On global dynamics in duffing equation with quasiperiodic perturbation

We consider Duffing equation with small perturbation consisting of time-independent nonconservative part similar to Van der Pol equation and quasiperiodic two-frequency part with irrational frequency ratio. Similarly to the analysis of time-periodic perturbation we apply analysis of resonances implying averaging. To study solutions near unperturbed separatrix we apply adapted Melnikov's method. We establish that the number of "partly passable" resonance levels is finite and qualitative behavior of solutions near other resonance levels is determined by the autonomous part of perturbation. We also study solutions corresponding to a limit cycle generated by the autonomous part of the perturbation. We demonstrate how solutions of the averaged system behave when a limit cycle corresponding to a three-dimensional torus of the original system goes through a neighborhood of a resonance level. In the case when the unperturbed system has a separatrix loop we use Melnikov formula to establish the transversal intersection of the stable and unstable manifolds of a saddle solution. This fact implies the existence of homoclinic solutions and nonregular dynamics in a neighborhood of the unperturbed separatrix. Applying all these techniques allows us to describe the global behavior of solutions.