Stochastic periodic orbits in fast-slow systems with self-induced stochastic resonance

TL;DR

This paper investigates noise-induced stochastic periodic orbits in fast-slow systems with self-induced stochastic resonance, providing a theoretical framework that matches simulations and extends to complex systems.

Contribution

It introduces a novel method to analytically predict stochastic periodic orbits caused by SISR in fast-slow systems, validated by simulations and applicable to complex models.

Findings

Theoretical predictions align with Monte Carlo simulations.

Method accurately predicts boundary crossing positions.

Applicable to complex FitzHugh-Nagumo systems.

Abstract

Noise is ubiquitous in various systems. In systems with multiple timescales, noise can induce various coherent behaviors. Self-induced stochastic resonance (SISR) is a typical noise-induced phenomenon identified in such systems, wherein noise acting on the fast subsystem causes stochastic resonancelike boundary crossings. In this paper, we analyze the stochastic periodic orbits caused by SISR in fast-slow systems. By introducing the notion of the mean first passage velocity toward the boundary, a distance matching condition is established, through which the critical transition position of boundary crossing can be calculated. The theoretical stochastic periodic orbit can be accordingly obtained via gluing the dynamics along the slow manifolds. It is shown that the theoretical predictions are in excellent agreement with the results of Monte Carlo simulations for a piecewise linear FitzHugh-Nagumo system even for large noise. Furthermore, the proposed method is extended to the original FitzHugh-Nagumo system and also found to exhibit consistent accuracy. These results provide insights into the mechanisms of coherent behaviors in fast-slow systems and will shed light on the coherent behaviors in more complex systems and large networks.