HIGHER-ORDER MELNIKOV THEORY FOR ADIABATIC SYSTEMS

In this paper, we study adiabatic Hamiltonian systems including those subject to small-amplitude forcing and damping. It is known that simpl e zeroes of the adiabatic Poincare-Arnold-Melnikov function imply the existence of primary intersection points of the stable and unstable ma nifolds of hyperbolic orbits. Hen, we present an Nth-order Melnikov fu nction whose simple zeroes correspond to Nth-order transverse intersec tion points and hence to N-pulse homoclinic orbits. Using this functio n, it can be shown that N-pulse homoclinic orbits arise in a plethora of adiabatic models, including systems with slowly varying potentials. The theory is illustrated on a damped Hamiltonian system with a slowl y varying cubic potential. In addition, the Nth-order adiabatic Melnik ov function is useful for showing the existence of multi-pulse resonan t periodic orbits in the special class of slow, time-periodic systems. (C) 1996 American Institute of Physics.