Kirill Morozov

We study nonconservative quasi-periodic (with $m$ frequencies) perturbations of two-dimensional Hamiltonian systems with nonmonotonic rotation. It is assumed that the perturbation contains the so-called parametric terms. The behavior of solutions in the vicinity of degenerate resonances is described. Conditions for the existence of resonance $(m + 1)$-dimensional invariant tori for which there are no generating ones in the unperturbed system are found. The class of perturbations for which such tori can exist is indicated. The results are applied to the asymmetric Duffing equation under a parametric quasi-periodic perturbation.

Quasi-periodic nonconservative perturbations of two-dimensional nonlinear Hamiltonian systems are considered. The definition of a degenerate resonance is introduced and the topology of a degenerate resonance zone is studied. Particular attention is paid to the synchronization process during the passage of an invariant torus through the resonance zone. The existence of so-called synchronization intervals is proved and new phenomena which have to do with synchronization are found. The study is based on the analysis of a pendulum-type averaged system that determines the dynamics near the degenerate resonance phase curve of the unperturbed system.