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极限环解析解确定平衡点吸引域的研究
引用本文:王超,张尧,夏成军,刘永强.极限环解析解确定平衡点吸引域的研究[J].华南理工大学学报(自然科学版),2008,36(6).
作者姓名:王超  张尧  夏成军  刘永强
作者单位:华南理工大学,电力学院,广东,广州,510640
基金项目:国家自然科学基金资助项目
摘    要:提出利用系统状态变量两两之间所有极限环的交集,确定高维系统在亚临界霍普夫分岔点附近平衡点吸引域的方法。首先利用改进中心流形降维的方法,对高维微分方程组在亚临界霍普夫分岔点进行降维,得到可进行极限环计算的形式;利用I.Bendxison定理推导极限环存在必要条件的解析表达式,为摄动增量法提供计算初值;然后利用摄动增量和谐波平衡法求取低维系统状态变量在分岔点附近不稳定极限环的近似解析解,用原变量替换近似解析解中的变量得到原系统变量的极限环;最后,将某一变量与其它所有变量形成的不稳定极限环投影到二维平面上取其交集,交集的边界即为该变量的稳定边界。该方法能够精确有效的分析算例中参数大幅变化下亚临界霍普夫点附近平衡点吸引域。

关 键 词:亚临界霍普夫分岔  中心流形  摄动增量法  不稳定极限环  吸引域  
收稿时间:2007-6-6
修稿时间:2007-7-23

Determination of Attraction Region of High-Dimension System Based on Analytical Solution to Unstable Limit Cycle
Wang Chao,Zhang Yao,Xia Cheng-jun,Liu Yong-qiang.Determination of Attraction Region of High-Dimension System Based on Analytical Solution to Unstable Limit Cycle[J].Journal of South China University of Technology(Natural Science Edition),2008,36(6).
Authors:Wang Chao  Zhang Yao  Xia Cheng-jun  Liu Yong-qiang
Abstract:the method to define the stable boundary of equilibrium points at the neighborhood of subcritical Hopf bifurcation point based on the intersection of all limit cycles was presented. Firstly, the method utilizes the improved center manifold method to simplify the differential equation sets of power system on the bifurcation point and works out the two-dimension differential equation sets that can be computed the approximate analytic solution. The analytic expression was deduced from the theory I. Bendixison, which supplied the initial values for computing the limit cycle. Secondly, make use of the perturbation-increment method and harmonic balance method to compute the unstable limit cycles of reduced system at the larger range of the bifurcation point. Thirdly, the variables were substituted by original variables to obtain the new sets of limit cycles. Finally, the intersection of the limit cycles projected on the plane coordinates will be as the variable’s stable region. The method was proved effective to analyze the equilibrium point’s attraction region near subcritical Hopf bifurcation point of the model.
Keywords:subcritical Hopf bifurcation  center manifold  perturbation-increment method  unstable limit cycle  attraction region
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