非线性切换系统的复杂行为及其机理分析

As a class of typical hybrid system, nonlinear swtiched system has wide engineering background, the complicated dynamical behaviors as well as the mechanism of which become a key problem in the research area of dynamical and control at home and abroad. In the project, we will focus on the nonlinear switched system to investigate different types of dynamical behaviors caused by the switching between smooth subsystems with different characteristics,especially between autonomous system and periodically excited system. Based on the analysis of the steady states as well as the bifurcations of the subsystems, the mechanism of different behaviors will be explored. Specificially, the dynamics of the switched systems will be discussed with different conditions such as the cases for subsystems with different characteristics and different switching forms, especially for the cases with different steady states involving in the behaviors, high codimensional bifurcations in the subsystems, the swithing point located at the critical position and different forms of internal resonance.The influence of the factors such as the variation of the parameters, the bifurcations in the subsystems, the small disturbance of the switching on the dynamical evolution of the switched system will be investigated.The difference of the behaviors as well as the mechanism caused by different forms of switching will be discussed to explore the dynamics as well as the mechanism of the switched system. The results obtained may serve for the advancement of the theory of nonlinear switched system as well as the practical enginneering applications.

作为一类典型的混杂系统，非线性切换系统具有广泛的工程背景，其复杂动力学行为及其机理研究是当前国内外动力学与控制领域的热点课题之一。本项目围绕非线性切换系统，考察具有不同特性的光滑子系统，尤其是在自治系统和周期激励系统之间的切换导致的各种动力学行为，通过分析子系统的各种平衡态及其分岔以及切换点的动力特性，探讨不同行为的产生原因，着重考察在具有不同特性的子系统及各种切换模式，特别是在诸如多平衡态参与切换，子系统存在高余为分岔、切换点处于临界位置以及存在各种共振关系等情形下切换系统的动力学特性，探讨各种因素诸如参数变化、子系统的分岔、切换模式的微扰等等对切换系统动力学演化过程的影响，研究不同切换模式所引起的系统行为差别及其产生原因，揭示切换系统的各种动力学行为及其产生机制，为发展非线性切换系统理论及实际工程应用提供服务。

作为一类典型的混杂系统，非线性切换系统具有广泛的工程背景，其复杂动力学行为及其机理研究是当前国内外动力学与控制领域的热点课题之一。（1）基于状态变量切换模式，分别给出了两个子系统参数空间诸如fold分岔、Hopf分岔等临界条件，得到了诸如2T-focus/cycle 型周期切换振荡、 4T-focus/cycle周期切换振荡、混沌切换振荡等复杂振荡行为，并揭示了其相应的产生机理。指出切换点数目成倍增加，会导致系统由倍周期分岔序列进入混沌。同时，解释了系统存在振荡周期减少序列等现象。(2) 基于时间切换模式，通过时间切换条件定义的局部截面以及子系统决定的局部映射，构造了整个时间切换系统的Poincaré映射，并根据多重打靶法和Runge-Kutta法计算得到Poincaré映射在给定参数下的不动点。通过单参数以及双参数分岔分析，指出切换系统的各种周期运动会经由鞍-结分岔，对称破缺分岔以及倍周期分岔等各种分岔通往混沌。此外，参数周期切换Lorenz系统的对称闭轨会经由鞍-结分岔后消失直接进入混沌振荡也会经由叉型分岔后失稳新产生一对非对称的同周期闭轨，进而这对非对称的周期闭轨就会各自经由倍周期分岔演化为混沌振荡。(3) 基于时间和状态变量混合切换模式，由局部截面和局部映射建立了整个时间状态混合切换系统的Poincaré映射，指出了由于状态切换条件导致系统周期解周期未知与时间切换周期解周期已知分析时的区别，并得到了其相应雅可比矩阵的形式表达式。根据Floquet乘子从不同的方向穿过单位元，得到了混合切换系统的双参数分岔曲线，将参数空间分割成具有不同吸引子的各个部分。研究表明，系统的周期解会经由倍周期分岔演化为混沌振荡，而fold分岔连接系统的周期3轨道和混沌运动。