A Wong-Zakai Approximation for Random Slow Manifolds with Application to Parameter Estimation

TL;DR

This paper develops a Wong-Zakai approximation for the random slow manifold in stochastic dynamical systems, enabling better understanding of long-term behavior and accurate parameter estimation.

Contribution

It introduces a novel Wong-Zakai approximation approach for the random slow manifold and demonstrates its effectiveness in parameter estimation within stochastic systems.

Findings

Existence of the random slow manifold for the approximation system.

The approximation system's slow manifold provides insights into the original system's dynamics.

The reduced system enables accurate parameter estimation.

Abstract

We study a Wong-Zakai approximation for the random slow manifold of a slow-fast stochastic dynamical system. We first deduce the existence of the random slow manifold about an approximation system driven by an integrated Ornstein-Uhlenbeck (O-U) process. Then we compute the slow manifold of the approximation system, in order to gain insights of the long time dynamics of the original stochastic system. By restricting this approximation system to its slow manifold, we thus get a reduced slow random system. This reduced slow random system is used to accurately estimate a system parameter of the original system. An example is presented to illustrate this approximation.