Nonlinear Equations. The Slaving Principle
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The main objective of this book is the study of dramatic macroscopic changes of systems. As seen in the introduction, this may happen when linear stability is lost. At such a point it becomes possible to eliminate very many degrees of freedom so that the macroscopic behavior of the system is governed by very few degrees of freedom only. In this chapter we wish to show explicitly how to eliminate most of the variables close to points where linear stability is lost. These points will be called critical points. It will be our goal to describe an easily applicable procedure covering most cases of practical importance. To this end the essential ideas are illustrated by a simple example (Sect. 7.1), followed by a presentation of our general procedure for nonlinear differential equations (Sects. 7.2–5). While the basic assumptions are stated in Sect. 7.2, the final results of our approach are presented in Sect. 7.4 up to (and including) formula (7.4.5). Section 7.3 and the rest of Sect. 7.4 are of a more technical nature. The rest of this chapter is devoted to an extension of the slaving principle to discrete noisy maps and to stochastic differential equations of the Îto (and Stratonovich) type (Sects. 7.6–9).
KeywordsNonlinear Equation Stochastic Differential Equation Linear Stability Formal Relation Iteration Procedure
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- Here, by an appropriate decomposition of the variables into rapidly oscillating parts and slowly varying amplitudes, the atomic variables were expressed by the field modes (order parameters) Other procedures are given inGoogle Scholar
- The latter procedures are based on rapidly converging continued fractions, at the expense that the slaved variables depend on the order parameters (unstable modes) at previous times (in higher order approximation). These papers included fluctuations of the Langevin type.Google Scholar
- In a number of special cases (in particular, if the fluctuations are absent), relations can be established to other theorems and procedures, developed in mathematics, theoretical physics, or other disciplines.Google Scholar
- A. Kelley: In Transversal Mappings and Flows, ed. by R. Abraham, J. Robbin (Benjamin, New York 1967)Google Scholar
- In contrast to the center manifold theorem, the slaving principle contains fluctuations, includes the surrounding of the center manifold, and provides a construction of s(u,ϕ,t).Google Scholar