Frequency spanning homoclinic families

Abstract

A family of maps or flows depending on a parameter $\nu$ which varies in an interval, spans a certain property if along the interval this property depends continuously on the parameter and achieves some asymptotic values along it. We consider families of periodically forced Hamiltonian systems for which the appropriately scaled frequency $\bar{\omega}(\nu)$ is spanned, namely it covers the semi-infinite line $[0,\infty).$ Under some natural assumptions on the family of flows and its adiabatic limit, we construct a convenient labelling scheme for the primary homoclinic orbits which may undergo a countable number of bifurcations along this interval. Using this scheme we prove that a properly defined flux function is $C^{1}$ in $\nu.$ Combining this proof with previous results of RK and Poje, immediately establishes that the flux function and the size of the chaotic zone depend on the frequency in a non-monotone fashion for a large class of Hamiltonian flows.