Synchronization of quasiperiodic oscillations in nearly Hamiltonian systems: The degenerate case

Quasiperiodic perturbations of two-dimensional nearly Hamiltonian systems with a limit cycle are considered. The behavior of solutions in a small neighborhood of a degenerate resonance is studied. Special attention is paid to the synchronization problem. Bifurcations of quasiperiodic solutions that arise when the limit cycle passes through the neighborhood of a resonance phase curve are investigated. The study is based on an analysis of an autonomous pendulum-type system, which is obtained by the method of averaging and determines the dynamics in the resonance zone. Two possible topological structures of the unperturbed averaged system are distinguished. For each case, the intervals of a control parameter that correspond to oscillatory synchronization are found. The results are applied to a Duffing-Van der Pol-type equation.