On Coupled Delayed Van der Pol-Duffing oscillators

TL;DR

This paper analyzes the dynamics and stability of coupled delay differential Duffing-Van der Pol oscillators, deriving conditions for bifurcations and stability using perturbation methods, with relevance to coupled laser oscillators.

Contribution

It introduces a detailed analytical approach to study stability and bifurcations in coupled delay differential oscillators, specifically the Duffing-Van der Pol type.

Findings

Derived in-phase mode solutions using Lindstedt's method

Established conditions for Hopf bifurcation and stability

Analyzed stability and bifurcations of the origin

Abstract

We investigate the dynamics of a delay differential coupled Duffing-Van der Pol oscillator equation. Using the Lindstedt's method, we derive the in-phase mode solutions and then obtain the slow flow equations governing the stability of the in-phase mode by employing the two variable perturbation method. We solve the slow flow equations using series expansion and obtain conditions for Hopf bifurcation and studied stability of the in-phase mode. Finally, we studied stability and bifurcations of the origin. Our interest in this system is due to the fact that it is related to the coupled laser oscillators.