Bifurcations in asymptotically autonomous Hamiltonian systems under oscillatory perturbations

TL;DR

This paper studies how decaying oscillatory perturbations affect the stability and bifurcation behavior of Hamiltonian systems near equilibrium, revealing phenomena like phase locking and drifting through averaging and Lyapunov methods.

Contribution

It introduces a stability analysis framework for Hamiltonian systems under oscillatory perturbations, highlighting bifurcations related to Lyapunov stability changes.

Findings

Identification of phase locking and drifting regimes

Bifurcation analysis of stability changes

Application of averaging and Lyapunov methods

Abstract

The effect of decaying oscillatory perturbations on autonomous Hamiltonian systems in the plane with a stable equilibrium is investigated. It is assumed that perturbations preserve the equilibrium and satisfy a resonance condition. The behaviour of the perturbed trajectories in the vicinity of the equilibrium is investigated. Depending on the structure of the perturbations, various asymptotic regimes at infinity in time are possible. In particular, a phase locking and a phase drifting can occur in the systems. The paper investigates the bifurcations associated with a change of Lyapunov stability of the equilibrium in both regimes. The proposed stability analysis is based on a combination of the averaging method and the construction of Lyapunov functions.