Hazime Mori

A Representative Example of Dissipative Structure

The examples of dissipative structures in nature are too numerous to count. In most cases, in addition to the complexity arising from the nonlinear nature of the governing laws, there are various external factors whose combined influence causes the behavior to become even more complex. For this reason, it is very important in an attempt to understand dissipative behavior that we carefully select a few model examples from the multitude and concentrate our study on such examples by employing well-controlled conditions. In so doing, it becomes possible to apprehend the general features of dissipative structures that transcend the specific features of individual systems. In this sense, the examples discussed in this chapter, Benard convection and the Belousov- Zhabotinskii reaction, are representative, and it is perhaps not inaccurate to state that the development seen in the research of nonlinear nonequilibrium systems in the context of chaos and dissipative structures has been made largely through the study of these two examples. In this chapter we discuss the prominent characteristics of the method used to study nonlinear nonequilibrium systems through the application of this method to these two specific problems. In so doing, we make several preliminary observations in preparation for later chapters.

Dynamics of Coupled Oscillator Systems

The discussion of dissipative structures appearing in the first five chapters was given in the context of spatio-temporal patterns existing in continuous media. In the present chapter we consider another important class of dissipative systems consisting of a large number of degrees of freedom, those composed of aggregates of isolated elements. A neural network consisting of intricately coupled excitable oscillators (neurons) is one example of such a system. In addition, there are many systems of this type composed of groups of cells exhibiting physiological activity. As the subject of study in nonlinear dynamics has broadened in recent years to include phenomena found in living systems, interest in coupled oscillator systems has grown. In what follows, we consider a relatively simple example forming the subject of a great deal of present study, that of a collection of limit cycles oscillators, and basing our investigation on the method of phase dynamics, we discuss the fundamental points regarding synchronization phenomena.

The Statistical Physics of Aperiodic Motion

What physical quantity can we use to capture and describe the multitude of invariant sets and the local structure of unstable manifolds W u that determine the form and structure of chaos? By introducing the expansion rate of neighboring orbits, which expresses the stretching and folding of segments of W u, and the local dimension, which describes the self-similarity of the nested structure of strange attractors, the geometric and statistical descriptions given in terms of chaotic orbits through the fluctuations of these quantities can be unified. We show that the chaotic bifurcations and tangency structure of unstable manifolds can be directly understood in terms of the spectrum ψ(Λ), and also that a fixed relationship exists between the spectra of this expansion rate and the local dimension.

On the Structure of Chaos

The type of chaos known as ‘on-off intermittency’ was discovered by Fujisaka and Yamada (1985) in the situation in which there exists a bifurcation from synchronous to asynchronous motion in a system of two coupled chaotic oscillators. With the elucidation of the geometric structure of this intermittent chaos due to Platt et al. (1993) and Ott and Sommerer (1994), the importance of this type of system has come to be recognized.

A Physical Approach to Chaos

The term ‘chaos’ is used in reference to unstable, aperiodic motion in dynamical systems and also to the state of a system which exhibits such motion. Almost any nonequilibrium open system will, when some bifurcation parameter characterizing the system is made sufficiently large, display chaotic behavior. It can be said that chaos is Nature’s universal dynamical form. Chaos is characterized by the coexistence of an infinite number of unstable periodic orbits that determine the form and structure of the chaotic behavior exhibited by any given system. In this chapter we consider the problem of identifying the descriptive signature of such a set of orbits, and we establish the point of view from which we will elucidate the nature of chaos.

Chaotic Bifurcations and Critical Phenomena

The nature of the behavior exhibited by a chaotic system is determined by the infinite number of invariant sets existing in its chaotic region. The great variety of chaotic phenomena that we observe results from the limitless variation in the types of invariant sets contained by the systems we encounter. The nature of the invariant sets that appear in any given system and the resulting behavior that it exhibits depend both on the type of system in question and the values of the various parameters characterizing it. For a nonequilibrium open system, as the values of such parameters are changed, the qualitative nature of the system’s behavior is seen to assume many forms, as it experiences the emergence, development and bifurcation of chaos.

Bifurcation Phenomena of Dissipative Dynamical Systems

The form and structure of an attractor are characterized by the types of unstable periodic points (saddle points) that it contains. Changing the value of a bifurcation parameter can cause various saddle points to enter and leave attractors. Chaotic bifurcations result from the collision of an attractor with such points and their resultant inclusion into the attractor.

Foundations of Reduction Theory

In the preceding chapters, we developed arguments concerning pattern dynamics based on relatively simple model equations. Many of these equations were derived phenomenologically. In the present chapter, we consider the theoretical foundation of perhaps the most important types of model equations considered in Part I, amplitude equations and phase equations. The degrees of freedom contained in the corresponding reduced equations are generally characterized by slow time development. For this reason, the remaining large number of degrees of freedom are eliminated, so to speak, adiabatically. For dissipative systems, at the foundation of this kind of reduction mechanism is a definite universal structure, and, as will become evident in this chapter, it is possible from the point of view presented here to gain a clear new understanding of the separately developed realizations of reduction theory that we have presented in previous chapters for the study of a variety of outwardly different types of situations. We begin here by considering a simple example through which the fundamental structure of reduction is clarified. We then see how this structure is actually realized in the derivation of amplitude and phase equations. Throughout this entire chapter, we wish to remove emphasis from the presentation of the reduction algorithm and place it, rather, on making clear the physical meaning contained in the reduction approach.

Amplitude Equations and Their Applications

In the preceding chapter, we saw how in the neighborhood of a bifurcation point at which a new spatial pattern arises, the equation describing the system in question can be reduced to a relatively simple form we refer to as an amplitude equation. Then, as a representative example of this reduction procedure, we derived the Newell-Whitehead (NW) equation using phenomenological considerations. In this chapter, we investigate how different types of amplitude equations are derived to describe a variety of physical conditions. We then study the properties of these equations.

Reaction—Diffusion Systems and Interface Dynamics

Existing studies on reaction—diffusion systems have been carried out with the Belousov-Zhabotinskii reaction acting as the standard example. Approaches based on amplitude equations are in general not suited to describe patterns peculiar to such ‘excitable’ systems because, while excitability originates in a particular property of global flow in phase space, amplitude equations are obtained by considering only local flow. In fact, BZ reaction systems display wave patterns that lack both the temporal and spatial smoothness displayed by solutions to amplitude equations, and thus it is much more natural to treat excitable systems using local interface structure representing the sudden change in state of a system over a small distance. Patterns containing interfaces arise out of the cooperative dynamics of degrees of freedom undergoing rapid and slow temporal change. Rapidly changing degrees of freedom form a bistable partial subsystem consisting of two states, an active (or excitable) state and an inactive (or nonexcitable) state. The speed with which the transition between these two states is carried out is controlled by the slowly changing degrees of freedom.