Abstract:
Mel'nikov's perturbation method for showing the existence of transversal intersections between invariant manifolds of saddle fixed points of dynamical systems is extended here to second order in a small parameter $\epsilon$. More specifically, we follow an approach due to Wiggins and derive a formula for the second order Mel'nikov vector of a class of periodically perturbed $n$-degree of freedom Hamiltonian systems. Based on the simple zero of this vector, we prove an $O(\epsilon^2)$ sufficient condition for the existence of isolated homoclinic (or heteroclinic) orbits, in the case that the first order Mel'nikov vector vanishes identically. Our result is applied to a damped, periodically driven 1-degree-of-freedom Hamiltonian and good agreement is obtained between theory and experiment, concerning the threshold of heteroclinic tangency.
