animals and even humans – these are only some examples of the fundamental natural phenomenon that is the subject of this book.

In this chapter we describe synchronization effects in chaotic systems. We start with a brief description of chaotic oscillations in dissipative dynamical systems, emphasizing the properties that are important for the onset of synchronization. Next, we describe different types of synchronization: phase, complete, generalized, etc. In studies of these phenomena computers are widely used, therefore in our illustrations we often use the results of computer simulations, but also show some real experimental data. Whenever possible, we try to underline a similarity to synchronization of periodic oscillations.

Chaotic oscillators

One of the most important achievements of nonlinear dynamics within the last few decades was the discovery of complex, chaotic motion in rather simple oscillators. Now this phenomenon is well-studied and is a subject of undergraduate and high-school courses; nevertheless some introductory presentation is pertinent. The term “chaotic” means that the long-term behavior of a dynamical system cannot be predicted even if there were no natural fluctuations of the system's parameters or influence of a noisy environment. Irregularity and unpredictability result from the internal deterministic dynamics of the system, however contradictory this may sound. If we describe the oscillation of dissipative, self-sustained chaotic systems in the phase space, then we find that it does not correspond to such simple geometrical objects like a limit cycle any more, but rather to complex structures that are called strange attractors (in contrast to limit cycles that are simple attractors).

In this chapter we describe synchronization of periodic oscillators by a periodic external force. The main effect here is complete locking of the oscillation phase to that of the force, so that the observed oscillation frequency coincides exactly with the frequency of the forcing.

We start our consideration with the case of small forcing. In Section 7.1 we use a perturbation technique based on the phase dynamics approximation. This approach leads to a simple phase equation that can be treated analytically. This equation is, however, nonuniversal, as its form depends on the particular features of the oscillator. Another analytic approach is presented in Section 7.2; here we assume not only that the force is small, but also that the periodic oscillations are weakly nonlinear. This enables us to use a method of averaging and to obtain universal equations depending on a few parameters. Historically, this is the first analytical approach to synchronization going back to the works of Appleton [1922], van der Pol [1927] and Andronov and Vitt [1930a,b]. The averaged equations can be analyzed in full detail, but their applicability is limited: in fact, quantitative predictions are possible only for small-amplitude self-sustained oscillations near the Hopf bifurcation point of their appearance.

Generally, when the forcing is not small and/or the oscillations are strongly nonlinear, we have to rely on the qualitative theory of dynamical systems. The tools used here are the annulus and the circle maps described in Section 7.3. This approach gives a general description up to the transition to chaos, it allows one to find limits of the analytical methods and provides a framework for numerical investigations of particular systems.