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.

中文翻译：

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中心鞍分岔系统的阻尼扰动

考虑平面上具有中心-鞍分岔的常微分方程的自治系统。研究了一类时间阻尼扰动的影响。考虑了趋于限制系统不动点的特定解。当未扰动系统的分岔参数取临界值和非临界值时，采用李雅普诺夫函数法分析这些解的稳定性。描述了确保扰动系统中分叉持续存在的条件。当分岔被打破时，在临界情况下会出现一对趋于极限系统退化不动点的解。结果表明，根据扰动的结构和参数，这些解决方案之一可以是稳定的、亚稳的或不稳定的，而另一种解决方案总是不稳定的。所提出的理论应用于研究具有二次变化驱动频率的系统中的自谐振捕获。