  The problem of the effect of two-frequency quasi-periodic perturbations on
systems close to arbitrary nonlinear two-dimensional Hamiltonian ones is
studied in the case when the corresponding perturbed autonomous systems have a
double limit cycle. Its solution is important both for the theory of
synchronization of nonlinear oscillations and for the theory of bifurcations of
dynamical systems. In the case of commensurability of the natural frequency of
the unperturbed system with frequencies of quasi-periodic perturbation,
resonance occurs. Averaged systems are derived that make it possible to
ascertain the structure of the resonance zone, that is, to describe the
behavior of solutions in the neighborhood of individual resonance levels. The
study of these systems allows determining possible bifurcations arising when
the resonance level deviates from the level of the unperturbed system, which
generates a double limit cycle in a perturbed autonomous system. The
theoretical results obtained are applied in the study of a two-frequency
quasi-periodic perturbed pendulum-type equation and are illustrated by
numerical computations.
