作　　者： ; ; ; ;

机构地区： 华南理工大学电力学院

出　　处： 《华南理工大学学报（自然科学版）》 2008年第6期133-137,共5页

摘　　要： 提出了一种在亚临界霍普夫分岔点附近,利用系统状态变量之间的不稳定极限环的交集确定高维系统吸引域的方法.首先,利用改进中心流形降维的方法,在亚临界霍普夫分岔点对高维微分方程组进行降维,得到符合计算极限环要求的数学表达式;其次,利用I.Bendixson定理推导极限环存在的必要条件,为计算提供初值;然后,采用摄动增量法和谐波平衡法求解降维系统在该分岔点附近的不稳定极限环的近似解析解,通过变量变换得到原系统的极限环;最后,将与某一变量相关的不稳定极限环投影到二维平面上,交集即为该变量的稳定区间.利用该方法可在参数于亚临界霍普夫分岔点大幅度变化时,精确分析算例系统中一类平衡点的吸引域. 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.

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

领　　域： [电气工程] [理学] [理学]