LYAPUNOV-SCHMIDT REDUCTION AND MELNIKOV INTEGRALS FOR BIFURCATION OF PERIODIC-SOLUTIONS IN COUPLED OSCILLATORS

We consider a system of autonomous ordinary differential equations dep ending on a small parameter such that the unperturbed system has an in variant manifold of periodic solutions. The problem addressed here is the determination of sufficient geometric conditions for some of the p eriodic solutions on this invariant manifold to survive after perturba tion. The main idea is to use a Lyapunov-Schmidt reduction for an appr opriate displacement function in order to obtain the bifurcation funct ion for the problem in a form which can be recognized as a generalizat ion of the subharmonic Melnikov function. Thus, the multidimensional b ifurcation problem can be cast in a form where the geometry of the pro blem is clearly incorporated. An important application can be made in case the uncoupled system of differential equations is a system of osc illators in resonance. In this case the invariant manifold of periodic solutions is just the product of the uncoupled oscillations. When eac h of the oscillators has one degree of freedom, the bifurcation functi on is computed by quadrature along the unperturbed oscillations. Addit ional applications include the computation of entrainment domains for a sinusoidally forced van der Pol oscillator and the computation of mu tual synchronization domains for a system of inductively coupled van d er Pol oscillators. (c) 1994 Academic Press, Inc.