平面切换微分系统的正规形及分岔

项目名称： 平面切换微分系统的正规形及分岔

项目编号： No.11471228

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 陈兴武

作者单位： 四川大学

项目金额： 68万元

中文摘要： 正规形是平面微分系统的最本质形式，其理论在微分系统的分岔及定性研究中极为重要。随着实际中越来越丰富的非光滑系统分岔现象的发现，大量工作从光滑系统转入到非光滑系统。相对光滑系统的正规形理论，切换微分系统等非光滑系统的正规形理论还是空白。本项目将对平面切换微分系统的正规形进行探索。由于系统中切换线、各切换区域内不同向量场的存在以及系统的不连续性，光滑微分系统中的经典方法已不适于用来获取切换系统的正规形。特别地，切换线上的不同向量场必须具有的相容性给正规形的获取带来极大困难。我们希望对切换系统量身构造一类分区域光滑的平面映射以克服系统的切换及向量场间的相容要求所产生的困难，获得平面切换微分系统的正规形。进一步，我们将利用正规形中的共振项定义一列广义李雅普诺夫量和一列广义周期量以研究平面非双曲切换微分系统的Hopf分岔和临界周期分岔，并给出一种突破经典方法中对这些量无关性要求的新方法。

中文关键词： 正规形；极限环；分岔；中心；非光滑微分系统

英文摘要： Normal forms are the essential forms of planar differential systems. Its theory is very important to the study of bifurcations and qualitative theory of differential systems. Since more and more plentiful bifurcation phenomena are found in non-smooth differential systems, more and more works are turned to non-smooth systems from the smooth case. Compared with the normal form theory of smooth systems, that of non-smooth systems such as switching differential ststems is still lacking. In this project we study normal forms of planar switching differential systems. Since there exist switching lines in the phase space and different vector fields in different regions, the classic methods of smooth systems are invalid for non-smooth ones. Specially, some great difficulty are caused by the required compability of all vector fields in all regions. In order to obtain the normal form, we hopt to construct a kind of planar smooth transformations in all switching regions to overcome such difficulty. Further, using resonance terms of normal forms, we define generalized Lyapunov quantities and generalized period ones to study Hopf bifurcations and critical period bifurcations of planar switching differential systems. Moreover, we give a new method, which is a breakthrough in the independence requirement of quantities for smooth systems, to determine the numbers of limit cycles and critical periods for planar non-hyperbolic switching differential systems.

英文关键词： normal form;limit cycle;bifurcation;center;non-smooth differential system