Splitting of the separatrices after a Hamiltonian-Hopf bifurcation under periodic forcing

TL;DR

This paper investigates how periodic forcing affects the splitting of invariant manifolds in a Hamiltonian-Hopf bifurcation, revealing the dependence on the forcing frequency's continued fraction properties.

Contribution

It provides a systematic analysis of the dominant harmonic changes in splitting functions based on the frequency's continued fraction expansion, covering various types of frequencies.

Findings

Detailed description of asymptotic splitting behaviour.

Dependence of dominant harmonics on frequency properties.

Methodology applicable to different frequency types.

Abstract

We consider the effect of a non-autonomous periodic perturbation on a 2-dof autonomous system obtained as a truncation of the Hamiltonian-Hopf normal form. Our analysis focuses on the behaviour of the splitting of the invariant 2-dimensional stable/unstable manifolds. We analyse the different changes of dominant harmonic in the splitting functions. We describe how the dominant harmonics depend on the quotients of the continuous fraction expansion of the periodic forcing frequency. We have considered different frequencies including quadratic irrationals, frequencies having continuous fraction expansion with bounded quotients and frequencies with unbounded quotients. The methodology used is general enough to systematically deal with all these frequency types. All together allow us to get a detailed description of the asymptotic splitting behaviour for the concrete perturbation considered.