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December  2003, 2(4): 567-577. doi: 10.3934/cpaa.2003.2.567

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

Richard H. Rand 1, and  Asok K. Sen 2,

1. 

Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, New York 14853, United States
2. 

Department of Mathematical Sciences, Indiana University, Indianapolis, IN 46202, United States

Received  August 2001 Revised  June 2003 Published  October 2003

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 ($\alpha$) and the amount of time-delay ($\tau$), the in-phase (out-of-phase) mode may be stable or unstable. There are also values of $\alpha$ and $\tau$ for which the in-phase and out-of-phase modes are both stable leading to birhythmicity. The results are illustrated in the $\alpha$-$\tau$ 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.
Keywords: time-delay, relaxation, Limit cycle, coupled oscillators..
Mathematics Subject Classification: 34K2.
Citation: Richard H. Rand, Asok K. Sen. A numerical investigation of the dynamics of a system of two time-delay coupled relaxation oscillators. Communications on Pure and Applied Analysis, 2003, 2 (4) : 567-577. doi: 10.3934/cpaa.2003.2.567
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