MELNIKOV TECHNIQUES FOR SINGULARLY PERTURBED SYSTEMS

The Melnikov method is an analytic technique for detecting Smale-chaos of periodically perturbed ordinary differential systems via the Smale -Birkhoff theorem by means of the Poincare map. The aim of the followi ng note is to study a class of singularly perturbed nonautonomous diff erential systems admitting an integral manifold in the space of motion such that the application of Melnikov's method to this class of dynam ical systems can be restricted to the corresponding system on the inte gral manifold. Our studies were initiated by a recent paper of FECKAN [1] who used the method of Lyapunov-Schmidt.