非光滑Filippov系统的局部和全局动力学行为研究

The bifurcation analysis as well as the complicity of Filippov systems is one of the hot and leading topics in the research area of nonlinear dynamics at home and abroad for the time being. The local and global behaviors of the Filippov systems with different dimension as well as different types of switching boundaries will be investigated in the project. The critical conditions, classifications as well as the normal forms of the local non-conventional bifurcations with codimension-1 and codimension-2 of boundary equilibrium points and the pseudo-equilibrium points under different types of tangent points will be discussed in details. The behaviors of different types of limit cycles on the boundaries will be analyzed to present the different bifurcation forms corresponding to the situations when the limit cycles move to or leave from the boundaries. The topological structures of the non-conventional bifurcations will be presented by employing the perturbation analysis of the unfolding parameters, which will be used to explore the mechanism of the bifurcations as well as their influence on the dynamics of the systems. Furthermore, the critical conditions for the existence of different types of homo-clinic orbit and the hetero-clinic orbit connecting equilibrium points will be presented, the topological structures of which under the perturbation of unfolding parameters will investigated. Meanwhile, the topological structures of the limit cycles for the condition when the Floquet multiplier, especially more than one Floquet multiplier, passes across the unit circle will be presented. Different routes to chaos will be obtained. The results of the project have not only scientific meaning to non-smooth bifurcation theory, but also significance to the practical applications in real systems, such as the model recognition, failure diagnosis, structure optimization as well as control.

Filippov系统的分岔及其复杂性研究是当前国内外非线性动力学领域内的热点和前沿课题之一。本项目深入分析不同维数Filippov系统在各种切换面下的局部和全局行为，研究不同切点性质下的边界平衡点和伪平衡点的余维一和余维二局部非常规分岔的临界条件、分类及其标准型，探讨各类极限环在滑动面上的行为，给出极限环进入滑动面或滑动部分离开滑动面的不同分岔模式，得到开折参数扰动下的不同分岔特性，揭示其相应的分岔机理及对系统整体动力学行为的影响。进而探讨连接不同形式和性质平衡点各种同、异宿轨道存在的临界条件，得到开折参数扰动下这些轨道的拓扑结构，同时，给出极限环相应Floquet乘子穿越单位圆，尤其是多次穿越时的拓扑结构，揭示系统通往混沌的不同道路。本项目的工作，对于发展非光滑分岔理论具有重要的科学意义，同时，对于实际非光滑工程系统的应用，如模型设别、故障诊断、结构优化及其控制等等也具有重要的应用价值。

本项目研究不同维数Filippov系统在各种切换面下的局部和全局行为，研究不同切点性质下的边界平衡点和伪平衡点的余维一和余维二局部非常规分岔的临界条件、分类及其标准型，探讨各类极限环在滑动面上的行为，给出极限环进入滑动面或滑动部分离开滑动面的不同分岔模式，得到开折参数扰动下的不同分岔特性，揭示其相应的分岔机理及对系统整体动力学行为的影响。进而探讨连接不同形式和性质平衡点各种同、异宿轨道存在的临界条件，得到开折参数扰动下这些轨道的拓扑结构，同时，给出极限环相应Floquet乘子穿越单位圆，尤其是多次穿越时的拓扑结构，揭示系统通往混沌的不同道路。对于平面Filippov系统及三维、四维Filippov系统，给出不同切点性质下的各种余维一和余维二局部分岔的临界条件，得到其相应的分岔行为及其机制，揭示不同分岔模式对系统动力学行为的影响规律。对于非光滑切换面，给出从常规和非常规分岔出来的极限环在滑动面上的行为，得到极限环进入滑动面或滑动部分离开滑动面的不同形式，揭示各种分岔特性及其对系统整体行为的影响。给出不同维数Filippov系统连接平衡点（包括真平衡点、边界平衡点及伪平衡点）的各种同、异宿轨道存在的临界条件，得到各种参数扰动下这些轨道的拓扑结构，得到了不同切换面下各种非常规分岔的存在条件，给出轨迹在不同切换面交点处存在分岔时的各种分岔行为，得到多参数扰动下系统的拓扑结构，给出不同切换面上非常规分岔之间的相互作用及其对系统行为的影响.本项目的工作，对于发展非光滑分岔理论具有重要的科学意义，同时，对于实际非光滑工程系统的应用，如模型设别、故障诊断、结构优化及其控制等等也具有重要的应用价值。