Higher order approximation of isochrons

TL;DR

This paper introduces a method to compute higher order derivatives of the phase function in oscillators, enabling more accurate approximations of isochrons, with potential applications in stochastic oscillator dynamics.

Contribution

It extends existing phase reduction techniques by providing a way to obtain higher order derivatives of the phase function for better isochron approximation.

Findings

Method applied to Stuart-Landau oscillator demonstrating second-order approximation.

Higher order derivatives enable improved analysis of oscillator dynamics under perturbations.

Potential application to stochastic oscillator studies in future work.

Abstract

Phase reduction is a commonly used techinque for analyzing stable oscillators, particularly in studies concerning synchronization and phase lock of a network of oscillators. In a widely used numerical approach for obtaining phase reduction of a single oscillator, one needs to obtain the gradient of the phase function, which essentially provides a linear approximation of isochrons. In this paper, we extend the method for obtaining partial derivatives of the phase function to arbitrary order, providing higher order approximations of isochrons. In particular, our method in order 2 can be applied to the study of dynamics of a stable oscillator subjected to stochastic perturbations, a topic that will be discussed in a future paper. We use the Stuart-Landau oscillator to illustrate the method in order 2.