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

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 invariant 2D stable/unstable manifolds. Due to the interaction of the intrinsic angle and the periodic perturbation the splitting behaves quasi-periodically on two angles. We analyse the different changes of the dominant harmonic in the splitting functions when the unfolding parameter of the bifurcation varies. We describe how the dominant harmonics depend on the quotients of the continuous fraction expansion (CFE) of the periodic forcing frequency. We have considered different frequencies including quadratic irrationals, frequencies having CFE with bounded quotients and frequencies with unbounded quotients. The methodology combines analytical and numeric methods with heuristic estimates of the role of the non-dominant harmonics. The approach is general enough to systematically deal with all these frequency types. Together, this allows us to get a detailed description of the asymptotic splitting behaviour for the concrete perturbation considered.