分段光滑系统中的幂零奇点及其分支

Bifurcation theory in piecewise smooth system has become one of the hot issues in dynamic behavior analysis of differential equations. In this project, we will study the bifurcation from a generalized nilpotent singular point in piecewise smooth system. Firstly, under the method of versal unfolding in the planar polynomial system, we study the Bogdanov-Takens bifurcation of the generalized cusp under a small perturbation in piecewise polynomial system. Secondly, by means of ingenious variable transformation and calculus techniques, we obtain the generalized Lyapunov constants and improve the present results in case that the singular point on the switching line is a generalized nilpotent center or focus. Finally, using the expression of Melnikov function in piecewise smooth system and the computational software, we derive its expansion and obtain all the possible simple zeros through changing the sign of dependent parameters. Meanwhile, we can have a better result concerning the number of limit cycles. What’s more, this work is an exploratory research on Bogdanov-Takens bifurcation in piecewise smooth system, and deeply investigate the degenerate Hopf bifurcation and the number of limit cycles. In conclusion, this work can enrich the bifurcation theory in piecewise smooth system and has the high academic value.

分段光滑系统的分支问题是近年来微分方程动力学行为分析领域的热点问题。本项目研究分段光滑系统中广义幂零奇点所出现的分支现象。首先，针对分段多项式系统中的广义尖点，利用平面多项式系统中普适开折的方法，研究广义尖点在小扰动下的Bogdanov-Takens分支现象。其次对于在切换轴上的广义幂零中心或焦点，结合更加巧妙的变量变换、微积分技巧求广义Lyapunov常数，改进现有的结果。最后利用分段Hamiltonian系统中Melnikov函数表达式，借助计算软件求得Melnikov展开式，通过改变自由参数符号得到尽可能多的Melnikov函数的单根，从而改进极限环存在个数的现有结果。本项目是对分段光滑系统中Bogdanov-Takens分支现象的探索，也是对退化Hopf分支、极限环个数问题的进一步深入研究，可以丰富分段光滑系统的分支理论，具有一定的理论价值。

分段光滑系统的分支问题是近年来微分方程动力学行为分析领域的热点问题之一。本项目主要研究分段光滑系统中所出现的分支问题。首先，通过计算焦点量和周期常数，研究切换系统中的中心及无穷远处的极限环分支，得到相较于光滑系统更多个数的极限环。其次利用分段Hamiltonian系统中Melnikov函数表达式，利用计算软件辅助求得Melnikov展开式，通过改变自由参数符号得到尽可能多的Melnikov函数的单根，从而改进现有的关于极限环可存在个数问题的结果。最后因为Z2-等变系统的特殊性，考虑其中的等时中心问题，通过适当的线性转换改进现有结果。本项目的主要研究工作是对分段光滑系统中新型分支现象的探索，也是对极限环个数问题的进一步深入研究，具有一定的理论价值和研究意义。