A mathematical model describing the initial stage of the capture into autoresonance for nonlinear oscillating systems with combined parametric and external excitation is considered. The solutions with unboundedly growing amplitude and limited phase mismatch correspond to the autoresonant capture. The paper investigates the existence, stability and bifurcations of such solutions in the presence of a weak dissipation in the system. Our technique is based on the study of particular solutions with power-law asymptotics at infinity and the construction of suitable Lyapunov functions.

中文翻译：

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结合激励和弱耗散的振动系统的自谐振

考虑了一个数学模型，该模型描述了结合参数和外部激励的非线性振荡系统捕获到自共振的初始阶段。具有无限增长的幅度和有限的相位失配的解对应于自谐振捕获。本文研究了在系统中存在弱耗散的情况下此类解决方案的存在性，稳定性和分支性。我们的技术基于对无穷大幂律渐近的特定解的研究以及适当Lyapunov函数的构造。