Contribution from the secondary harmonics of a disturbance to the separatrix map of the Hamiltonian system

Abstract

The special role of low-frequency secondary harmonics with frequencies that are sums of and differences between primary frequencies entering into the Hamiltonian in explicit form has been already discussed in the literature. These harmonics are of the second order of smallness and constitute a minor fraction of the disturbance. Nevertheless, under certain conditions, their contribution to the amplitude of the separatrix map of the system may be several orders of magnitude higher than the contributions from primary harmonics and, thereby, govern the formation of dynamic chaos. This work generalizes currently available theoretical and numerical data on this issue. The role of secondary harmonics is demonstrated with a pendulum the disturbance of which in the Hamiltonian is represented by two asymmetric closely spaced high-frequency harmonics. An analytical expression for the contribution of the secondary harmonics to the separatrix map amplitude for this system is derived, and the range of very low secondary frequencies not studied earlier is considered using this equation. The domains where the separatrix map amplitude linearly grows with frequency and the chaotic layer size is frequency-independent are indicated. Theoretical predictions are compared with numerical data.