Kopel系统中Neimark-Sacker分岔的不变曲线  

Invariant Curve of Neimark-Sacker Bifurcation in the Kopel System

作　　者：袁利国[1,2] 魏周超 郭军[1] Yuan Liguo;Wei Zhouchao;Guo Jun(College of Mathematics and Informatics,South China Agricultural University,Guangzhou 510642,China;School of Mathematical Sciences,Anhui University,Hefei 230601,China;School of Mathematics and Physics,China University of Geosciences(Wuhan),Wuhan 430074,China)

机构地区：[1]华南农业大学数学与信息学院,广州510642 [2]安徽大学数学科学学院,合肥230601 [3]中国地质大学(武汉)数学与物理学院,武汉430074

出　　处：《动力学与控制学报》2023年第1期81-88,共8页Journal of Dynamics and Control

基　　金：国家自然科学基金资助项目(12172340,11772306);天津市自然科学基金重点项目(19JCZDJC32300)。

摘　　要：首先阐述了Kopel系统的复杂动力学的研究进展.再基于分岔与规范型理论,给出了Kopel系统的Neimark-Sacker(N-S)分岔的一类新证明,并得到N-S分岔所产生不变曲线的近似表达式.最后,对N-S分岔及不变曲线进行了数值模拟分析,验证了理论推导所得结果,并对N-S分岔中的扰动参数与不变曲线的近似表达式中参数之间的相互影响进行了分析.所有这些分析对Kopel系统的已有动力学研究是一个补充,对其经济现象的内在规律提供了理论支撑.Firstly,research progress of complex dynamics of the Kopel system is described.Based on the bifurcation and normal form theory,a new proof of Neimark-Sacker(N-S)bifurcation of Kopel system is given,and approximate expression of the invariant curve generated by N-S bifurcation is obtained.Finally,the N-S bifurcation and invariant curve are numerically simulated,which verifies the theoretical results,and the mutual effect between the perturbation parameters in N-S bifurcation and the parameters in the approximate expression of invariant curve is analyzed.All analyses are supplement to the existing research of the Kopel system and provide a theoretical support for the intrinsic law of economic phenomenon.

关 键 词：Kopel系统 NEIMARK-SACKER分岔 不变曲线 不动点 

分 类 号：O193[理学—数学] O317[理学—基础数学] O322