On two-frequency quasi-periodic perturbations of systems close to two-dimensional Hamiltonian ones with a double limit cycle

TL;DR

This paper investigates how two-frequency quasi-periodic perturbations affect systems near two-dimensional Hamiltonian systems with a double limit cycle, focusing on resonance phenomena, bifurcations, and the structure of resonance zones.

Contribution

It derives averaged systems to analyze resonance zones and bifurcations in perturbed Hamiltonian systems with double limit cycles, providing new insights into their dynamical behavior.

Findings

Resonance zones are characterized by the structure of averaged systems.

Bifurcations leading to double limit cycles are identified.

Numerical simulations confirm theoretical predictions.

Abstract

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.