Hopf Bifurcation, Approximate Periodic Solutions and Multistability of Some Nonautonomous Delayed Differential Equations

数学 霍普夫分叉 分岔图 多稳态 干草叉分叉 分叉理论的生物学应用 鞍结分岔 分叉 时滞微分方程 倍周期分岔 数学分析 控制理论(社会学) 范德波尔振荡器 跨临界分岔 应用数学 微分方程 非线性系统 物理 计算机科学 控制(管理) 量子力学 人工智能
作者
Wenxin Zhang,Lijun Pei,Yueli Chen
出处
期刊:International Journal of Bifurcation and Chaos [World Scientific]
卷期号:33 (11) 被引量:1
标识
DOI:10.1142/s0218127423501274
摘要

Research on nonautonomous delayed differential equations (DDEs) is crucial and very difficult due to nonautonomy and time delay in many fields. The main work of the present paper is to discuss complex dynamics of nonautonomous DDEs, such as Hopf bifurcation, periodic solutions and multistability. We consider three examples of nonautonomous DDEs with time-varying coefficients: a harmonically forced Duffing oscillator with time delayed state feedback and periodic disturbance, generalized van der Pol oscillator with delayed displacement difference feedback and periodic disturbance, and an electro-mechanical system with delayed and periodic disturbance. Firstly, we obtain the amplitude equations of these three examples by the method of multiple scales (MMS), and then analyze the stability of approximate solutions by the Routh–Hurwitz criterion. The obtained amplitude equations are used to construct the bifurcation diagrams, so that we can observe the occurrence of the Hopf bifurcation and judge its type (super- or subcritical) from the bifurcation diagrams. We discover rich dynamic phenomena of the three systems under consideration, such as Hopf bifurcation, quasi-periodic solutions and the coexistence of multiple stable solutions, and then discuss the impact of some parameter changes on the system dynamics. Finally, we validate the correctness of these theoretical conclusions by software WinPP, and the numerical simulations are consistent with our theoretical findings. Therefore, the MMS we use to analyze the dynamics of nonautonomous DDEs is effective, which is of great significance to the research of nonautonomous DDEs in many fields.
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