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.

In this chapter we specify in more detail the notion of self-sustained oscillators that was sketched in the Introduction. We argue that such systems are ubiquitous in nature and engineering, and introduce their universal description in state space and their universal image – the limit cycle. Next, we discuss the notion and the properties of the phase, the variable that is of primary importance in the context of synchronization phenomena. Finally, we analyze several simple examples of self-sustained systems, as well as counter-examples. In this way we shall illustrate the features that make self-sustained oscillators distinct from forced and conservative systems; in the following chapters we will show that exactly these features allow synchronization to occur. Our presentation is not a systematic and complete introduction to the theory of self-sustained oscillation: we dwell only on the main aspects that are important for understanding synchronization phenomena.

The notion of self-sustained oscillators was introduced by Andronov and Vitt [Andronov et al. 1937]. Although Rayleigh had already distinguished between maintained and forced oscillations, and H. Poincare had introduced the notion of the limit cycle, it was Andronov and Vitt and their disciples who combined rigorous mathematical methods with physical ideas. Self-sustained oscillators are a subset of the wider class of dynamical systems. The latter notion implies that we are dealing with a deterministic motion, i.e., if we know the state of a system at a certain instant in time then we can unambiguously determine its state in the future.

This chapter is devoted to the simplest case of synchronization: entrainment of a self-sustained oscillator by an external force. We can consider this situation as a particular case of the experiments with interacting pendulum clocks described in Chapter 1. Let us assume that the coupling between clocks is unidirectional, so that only one clock influences another, and that is exactly the case of external forcing we are going to study in detail here. This is not only a simplification that makes the presentation easier; there are many real world effects that can be understood in this way. Probably, the best example in this context is not the pendulum clock, but a very modern device, the radio-controlled watch. Its stroke is corrected from time to time by a very precise clock, and therefore the watch is also able to keep its rhythm with very high precision. We have also previously mentioned examples of a living nature: the biological clocks that govern the circadian rhythms of cells and the organisms controlled by the periodic rhythms originating from the rotation of the Earth around its axis and around the Sun. Definitely, such actions are also unidirectional. We consider several other examples below.

We start by considering a harmonically driven quasilinear oscillator. With this example we explain entrainment by an external force and discuss in detail what happens to the phase and frequency of the driven system at the synchronization transition. Next, we introduce the technique of stroboscopic observation; in the following we use it to study synchronization of strongly nonlinear systems, entrainment of an oscillator by a pulse train, as well as synchronization of order n : m.

In this chapter we describe synchronization by external forces; we shall discuss effects other than those presented in Chapters 7 and 10. The content is not homogeneous: different types of systems and different types of forces are discussed here. Nevertheless, it is possible to state a common property of all situations: synchronization occurs when the driven system loses its own dynamics and follows those of the external force. In other words, the dynamics of the driven system are synchronized if they are stable with respect to internal perturbations. Quantitatively this is measured by the largest Lyapunov exponent: a negative largest exponent results in synchronization (note that here we are speaking not about a transverse or conditional Lyapunov exponent, but about the “canonical” Lyapunov exponent of a dynamical system). This general rule does not depend on the type of forcing or on the type of system; nevertheless, there are some problem-specific features. Therefore, we consider in the following sections the cases of periodic, noisy, and chaotic forcing separately. In passing, it is interesting to note that we can also interpret phase locking of periodic oscillations by periodic forcing (Chapter 7) as stabilization of the dynamics: a nonsynchronized motion (unforced, or outside the synchronization region) has a zero largest Lyapunov exponent, while in the phase locked state it is negative.

Another important concept we present in this chapter is sensitivity to the perturbation of the forcing. In contrast to sensitivity to initial conditions, which is measured via the Lyapunov exponent, sensitivity to forcing has no universal quantitative characteristics.

Synchronization in historical perspective

The Dutch researcher Christiaan Huygens (Fig. 1.1), most famous for his studies in optics and the construction of telescopes and clocks, was probably the first scientist who observed and described the synchronization phenomenon as early as in the seventeenth century. He discovered that a couple of pendulum clocks hanging from a common support had synchronized, i.e., their oscillations coincided perfectly and the pendula moved always in opposite directions. This discovery was made during a sea trial of clocks intended for the determination of longitude. In fact, the invention and design of pendulum clocks was one of Huygens’ most important achievements. It made a great impact on the technological and scientific developments of that time and increased the accuracy of time measurements enormously. In 1658, only two years after Huygens obtained a Dutch Patent for his invention, a clock-maker from Utrecht, Samuel Coster, built a church pendulum clock and guaranteed its weekly deviation to be less than eight minutes.

