Secondary homoclinic bifurcation theorems.

We develop criteria for detecting secondary intersections and tangencies of the stable and unstable manifolds of hyperbolic periodic orbits appearing in time-periodically perturbed one degree of freedom Hamiltonian systems. A function, called the "Secondary Melnikov Function" (SMF) is constructed, and it is proved that simple (resp. degenerate) zeros of this function correspond to transverse (resp. tangent) intersections of the manifolds. The theory identifies and predicts the rotary number of the intersection (the number of "humps" of the homoclinic orbit), the transition number of the homoclinic points (the number of periods between humps), the existence of tangencies, and the scaling of the intersection angles near tangent bifurcations perturbationally. The theory predicts the minimal transition number of the homoclinic points of a homoclinic tangle. This number determines the relevant time scale, the minimal stretching rate (which is related to the topological entropy) and the transport mechanism as described by the TAM, a transport theory for two-dimensional area-preserving chaotic maps. The implications of this theory on the study of dissipative systems have yet to be explored. (c) 1995 American Institute of Physics.