组合激励下振荡系统中自谐振模式的分岔
Bifurcations of autoresonant modes in oscillating systems with combined excitation
AI总结：
本文研究组合参数与外周期缓变激励下非线性系统进入自谐振的捕获模型，分析激励参数过分岔值时不同自谐振模式的稳定性，并利用改进平均化方法与Lyapunov函数给出长时间渐近近似。
摘要：
本文考虑了一个数学模型，该模型描述非线性系统在参数激励与外部周期缓变扰动共同作用下被捕获进入自谐振的过程。自谐振现象与具有无限增长振幅和有限相位失配的解相关。本文研究了当激励参数经过分岔值时这类解的行为。特别地，分析了不同自谐振模式的稳定性，并通过一种改进的平均化方法，结合所构造的Lyapunov函数，提出了自谐振解在渐近长时间区间上的渐近近似。
英文摘要：
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.
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