Damped perturbations of systems with centre-saddle bifurcation

TL;DR

This paper studies how time-damped power-law perturbations affect the stability and bifurcation structure of a planar system with a centre-saddle bifurcation, revealing conditions for bifurcation persistence and solution stability.

Contribution

It provides new conditions for the persistence of bifurcations under damped power-law perturbations and analyzes the stability of solutions near degenerate fixed points.

Findings

Conditions for bifurcation persistence under perturbations

Existence of solutions tending to fixed points at infinity

Stability classification of solutions depending on perturbation parameters

Abstract

An autonomous system of ordinary differential equations in the plane with a centre-saddle bifurcation is considered. The influence of time damped perturbations with power-law asymptotics is investigated. The particular solutions tending at infinity to the fixed points of the limiting system are considered. The stability of these solutions is analyzed when the bifurcation parameter of the unperturbed system takes critical and non-critical values. Conditions that ensure the persistence of the bifurcation in the perturbed system are described. When the bifurcation is broken, a pair of solutions tending to a degenerate fixed point of the limiting system appears in the critical case. It is shown that, depending on the structure and the parameters of the perturbations, one of these solutions can be stable, metastable or unstable, while the other solution is always unstable.