高维多项式微分系统Zero-Hopf分岔分析及算法推导

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

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.

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.