Foundations of Reduction Theory

Abstract

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.