Quasi-Periodic Parametric Perturbations of Two-Dimensional Hamiltonian Systems: Degenerate Resonances and Synchronization

We consider nonconservative quasi-periodic perturbations of two-dimensional Hamiltonian systems with nonmonotonic rotation. The peculiarity of these systems is that degenerate resonances can take place. Special focus is on those systems for which the corresponding autonomous perturbed system has a structurally stable limit cycle. If the cycle appears in the neighborhood of a nonresonance phase curve, then it corresponds to an invariant torus in the initial system. In this regard, the problem of synchronization arises when the invariant torus passes through a resonance zone. In this paper, we distinguish a class of perturbations (the so-called parametric perturbations) under which synchronization can be violated. Also, we introduce the concept of generalized synchronization and give conditions for this type of synchronization to occur. As an example, we study a Duffing-like equation with an asymmetric potential function.