# American Institute of Mathematical Sciences

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On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities
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

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