A mathematical model describing the capture of nonlinear systems into the autoresonance by a combined parametric and external periodic slowly varying perturbation is considered. The autoresonance phenomenon is associated with solutions having an unboundedly growing amplitude and a limited phase mismatch. The paper investigates the behaviour of such solutions when the parameters of the excitation pass through bifurcation values. In particular, the stability of different autoresonant modes is analyzed and the asymptotic approximations of autoresonant solutions on asymptotically long time intervals are proposed by a modified averaging method with using the constructed Lyapunov functions.

中文翻译：

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组合激励振荡系统中自谐振模式的分岔

考虑了一个数学模型，该模型描述了通过参数化和外部周期性慢变扰动的组合将非线性系统捕获到自谐振中。自谐振现象与具有无限增长幅度和有限相位失配的解决方案有关。该论文研究了当激励参数通过分岔值时这种解的行为。特别地，分析了不同自谐振模式的稳定性，并通过使用构造的李雅普诺夫函数的改进平均方法提出了渐近长时间间隔上自谐振解的渐近近似。