A numerical investigation of the dynamics of a system of two time-delay coupled relaxation oscillators

Abstract

In this paper we examine the dynamics of two time-delay coupled relaxationoscillators of the van der Pol type. By integrating the governingdifferential-delay equations numerically, we find the various phase-lockedmotions including the in-phase and out-of-phase modes. Our computationsreveal that depending on the strength of coupling ($\alpha$) and the amountof time-delay ($\tau$), the in-phase (out-of-phase) mode may be stable orunstable. There are also values of $\alpha$ and $\tau$ for which thein-phase and out-of-phase modes are both stable leading to birhythmicity.The results are illustrated in the $\alpha$-$\tau$ parameter plane. Nearthe boundaries between stability and instability of the in-phase(out-of-phase) mode, many other types of phase-locked motions can occur.Several examples of these phase-locked states are presented.