Pattern Formation

Abstract

This chapter consists of two parts. In the first part, the fundamental mechanism of pattern formation will be explained. In this context, the notion of a pattern will be defined explicitly. In fact, it will be shown that pattern formation involves two different but related types of patterns: attractor and repellor patterns, on the one hand, and basis patterns, on the other. Attractor and repellor patterns can come, for example, as fixed point patterns. In this case, we are dealing with fixed point attractor and fixed point repellor patterns. Attractor patterns correspond to the patterns that we humans form with our bodies (e.g., when walking) or to brain activity patterns in the human brain. Attractor patterns corresponds to the patterns that are usually observed in experiments in the first place. Attractor patterns can turn into repellor patterns at bifurcation points, and in this context are like two sides of the same coin. Attractor patterns are composed of elementary units: the basis patterns. In special cases, an attractor pattern may be composed of a single basis pattern. In such cases, an attractor pattern is identical to a basis pattern. In the context of the basis patterns the abstract concept of eigenvalues will be introduced. By definition, these eigenvalues determine how quickly basis patterns emerge and disappear. However, taking all eigenvalues together as a set, the set of eigenvalues constitutes an eigenvalue spectrum. This spectrum is the key for understanding pattern formation in general, and transitions between attractor patterns, the emergence of attractor patterns, and bifurcations, in particular. That is, eigenvalues play a key role in the theory of pattern formation and synergetics. Finally, in the context of basis patterns also the concept of pattern amplitudes will be introduced. As will be explained below, pattern amplitudes describe how much basis patterns contribute to attractor patterns. Pattern amplitudes, in general, and the so-called reduced amplitude space, in particular, allow for a convenient description of the formation of patterns and transitions between attractor patterns. The second part of this chapter is devoted to the Lotka-Volterra-Haken amplitude equations. These equations describe pattern formation in the aforementioned reduced amplitude space. The Lotka-Volterra-Haken amplitude equations correspond to a general class of amplitude equations in the theory of pattern formation. They will be used in all applications of this book to describe the formation of brain and body activity patterns in humans and animals (i.e., “perception”, “cognition”, and “behavior”).