Interaction of Lower and Higher Order Hamiltonian Resonances

Abstract

The tools of normal forms and recurrence are used to analyze the interaction of low and higher order resonances in Hamiltonian systems. The resonance zones where the short-periodic solutions of the low order resonances exist are characterized by small variations of the corresponding actions that match the variations of the higher order resonance; this yields cases of embedded double resonance. The resulting interaction produces periodic solutions that in some cases destabilize a resonance zone. Applications are given to the three dof 1: 1: 4 resonance and to periodic FPU-chains producing unexpected nonlinear stability results and quasi-trapping phenomena.