Bautin系统的Lyapunov量复算法

Complex Algorithm of Lyapunov Values for Bautin System

Lyapunov量(及与之等价的焦点量)在平面向量场的定性理论和分岔理论中占有非常的地位对研究微分方程的稳定性有重要作用;它是判断原点是否为细焦点或中心的一种经典手段;也可以用来判断由退化Hopf分岔所产生的极限环个数,与著名的Hilbert第16问题有密切的关系。Lyapunov量复算法是得到焦点的一种好方法。本文主要研究Bautin系统的Lyapunov量复算法。借助于计算工具Maple数学软件,运用Lyapunov量复算法计算了这一系统的Lyapunov量,并证明了细焦点的阶数最高为3。本文的研究所具有的优点是采用有效简捷的算法,给出相关结果的新的证明。本文结果可用于该系统在原点的极限环个数的判定,对该系统的极限环分岔研究有重要的理论指导意义。

Since Lyapunov values（or equivalent focal values） have very important status in qualitative theory and bifurcation theory of plane vector fields. The research on Lyapunov values plays a vital role in stability of differential equation. It is one classical approach to determining the origin whether for a weak focus or a center. The focal values can judge the number of limit cycles occurring in degenerate Hopf bifurcation which has the close relation to the famous Hilbert 16th problem. The complex algorithm of Lyapunov values is a good method for obtaining Lyapunov values. The complex algorithm of Lyapunov values for Bautin system was mainly investigated. With the aid of the computational tool - Maple mathematical software, the Lyapunov values for this system were computed by using the complex algorithm of Lyapunov values under given program. It also provs that the highest order of fine focus is 3. The research merit is that the new proofs about relevant results are given by utilizing effective and simple algorithm. The results can judge the number of limit cycles occurring at origin and have significant guiding role to research the bifurcations of limit cycles of this system.