Dead zones and phase reduction of coupled oscillators

TL;DR

This paper investigates the occurrence and effects of dead zones in the phase interactions of coupled dynamical systems, especially weakly coupled oscillators, revealing how they influence emergent behaviors.

Contribution

It provides criteria for dead zone formation in phase interaction functions and explores their implications in multiscale relaxation oscillators.

Findings

Dead zones can arise from coupling on specific oscillation branches.

Dead zones significantly affect the dynamics of coupled oscillators.

Application to multiscale oscillators demonstrates practical relevance.

Abstract

A dead zone in the interaction between two dynamical systems is a region of their joint phase space where one system is insensitive to the changes in the other. These can arise in a number of contexts, and their presence in phase interaction functions has interesting dynamical consequences for the emergent dynamics. In this paper, we consider dead zones in the interaction of general coupled dynamical systems. For weakly coupled limit cycle oscillators, we investigate criteria that give rise to dead zones in the phase interaction functions. We give applications to coupled multiscale oscillators where coupling on only one branch of a relaxation oscillation can lead to the appearance of dead zones in a phase description of their interaction.