The quasi-periodic Hamiltonian Hopf bifurcation

Abstract

We consider the quasi-periodic dynamics of non-integrable perturbations of a family of integrable Hamiltonian systems with normally 1 : -1 resonant invariant tori. In particular, we focus on the supercritical quasi-periodic Hamiltonian Hopf bifurcation and address the persistence problem of the singular foliation into invariant quasi-periodic tori near the bifurcation point. With the help of KAM theory, we show that the singular torus foliation survives a small perturbation and that the persisting tori in this foliation form Cantor families. A leading example is the Lagrange top near gyroscopic stabilization weakly coupled with a quasi-periodic oscillator.