Invariant manifold reduction for stochastic dynamical systems

TL;DR

This paper develops a method for reducing stochastic dynamical systems to invariant manifolds, enabling simpler analysis of complex nonlinear stochastic behaviors through two invariance concepts.

Contribution

The authors derive an invariant manifold reduction principle for stochastic differential equations, considering both cocycle-based and almost sure invariance concepts.

Findings

Derived reduction principle for stochastic systems

Analyzed random and deterministic invariant manifolds

Provided a framework for system dimension reduction

Abstract

Invariant manifolds facilitate the understanding of nonlinear stochastic dynamics. When an invariant manifold is represented approximately by a graph for example, the whole stochastic dynamical system may be reduced or restricted to this manifold. This reduced system may provide valuable dynamical information for the original system. The authors have derived an invariant manifold reduction or restriction principle for systems of Stratonovich or Ito stochastic differential equations. Two concepts of invariance are considered for invariant manifolds. The first invariance concept is in the framework of cocycles -- an invariant manifold being a random set. The dynamical reduction is achieved by investigating random center manifolds. The second invariance concept is in the sense of almost sure -- an invariant manifold being a deterministic set which is not necessarily attracting. The restriction of the original stochastic system on this deterministic local invariant manifold is still a stochastic system but with reduced dimension.