HIGHER-ORDER CHANGE OF THE HAMILTONIAN BETWEEN SADDLE APPROACHES BY PHASE PLANE MATCHING

Autonomous Hamiltonian systems with a homoclinic orbit connecting a sa ddle point to itself are analyzed under small symmetric perturbations, which correspond to dissipative perturbations for conservative system s. Since the differential equations are autonomous, phase plane variab les are used. The usual asymptotic expansion of a nearly homoclinic or bit is shown to fail as the surrounding saddle regions are approached. A non-linear analysis of the surrounding saddle regions is derived. B y matching (to sufficiently high order) the asymptotic expansion of th e nearly homoclinic orbit to the solution in the surrounding saddle re gions, we obtain the change in the Hamiltonian from one saddle approac h to the next. In this way, we derive a new higher-order logarithmic c orrection to the well-known Melnikov integral for the leading order ch ange of the Hamiltonian. (C) 1996 Elsevier Science Ltd.