A Numerical Investigation of the Dynamics of a System of Two Time-delay Coupled Relaxation Oscillators

In this paper we examine the dynamics of two time-delay coupled relaxation oscillators of the van der Pol type. By integrating the governing differential-delay equations numerically, we find the various phase-locked motions including the in-phase and out-of-phase modes. Our computations reveal that depending on the strength of coupling (α) and the amount of time-delay (τ), the in-phase (out-of-phase) mode may be stable or unstable. There are also values of α and τ for which the in-phase and out-of-phase modes are both stable leading to birhythmicity. The results are illustrated in the α-τ parameter plane. Near the 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.