Abstract
In this paper, we study adiabatic Hamiltonian systems including those subject to small-amplitude forcing and damping. It is known that simple zeroes of the adiabatic Poincare–Arnold–Melnikov function imply the existence of primary intersection points of the stable and unstable manifolds of hyperbolic orbits. Here, we present an Nth-order Melnikov function whose simple zeroes correspond to Nth-order transverse intersection points and hence to N-pulse homoclinic orbits. Using this function, 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 slowly varying cubic potential. In addition, the Nth-order adiabatic Melnikov function is useful for showing the existence of multi-pulse resonant periodic orbits in the special class of slow, time-periodic systems.
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