A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

TL;DR

This paper introduces a rigorous framework for defining stochastic isochrons as random stable manifolds in noisy dynamical systems, linking classical and recent approaches to understanding stochastic oscillations.

Contribution

It proposes a new definition of stochastic isochrons within the random dynamical systems framework, connecting stable manifolds with isochron maps under noise.

Findings

Defined stochastic isochrons as random stable manifolds.

Established a random version of isochron maps.

Linked the dynamical systems approach with expected return time methods.

Abstract

For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.