After this invention, Huygens continued his efforts to increase the precision and stability of such clocks. He paid special attention to the construction of clocks suitable for use on ships in the open sea. In his memoirs Horologium Oscillatorium (The Pendulum Clock, or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks), where he summarized his theoretical and experimental achievements, Huygens [1673] gave a detailed description of such clocks.

In Chapter 3 we studied in detail synchronization of an oscillator by an external force. Here we extend these ideas to more complicated situations when two or several oscillators are interrelated.

We start with two mutually coupled oscillators. This case covers the classical experiments of Huygens, Rayleigh and Appleton, as well as many other experiments and natural phenomena. We describe frequency and phase locking effects in these interacting systems, as well as in the presence of noise. Further, we illustrate some particular features of synchronization of relaxation oscillators and briefly discuss the case when several oscillators interact. Here we also discuss synchronization properties of a special class of systems, namely rotators.

This chapter also covers synchronization phenomena in large ordered ensembles of systems (chains and lattices), as well as in continuous oscillatory media. An interesting effect in these systems is the formation of synchronous clusters.

We proceed with a description and qualitative explanation of self-synchronization in large populations of all-to-all (globally) coupled oscillators. An example of this phenomenon – synchronous flashing in a population of fireflies – was described in Chapter 1; further examples are presented in this chapter. We conclude this chapter by presenting diverse examples.

Mutual synchronization of self-sustained oscillators

In this section we discuss synchronization of mutually coupled oscillators. This effect is quite similar to the case of external forcing that we described in detail in Chapter 3. Nevertheless, there are some specific features, and we consider them below. We also briefly mention the case when several oscillators interact.

The goal of this chapter is to describe the basic properties of the complete synchronization of chaotic systems. Our approach is the following: we take a system as simple as possible and describe it in as detailed a way as possible. The simplest chaotic system is a one-dimensional map, it will be our example here. We start with the construction of the coupled map model, and describe phenomenologically, what complete synchronization looks like. The most interesting and nontrivial phenomenon here is the synchronization transition. We will treat it as a transition inside chaos, and follow a twofold approach. On one hand, we exploit the irregularity of chaos and describe the transition statistically. On the other hand, we explore deterministic regular properties of the dynamics and describe the transition topologically, as a bifurcation. We hope to convince the reader that these two approaches complement each other, giving the full picture of the phenomenon. In the next chapter, where we discuss many generalizations of the simplest model, we will see that the basic features of complete synchronization are generally valid for a broad class of chaotic systems.

The prerequisite for this chapter is basic knowledge of the theory of chaos, in particular of Lyapunov exponents. For the analytical statistical description we use the thermodynamic formalism, while for the topological considerations a knowledge of bifurcation theory is helpful. One can find these topics in many textbooks on nonlinear dynamics and chaos [Schuster 1988; Ott 1992; Kaplan and Glass 1995; Alligood etal. 1997; Guckenheimer and Holmes 1986] as well as in monographs [Badii and Politi 1997; Beck and Schlögl 1997].

In this Appendix we present translations of the original texts of Christiaan Huygens where he describes the discovery of synchronization [Huygens 1967a,b].

A letter from Christiaan Huygens to his father, Constantyn Huygens

26 February 1665.

While I was forced to stay in bed for a few days and made observations on my two clocks of the new workshop, I noticed a wonderful effect that nobody could have thought of before. The two clocks, while hanging [on the wall] side by side with a distance of one or two feet between, kept in pace relative to each other with a precision so high that the two pendulums always swung together, and never varied. While I admired this for some time, I finally found that this happened due to a sort of sympathy: when I made the pendulums swing at differing paces, I found that half an hour later, they always returned to synchronism and kept it constantly afterwards, as long as I let them go. Then, I put them further away from one another, hanging one on one side of the room and the other one fifteen feet away. I saw that after one day, there was a difference of five seconds between them and, consequently, their earlier agreement was only due to some sympathy that, in my opinion, cannot be caused by anything other than the imperceptible stirring of the air due to the motion of the pendulums. Yet the clocks are inside closed boxes that weigh, including all the lead, a little less than a hundred pounds each.

