Stability and bifurcation phenomena in asymptotically Hamiltonian systems

TL;DR

This paper studies how time-dependent, decaying perturbations affect the stability and bifurcation behavior of autonomous Hamiltonian systems with equilibrium points, revealing how such perturbations can lead to new attractors or repellers.

Contribution

It analyzes bifurcations and stability changes in asymptotically Hamiltonian systems under decaying perturbations, highlighting the influence of perturbation structure.

Findings

Bifurcations depend on the structure of decaying perturbations

Long-term stability is influenced by nonlinear and non-autonomous terms

Emergence of new attracting or repelling states in perturbed systems

Abstract

The influence of time-dependent perturbations on an autonomous Hamiltonian system with an equilibrium of center type is considered. It is assumed that the perturbations decay at infinity in time and vanish at the equilibrium of the unperturbed system. In this case the stability and the long-term behaviour of trajectories depend on nonlinear and non-autonomous terms of the equations. The paper investigates bifurcations associated with a change of Lyapunov stability of the equilibrium and the emergence of new attracting or repelling states in the perturbed asymptotically autonomous system. The dependence of bifurcations on the structure of decaying perturbations is discussed.