摘要
在 Stuart- L andau系统中 ,通过系统每个变量到自身的时滞反馈 ,建立 Stuart- L andau时滞模型 ,研究时滞和反馈增益对该系统联合作用的影响规律。确定在时滞和反馈增益系数两参数表明的空间中系统平凡解的线性稳定性条件 ,利用 Hopf分岔定理得到系统出现 1∶ 2双 Hopf分岔的充分必要条件。借助中心流形和规范型方法 ,将系统约化到四维中心流形。从理论上预测由时滞和反馈增益导致的双 Hopf分岔点附近的动力学行为 ,得到双Hopf分岔引起的各种不同拓扑结构的周期解的解析形式 ,数值模拟与理论分析结果完全一致。结果表明 :时滞和反馈增益不仅可以使系统的运动进入所谓的“静默区”,而且可以导致非共振双 Hopf分岔和它产生的不同拓扑结构的周期运动和多稳态周期运动。
A Stuart-Landau system with delayed state feedback is proposed to investigate effects of the time delay and feedback gain on the system. Its local linear stability for trivial equilibrium is analyzed and critical condition is represented in two parameters, namely, time delay and feedback gain. Moreover, sufficient and necessary condition for a 1:√2 double Hopf bifurcation occurring in the system is obtained by using Hopf bifurcation theorem. To classify various bifurcating solutions derived from the double Hopf bifurcation, the system is reduced on a 4-dimensional center manifold. These solutions with the distinct topological structures are expressed in a closed form and the dynamics induced by time delay and gain is predicted analytically. The analytical results are in good agreement with those from numerical simulation. This confirms the analytical prediction. The obtained results show that the time delay and feedback gain may not only lead to the called death island, but also to double Hopf bifurcation from which various periodic motions with distinct topological structures occur in the system under consideration. These results have some potential applications, such as controlling vibration and synchronization of the system.
出处
《振动工程学报》
EI
CSCD
北大核心
2005年第1期24-29,共6页
Journal of Vibration Engineering
基金
国家自然科学基金资助项目 (0 4 72 0 83)
