Bifurcations in the Dynamics of Two Coupled Van Der Pol Oscillators with Delay Coupling

We investigate the dynamics of a system of two van der Pol oscillators with delayed velocity coupling. We use the method of averaging to reduce the problem to the study of a slowow in three dimensions. We study the steady state solutions of this slow ow, with special attention given to the bifurcations accompanying their change in number and stability. Our interest in this system is due to its relevance to coupled laser oscillators. INTRODUCTION This work is concerned with the mutual interaction of two limit cycle oscillators. It is motivated by applications to laser dynamics (York and Compton, 1992), (York 1993), (Lynch and York, 1995), (Lynch, 1995). When two lasers operate physically close to one another, the light output of each may a ect the behavior of the other. Since the frequencies of some lasers are in the 10 GHz range, the time for light to travel from one laser to the other, a distance of the order of cm, represents a substantial portion of the period of the uncoupled laser oscillator. This immediately leads to the inclusion of delay e ects in the coupling terms. In this paper we investigate the dynamics of two weakly coupled van der Pol oscillators in which the coupling terms have time delay . We use the method of averaging to obtain an approximate simpli ed system of three slow ow equations and then investigate the stability and bifurcation of their equilibria (corresponding to periodic motions in the original system). This work is an extension of a previous paper by the present authors (Wirkus and Rand, 1997) and is also related to previous studies of coupled van der Pol oscillators in which the coupling terms omitted delay e ects (Rand and Holmes, 1980), (Storti and Rand, 1982), (Storti and Rand, 1986),(Chakraborty and Rand, 1988). DERIVATION OF SLOW FLOWEQUATIONS We investigate two van der Pol oscillators with delay coupling (York 1993),(Lynch and York, 1995):