关键词: 亚临界霍普夫分岔, 不稳定极限环, 解析解, 吸引域, 中心流形, 摄动增量法, 谐波平衡法

Abstract:

This paper proposes a new method to determine the attraction region of the high-dimension system by using the intersection of the unstable limit cycles among system state variables near the subcritical Hopf bifurcation point. In this method, first, an improved center manifold method is used to reduce the dimension of the high-dimension differential equation sets at the subcritical Hopf bifurcation point, thus obtaining an appropriate mathematic representation meeting the requirements of limit cycle computation. Next, the necessary condition for the existence of limit cycle is deduced based on the I. Bendixson theory, which supplies the initial values for the computation. Then, the perturbation-increment method and the harmonic balance method are both adopted to solve the approximate analytical solution to the unstable limit cycles of the dimension-reduced system near the bifurcation point, and the limit cycle of the original system is obtained via variable transformation. Finally, the unstable limit cycles related to a variable are projected on a two-dimension plane, the intersection being the stable region of the variable. It is found that the proposed method helps to accurately analyze the attraction region of a class of equili-brium points of the system when the parameter greatly changes at the subcritical Hopf bifurcation point.

Key words: subcritical Hopf bifurcation, unstable limit cycle, analytical solution, attraction region, center manifold, perturbation-increment method, harmonic balance method