Degenerate Resonances and Synchronization in Nearly Hamiltonian Systems Under Quasi-periodic Perturbations

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.