高维多项式微分系统Zero-Hopf分岔分析及算法推导

doi:10.12341/jssms21399

系统科学与数学 ›› 2021, Vol. 41 ›› Issue (12): 3280-3298.DOI: 10.12341/jssms21399

高维多项式微分系统Zero-Hopf分岔分析及算法推导

黄博, 韩德仁

北京航空航天大学数学科学学院, 信息与行为教育部重点实验室, 北京 100191

收稿日期:2021-07-23 修回日期:2021-12-02 出版日期:2021-12-25 发布日期:2022-03-16
基金资助:
国家自然科学基金（12101032，11625105，12131004）资助课题.

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黄博, 韩德仁. 高维多项式微分系统Zero-Hopf分岔分析及算法推导[J]. 系统科学与数学, 2021, 41(12): 3280-3298.

HUANG Bo, HAN Deren. Analysis of Zero-Hopf Bifurcation in High Dimensional Polynomial Differential Systems with Algorithm Derivation[J]. Journal of Systems Science and Mathematical Sciences, 2021, 41(12): 3280-3298.

Analysis of Zero-Hopf Bifurcation in High Dimensional Polynomial Differential Systems with Algorithm Derivation

HUANG Bo, HAN Deren

LMIB, School of Mathematical Sciences, Beihang University, Beijing 100191

Received:2021-07-23 Revised:2021-12-02 Online:2021-12-25 Published:2022-03-16

针对高维多项式微分系统的Zero-Hopf分岔进行分析.首先，我们将分岔分析问题约化为代数问题，并基于半代数系统求解的符号算法给出微分系统存在~Zero-Hopf~分岔点的判定方法.然后，基于二阶平均方法推导出微分系统Zero-Hopf分岔分析的算法框架，并利用符号计算方法通过具体算例开展了极限环分岔研究，得到了一些新结果.最后提出几个相关的研究问题.

关键词： 微分系统, 分岔分析, 半代数系统, 符号计算, 平均方法

This paper deals with the Zero-Hopf bifurcation in high dimensional polynomial differential systems. First, we reduce the problem of bifurcation analysis to an algebraic problem, and we give a method for determining the bifurcation set of the Zero-Hopf bifurcation points of differential systems by using symbolic algorithm for solving semi-algebraic systems. Then, based on the second order averaging method, the algorithmic framework of the Zero-Hopf bifurcation analysis of differential systems is derived, and the limit cycle bifurcation problem is studied through specific examples by using the methods of symbolic computation, and some new results are obtained. Finally, we propose several related research problems.

Key words: Differential systems, bifurcation analysis, semi-algebraic systems, symbolic computation, averaging method

MR(2010)主题分类:

34C07
92D25

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