平面分段光滑Liénard微分系统的动力学分析

分段光滑微分系统的动力学分析是微分方程定性理论研究中的热点问题之一。与光滑微分系统相比，分段光滑微分系统的动力学研究无论从方法还是理论上都还很不完善。.本项目研究平面分段光滑Liénard微分系统的动力学行为。内容主要包括：1、研究分段线性Liénard微分系统的极限环个数，有限奇点和无穷远点的定性性质，进而给出系统的全局相图。 2、讨论分段光滑Liénard微分系统的Hopf分支和Poincaré分支，得到此类系统的极限环最大个数，从而补充和改进现有结果。3、推导分段光滑Liénard快慢系统的焦点量和慢发散量积分公式，为探讨鸭极限环的环性问题提供理论工具。.分段光滑微分系统具有丰富的动力学行为，其极限环分支问题与弱化Hilbert第16问题密切相关。此外，分段光滑Liénard微分系统广泛存在于电气工程、自动控制、种群生物学领域。因此本项目的研究有着重要的理论意义、学术价值和应用前景。

Dynamic analysis of piecewise smooth differential systems is one of the hot topics in qualitative theory of differential equations. Compared with the smooth differential systems, dynamic research of piecewise smooth differential systems is far from perfect both in method and theory..This project investigates the dynamical behaviors of planar piecewise smooth Liénard differential systems.The contributions of this project are presented as follows: 1、Consider the number of limit cycles of piecewise linear Liénard differential systems, discuss the qualitative properties of finite and infinite equilibrium, thus we can obtain the global phase portraits of these systems. 2、Study the maximun number of limit cycles for piecewise smooth Liénard differential systems via the Hopf bifurcation and Poincaré bifurcation, then supplement and improve the known results. 3、Deduce the focal values and slow divergence integral of piecewise smooth Liénard slow-fast systems, which can be used to studing the cyclicity of canard limit cycles..Piecewise smooth differential systems have abundant dynamical behaviors, and bifurcation of limit cycles is closely related to weak Hilbert's 16th problem. In addition, piecewise smooth Liénard differential systems are widely used in electrical engineering、automatic control、mathematical biology and other fields. Therefore, the research of this project has important theoretical significance, academic value and application prospects.