Approximation of invariant foliations for stochastic dynamical systems

TL;DR

This paper develops an asymptotic analysis method to approximate invariant foliations in stochastic dynamical systems, representing them as perturbations of deterministic structures with estimated errors.

Contribution

It introduces a novel technique to approximate random invariant foliations for stochastic systems using asymptotic analysis, extending deterministic methods.

Findings

Approximate random invariant foliations as perturbations of deterministic ones.

Quantified deviation errors in the approximation.

Applicable to systems with small noisy perturbations.

Abstract

Invariant foliations are geometric structures for describing and understanding the qualitative behaviors of nonlinear dynamical systems. For stochastic dynamical systems, however, these geometric structures themselves are complicated random sets. Thus it is desirable to have some techniques to approximate random invariant foliations. In this paper, invariant foliations are approximated for dynamical systems with small noisy perturbations, via asymptotic analysis. Namely, random invariant foliations are represented as a perturbation of the deterministic invariant foliations, with deviation errors estimated.