几类分片光滑动力系统的极限环分支

In many disciplines, such as electronics, control theory, mechanical systems with dry frictions and so on, rich dynamical models are described by piecewise smooth differential systems. In the known results their dynamics has been discussed from different points of view, including limit cycle bifurcations. This project will carry out the following studies on bifurcation of limit cycles for several kinds of piecewise differential systems with important theoretical significance. Firstly, for a planar piecewise smooth system with a focus of Fold-Fold type, we consider Hopf bifurcation near the focus by establishing a succession function. Secondly, we suppose that a planar piecewise smooth Hamiltonian system has a family of periodic orbits near a homoclinic loop with a singular point of Saddle-Fold type. Under small perturbations, we study limit cycle bifurcations from the closed orbits by using the first Melnikov function. Thirdly, for two kinds of heteroclinic loops with a saddle and Fold-Regular points in a planar piecewise smooth system, we first present some stability criteria for the heteroclinic loops. Then, based on the theory of stability-changing, we discuss limit cycle bifurcations from the heteroclinic loops. At last, under polynomial perturbations, we consider limit cycle bifurcations from parallel flows on an invariant cone of a piecewise smooth linear three-dimensional system, and discuss the least upper bound of the number of limit cycles appearing on the cone by using the first averaging method. From discussion above, we can obtain new dynamical properties for piecewise smooth systems.

分片光滑系统可描述许多学科中的动力学模型，如电子学、控制论、具干摩擦的力学系统等。现有成果已从不同角度讨论了其动力学行为，其中包括极限环分支。本项目针对几类具有一定理论意义的分片光滑系统，开展如下四个方面的极限环分支问题的研究：(1) 对于具Fold-Fold型焦点的平面分片光滑系统，通过建立焦点附近的后继函数，研究其Hopf分支；(2) 假设一平面分片光滑哈密顿系统有一闭轨族，位于具Saddle-Fold型奇点的同宿环附近，应用一阶Melnikov函数法，研究这族闭轨分支出极限环个数；(3) 对于具双曲鞍点和Fold-Regular奇点的平面分片光滑一般系统的两类异宿环，给出其稳定性的判别量，进而研究其极限环分支；(4) 应用一阶平均法，研究一类三维分片光滑线性系统，在多项式扰动下，不变锥上平行流分支出极限环的最大个数。通过上述问题的研究，可得到分片光滑系统新的动力学性质。

本项目主要研究了几类光滑和非光滑系统的极限环分支问题，完成了预定目标。项目组已公开发表标注基金资助的论文11篇，另有1篇已被接收，其中SCI论文8篇；项目在研期间，有6名硕士生毕业并取得硕士学位，现有在读硕士生4名。. 主要研究成果如下：(1) Hopf分支 本项目研究了一类较有代表性的φ-Laplacian算子的Liénard 系统的Hopf分支，给出了与普通焦点量等价的两组焦点量，分析了细焦点分支和中心的扰动分支，以及4个具体系统的Hopf环性数。对于零次4区域系统，得到了n次多项式扰动下分支出极限环个数的上、下界。(2)分片光滑近哈密顿系统一阶Melnikov函数方法的推广及应用 本项目研究了具Saddle-Fold等类型奇点的广义同宿分支和以折线为分界线闭轨族分支，给出了对应系统一阶Melnikov函数展开式和前面若干项的系数公式，得到了以直线为分界线分片光滑二次系统同宿环附近可分支出5个极限环，与光滑系统相比多得到3个极限环。 证明了一类函数组为Chebyshev系统，进而改进了已有文献中相关问题闭轨族分支出极限环个数的上界。(3) 具奇直线的平面多项式系统的多参数分支 对于n次多项式扰动下的具一条奇直线的二次系统和两条奇直线的三次系统，改进了已有文献的关于此类系统极限环个数的下界，分别多得到了1个极限环。（4）高维系统固定时间同步和周期解的存在性 对具混合时滞的非连续Cohen-Grossberg中立型神经网络的固定时间同步问题，给出了一个新的非连续反馈控制方法，得到了使得所研究系统固定时间同步的一些充分条件。对于具奇性Rayleigh系统，得到的周期解的存在条件改进和完善了已有相关结果。