分段光滑微分系统的定性理论与极限环分支的若干问题

In the past several decades, piecewise smooth differential systems have being continuously appeared in various fields of science, such as mechanical engineering with imipact, electronic circuits with switches, mechanical system with dry friction, biology and so on. With the deepening knowledge of the world，the research on piecewise smooth differential systems which has its own theoretical significance and potential applications, has attracted more and more attentions in the past several decades. The qualitative and bifurcations theories in piecewise smooth differential systems have been rapidly developed in the past 30 years, and there are some important results for qualitative properties, limit cycle bifurcations and nomal form. But most theoretical studies are still preliminary, and there exist some difficult problems to be studied. In this project, we plan to further study the qualitative theory and limit cycle bifurcations for those problems, and obtain some important dynamical phenomena. The main research directions are in three aspets：(1) Limit cycle bifurcations, including the method of the Melnikov function and the averaging theory for high dimensional piecewise smooth differential systems, and the limit cycle bifurcation near a homoclinic loop with degenerate point in planar smooth differential systems. (2) Normal forms of planar piecewise smooth differential systems near high co-dimensional singularity under topological equivalence. (3) The dynamical analysis near high co-dimensional manifold of discontinuous set in piecewise smooth differential systems.

分段光滑微分系统广泛发现在发生碰撞的机械工程、伴随有开关的电路、具有干摩擦的力学系统、生物学等各领域中对工程技术或自然现象的建模中。对它的研究有着重要理论意义和应用价值。尽管近二、三十年分段光滑微分系统的定性与分支理论得到迅速的发展，其定性性质、极限环分支和正规形等方面都已得到一系列重要结果，但其理论研究仍只是初步，仍遗留很多困难的问题有待解决。本项目计划在这些方面进一步丰富和发展分段光滑微分系统的理论，获得一些重要的动力学性质，具体为：（1）极限环分支理论，其中包含高维分段光滑微分系统的 Melnikov 函数法和平均法理论，以及平面光滑微分系统在退化同宿环附近的极限环分支。（2）平面分段光滑微分系统在不连续线上高余维奇点邻域内的拓扑等价正规形。（3）分段光滑微分系统在不连续集中高余维子流形附近的动力学分析。

本项目主要研究常微分方程和动力系统的分支理论及弱化Hilbert第16问题，发展和补充了相关的数学理论。项目负责人按照申请书的研究计划开展研究，得到一些有意义的结果。主要结果发表在Nonlinearity, J. Differential Equations和J. Dynam. Differential Equations等国际核心期刊上。具体的研究结果如下：1. 解析的近哈密顿系统中在(双)同、异宿环和初等中心附近的极限环分支。在已有结果的基础上，补充了一阶Melnikov 函数在四类奇异环附近渐近展开式中低次项系数的具体表达式，并借用初等中心给出了高次项系数的一般特征。基于一阶Melnikov 函数展开式中各项系数表达式和一般特征，建立了相应的极限环分支理论。 2. 与Arnold 提出的弱化Hilbert 16问题相关的Abel积分零点个数估计。给出四类m+1次Lienard系统(3次未绕系统)相应Abel积分零点个数与多项式次数的线性估计。结论显示当未绕系统中含有双同宿环时孤立零点个数最多，且是新结果。3. 分段光滑微分系统中的极限环。结合Maple程序，利用平均法理论研究两类退化(k,l)中心扰动后产生的极限环个数。