Slow manifolds for stochastic systems with non-Gaussian stable L\'evy noise

TL;DR

This paper constructs slow manifolds for stochastic systems driven by non-Gaussian stable Lévy noise, demonstrating their convergence to critical manifolds as the noise scale diminishes, aiding long-term dynamic analysis.

Contribution

It introduces a method to construct slow manifolds in non-Gaussian stochastic systems and analyzes their convergence and approximation properties.

Findings

Slow manifolds with exponential tracking are constructed.

Slow manifolds converge to critical manifolds as noise scale approaches zero.

Error estimates for slow manifold approximations are provided.

Abstract

This work is concerned with the dynamics of a class of slow-fast stochastic dynamical systems with non-Gaussian stable L\'evy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, eliminating the fast variables to reduce the dimension of these coupled dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps understand long time dynamics. The approximation of slow manifolds with error estimate in distribution are also considered.