So far we have considered synchronization of periodic oscillators, also in the presence of noise, and have described phase locking and frequency entrainment. In this chapter we discuss similar effects for chaotic systems. The main idea is that (at least for some systems) chaotic signals can be regarded as oscillations with chaotically modulated amplitude and with more or less uniformly rotating phase. The mean velocity of this rotation determines the characteristic time scale of the chaotic system that can be adjusted by weak forcing or due to weak coupling with another oscillator. Thus, we expect to observe phase locking and frequency entrainment for this class of systems as well. It is important that the amplitude dynamics remain chaotic and almost unaffected by the forcing/coupling.

The effects discussed in this chapter were initially termed “phase synchronization” to distinguish them from the other phenomena in coupled (forced) chaotic systems (these phenomena are presented in Part III). As these effects are a natural extension of the theory presented in previous sections, here we refer to them simply as synchronization.

In the following sections we first introduce the notion of phase for chaotic oscillators. Next, we present the effect of phase synchronization of chaotic oscillators and discuss it from statistical and topological viewpoints. We consider both entrainment of a chaotic oscillator by a periodic force and mutual synchronization of two nonidentical chaotic systems. Our presentation assumes that the reader is familiar with the basic aspects of the theory of chaos.

In this chapter we dwell on techniques of experimental studies of synchronization and give some practical hints for experimentalists. Previously, presenting different features of this phenomenon, we illustrated the theory with the results of a number of experiments and observations. In those examples the presence (or absence) of synchronization was quite obvious, but this is not always the case. Actually, detection of synchronization of irregular oscillators is not an easy task. A simple visual inspection of signals, as was done by Huygens in his experiments with clocks, is not always sufficient, and special techniques of data analysis are required. Indeed, the mere estimation of phase and frequency from a complex time series, especially from a nonstationary one, is a complicated problem, and we begin with its discussion. Next, we proceed in two directions: first, we summarize how to determine the synchronization properties of oscillator(s) experimentally; second, we use the idea of synchronization to analyze the interdependence between two (or more) scalar signals. Some technical details of data processing are given in Appendix A2.

Estimating phases and frequencies from data

Synchronization arises as the appearance of a relationship between phases and frequencies of interacting oscillators. For periodic oscillators these relations (phase and frequency locking) are rather simple (see Eqs. (3.3) and (3.2)); for noisy and chaotic systems the definition of synchronization is not so trivial. Anyway, in order to analyze synchronization in an experiment, we have to estimate phases and frequencies from the data we measure. To be not too abstract, we consider a human electrocardiogram (ECG) and a respiratory signal (air flow measured at the nose of the subject) as examples (Fig. 6.1).

The theory of electric transport in semiconductors describes how charge carriers interact with electric and magnetic fields, and move under their influence. In general, we will consider the regime far from thermodynamic equilibrium, under which linear relations between current and voltage do not hold, and more sophisticated modeling of the microscopic charge-transport processes is required. There are various levels at which semiconductor transport can be modeled, depending upon the specific structures under consideration as well as the operating conditions. In this chapter we will survey a hierarchy of approaches to nonlinear charge transport and discuss the regimes of validity. This hierarchy of models will form the physical basis for the instabilities and spatio-temporal pattern formation processes to be discussed in the subsequent chapters. It will also serve to introduce some systematics and give guidance concerning the confusing variety of nonlinear dynamic models that have been developed and studied in the field of semiconductor instabilities within the recent past.

Introduction

With the advent of modern semiconductor-growth technologies such as molecular beam epitaxy (MBE) and metal–organic chemical vapor deposition (MOCVD), artificial structures composed of different materials with layer widths of only a few nanometers have been grown, and additional lateral patterning by electron-beam lithography or other lithographic or etching techniques (ion-beam, X-ray, and scanning-probe microscopies) can impose lateral dimensions of quantum confinement in the 10 nm regime. Alternatively, lateral structures can be induced by the Stranski–Krastanov growth mode in strained material systems, which leads to the self-organized formation of islands (“quantum dots”) a few nanometers in diameter (Bimberg et al. 1999).