分段光滑系统的非常规分岔研究

分段光滑系统中非常规分岔的动力特性是当前国内外动力学研究领域内的热点课题之一。本项目利用微分包含理论，深入分析分段光滑系统边界平衡点和伪平衡点在不同切点性质下的各种余维一和余维二分岔临界条件、分类、标准型及其开折，揭示其相应的分岔特性，进而考察极限环各种分岔行为的分类，给出其相应标准型及其开折参数下的拓扑结构，同时，得到鞍点同、异宿轨道，伪鞍结点同、异宿轨道，伪鞍点同、异宿轨道的存在条件，分析参数的影响，给出不同轨道的拓扑结构，并按照拓扑结构进行分类，揭示其相应的参数空间中的不同分岔行为及其对分段光滑系统其复杂性道路的影响。考察不同维数和单、多切换面下分段光滑系统的各种非常规局部分岔和全局分岔的拓扑结构和分岔特性，分析比较其与平面非光滑系统中相应非常规分岔之间的区别，研究随参数变化系统的动力学行为，揭示各种非光滑分岔对系统行为的影响及其动力学演化本质，发展分段光滑系统分岔理论。

The dynamics of non-conventional bifurcations in piecewise smooth system is one of the hot topics in the area of nonlinear systems at home and abroad. By employing the differential inclusion theory, the co-dimension one and co-dimension two bifurcations of the boundary equilibrium points and the pseudo equilibrium points under different properties of the tangency will be investigated in the project. Critical conditions, classifications, normal forms as well as the universal unfolding forms of the bifurcations will be presented, based on which the characteristics of the bifurcation will be obtained. The classifications of the bifurcations of limit cycles will be discussed to investigate the corresponding normal forms and the topological structures with the disturbance of the parameters. Meanwhile, the existence conditions for homo-clinic and hetero-clinic orbits related to saddle, pseudo saddle-node and pseudo saddle will be derived. The topological structures will be obtained with the variation of parameters, which are used to classify the types of the non-conventional global bifurcations. The influence of the bifurcations on the dynamics of piecewise smooth system will be analyzed, which reveals the complex route of the system. Furthermore, The bifurcations for the cases with different dimension of the system and transitions with single and multiple boundaries will be explored. The difference of the non-smooth bifurcations between the planar vector field and higher dimensional system will be presented. Upon the analysis of the behaviors with the variation of the parameters, the influence of non-smooth bifurcations on the dynamics and the properties of the dynamical evolutions will be explored. The project may help to the development of bifurcation theory of piecewise smooth system.