Elementary Symbolic Dynamics and Chaos in Dissipative Systems

The following sections are included:

• Preface

• Contents

Throughout this book we shall use the term “physics” in a broad sense, understanding it as a synonym for the physical sciences. In this chapter a few physical systems will be analyzed and modelled by nonlinear equations which may exhibit chaotic behaviour in a suitable range of the control parameters. We shall use some of these models in subsequent chapters to show one or other aspects of a mathematical theory or a certain numerical method, and to gain deeper insight into the phenomenon of chaos. We start with a question which has been asked by everyone who comes across the somewhat fuzzy word chaos in the context of an exact science like physics.

In the last chapter we have seen a number of simple one-dimensional mappings which may exhibit quite complex dynamical behaviour. Unlike many other one-dimensional models in physics which are either too specific or too trivial to have any practical other than pedagogical meaning, one-dimensional mappings are very rich in content and, at the same time, simple enough to be accessible to analytical means and not very time-consuming in numerical studies. For this reason we shall concentrate in this book on a few typical one-dimensional mappings. Although one-dimensional mappings cannot cover all the properties of higher dimensional nonlinear systems, they do have many features in common. Therefore, one-dimensional mappings may provide a good beachhead to enter the vast Chaosland, and it is quite rewarding to return to them when studying more complicated dynamical systems.

In this chapter, we study the problem of labelling and ordering all the periodic windows in one-dimensional mappings. This can be accomplished by the method of symbolic dynamics, an algebraic approach which originated from the abstract topological theory of dynamical systems, developed in the 1930’s¹ and made concrete, when applied to one-dimensional mappings, since the 1970’s. Although symbolic dynamics provides the most rigorous way to define chaotic motion in dynamical systems, its abstract formulation (see, e.g., Alekseev and Yakobson, 1981) still prevents a non-mathematically minded scientist from appreciating the power of this beautiful theory. An easier way to overcome this difficulty is just to learn the game by playing it with simple examples…

Circle mappings and higher dimensional maps are in their own rights rich subjects; a detailed study would go beyond the scope of this book. We include this chapter in order to link one-dimensional mappings and differential equations. Therefore, instead of a comprehensive review, we shall only touch on those aspects of the problem which either are needed in the sequel or contain new contributions by the Chinese researchers…

Many interesting models in the physical sciences are described by systems of ordinary differential equations. While bifurcation and chaos in discrete mappings can be studied more or less thoroughly by both analytical and numerical means, similar tasks for differential equations may encounter great difficulty. Analytical tools can be of some help only in very limited cases, e.g., when the criterion of Melnikov (see, e.g., Chapter 4 of Guckenheimer and Holmes, B1983) or Silnikov (see, e.g., Gaspard and Nicolis, 1983) is applicable. These methods usually only provide a criterion for the existence of chaotic motion and give an estimate for the threshold of transition to chaos, but cannot help with the exploration of the global structure of the parameter space. Although numerical studies are very time-consuming in most cases, nevertheless, our present understanding of chaotic behaviour in differential equations relies heavily on numerical work. Therefore, we shall basically devote this chapter to numerical studies of ordinary differential equations…

Our present understanding of chaos in nonlinear systems would be unthinkable without using computers. However, there might be some serious objections to the use of computers. Anyway, on a computer with a finite word length and in a finite run time, one cannot realize a truly quasiperiodic motion, let alone chaotic trajectories. In principle, it is impossible to distinguish a very long periodic orbit from a quasiperiodic or chaotic one by merely looking at the trajectories. In addition, inevitable round-off errors place doubts on any claim that the observed erratic motion is deterministic chaos. Nevertheless, the situation is far from being hopeless, as it might appear to be at a first glance. The reliability of our knowledge on chaos gained, from computer studies, is theoretically guaranteed by some rigorous mathematical results, and, in practice, by a pursuit of a correct research strategy…

So far, we have been studying the asymptotic behaviour of nonlinear systems as the time goes to infinity. In order to settle down at a stationary regime, one always has to throwaway a certain number of iterates or periods while carrying out the numerical work. However, how could one be confident that the transients have really faded out? In studies of critical phenomena and phase transitions in physical systems, it has been known since a long time that there exists so-called critical slowing down: it would take a very very long time for a system to reach a new equilibrium state, after having been perturbed from an equilibrium state, if it had been very close to or was right at the critical point. The same happens with the time evolution of nonlinear systems, when they are close to a bifurcation point (see Section 7.1 below). We already knew of some consequence of this slowing down, when introducing the “operational dimension” in Section 6.7…

The following sections are included:

• References
  • Books, Conference Proceedings, and Collection of Papers
  • Papers, including Reviews

• Subject Index