Phase models and oscillators with time delayed coupling

Sue Ann Campbell,Ilya Kobelevskiy

doi:10.3934/dcds.2012.32.2653

Abstract

We consider two identical oscillators with time delayed coupling, modelledby a system of delay differential equations. We reduce the system of delaydifferential equations to a phase model where the time delay enters as aphase shift. By analyzing the phase model, we show how the time delayaffects the stability of phase-locked periodic solutions and causesstability switching of in-phase and anti-phase solutions as the delayis increased. In particular, we show how the phase model can predict whenthe phase-flip bifurcation will occur in the original delay differentialequation model. The results of the phase model analysis are applied topairs of Morris-Lecar oscillators with diffusive or synaptic coupling andcompared with numerical studies of the full system of delay differentialequations.

Highlights

Coupled oscillators occur as models for many systems including neural networks [1, 2], laser arrays [3, 4], flashing of fireflies [5], cardiac pacemaker cells [6] and even movement of a slime mold [7]
In this paper we studied a phase model for two coupled identical oscillators with delayed connections
H, we showed that as the delay is increased, a countable number of stability switches of the inphase and anti-phase solutions must occur

Summary

Introduction

Coupled oscillators occur as models for many systems including neural networks [1, 2], laser arrays [3, 4], flashing of fireflies [5], cardiac pacemaker cells [6] and even movement of a slime mold [7]. The previous proposition proves that the phase-flip bifurcation occurs in system (8), if the coupling is sufficiently weak and the interaction function satisfies (17), and gives a formula for the frequency jump at the bifurcation This bifurcation is degenerate since two equilibrium points change stability simultaneously. (For example, in the model of [8] they use five modes to get a good representation of H.) if the coefficients of the higher Fourier modes are small enough that H is close to (17) one might still expect to see something similar to a phase-flip bifurcation where the change of stability of the in-phase and anti-phase solutions occur close together but not at the same value of the delay.

Example

10 Numerically calculated

Conclusions