Isostables for stochastic oscillators

TL;DR

This paper extends the phase-amplitude framework for stochastic oscillators by defining stochastic isostables as eigenfunctions associated with the least negative nontrivial eigenvalues, providing a comprehensive description of noisy limit cycle dynamics.

Contribution

It introduces the stochastic isostable coordinate, completing the phase-amplitude analysis for noisy oscillators and encompassing noise-induced oscillations.

Findings

Defines stochastic isostables as eigenfunctions with least negative nontrivial eigenvalues.

Provides a framework for stochastic limit cycle dynamics including noise effects.

Complements the existing phase description for stochastic oscillators.

Abstract

Thomas and Lindner (2014, Phys.Rev.Lett.) defined an asymptotic phase for stochastic oscillators as the angle in the complex plane made by the eigenfunction, having a complex eigenvalue with a least negative real part, of the backward Kolmogorov (or stochastic Koopman) operator. We complete the phase-amplitude description of noisy oscillators by defining the stochastic isostable coordinate as the eigenfunction with the least negative nontrivial real eigenvalue. Our results suggest a framework for stochastic limit cycle dynamics that encompasses noise-induced oscillations.