高频激励下的次谐波同步与极限环调制：重整化群方法
Subharmonic entrainment and limit cycle modulation by high frequency excitation: A Renormalization group approach
AI总结：
本文利用 Blekhman 微扰法和重整化群技术，研究 van der Pol-Duffing 振荡器在快慢双频激励下，以高频信号振幅为独立控制参数实现次谐波同步与极限环调控，并给出阈值条件，结果经数值模拟验证。
摘要：
本文探讨了在同时受到慢参数驱动和快外部强迫两种频率激励的 van der Pol-Duffing 振荡器中，通过改变外部快信号的振幅，实现次谐波（1:2）同步和超临界 Hopf 分岔的可能性。我们还推导了产生次谐波振荡所需的阈值参数强度条件。采用 Blekhman 微扰法（运动的直接分解）和重整化群技术，研究了信号振幅如何在调制极限环动力学以及次谐波产生中发挥关键作用。对此类受驱动非线性系统的非线性响应和分岔的研究通常将快驱动强度作为控制参数。本文表明，高频信号的振幅除了能够使慢、快分量得到良好分离从而便于研究动力学之外，还可以作为一个独立的控制参数，用于同时控制振荡器中的极限环行为和次谐波振荡的起始。我们的解析估计得到了数值模拟的有力支持。
英文摘要：
In this article, we explore the possibility of a sub-harmonic $(1{:}2)$ entrainment and supercritical Hopf bifurcation in a van der Pol-Duffing oscillator that has been excited by two frequencies, comprising a slow parametric drive and a fast external forcing, through the variation of the amplitude of the external fast signal. We also deduce the condition for the threshold parametric strength required to generate sub-harmonic oscillation. The Blekhman perturbation (direct partition of motion) and the Renormalization group technique have been employed to study how the signal amplitude plays a pivotal role in modulating the limit cycle dynamics as well as the subharmonic generation. Studies of nonlinear responses and bifurcations of such driven nonlinear systems are usually done by treating the strength of the fast drive as the control parameter. Here we show that, beyond its role in allowing one to study the dynamics with the slow and fast components nicely separated, the amplitude of the high-frequency signal can also be treated as an independent control parameter for controlling both the limit cycle behavior and the onset of subharmonic oscillation in the oscillator. Our analytical estimations are well supported by numerical simulations.
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