Damped Perturbations of Systems with Center-Saddle Bifurcation

An autonomous system of ordinary differential equations on the plane with a center-saddle bifurcation is considered. The influence of a class of time damped perturbations is investigated. The particular solutions tending to the fixed points of the limiting system are considered. The stability of these solutions is analyzed by Lyapunov function method when the bifurcation parameter of the unperturbed system takes critical and noncritical 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. The proposed theory is applied to the study of autoresonance capturing in systems with quadratically varying driving frequency.