非光滑连续系统的簇发振荡模式及其分岔机理研究

As the non-smooth dynamical systems are becoming more and more important in engineering fields, they draw attention of many scientific workers and engineers. The non-smooth dynamical systems have been one of the hot spot of nonlinear dynamics study in recent years. This subject will study the bursting behavior and the associated induced mechanisms in non-smooth continuous systems based on non-smooth bifurcation analysis. Its main contents are presented as follows: In the autonomous case, bursting of point-point type and bursting of point-cycle type will be investigated in this subject. The bifurcation of a rest state leading to repetitive spiking (the bifurcation for the emergence of a spiking state) and the bifurcation of a repetitive spiking state leading to the rest state (the bifurcation for the termination of a spiking state) will be explored to distinguish the topological classification of bursting behaviors. Especially, there will be an analysis of the interaction between the bifurcations of equilibria at different switching boundaries as well as the co-dimension two bifurcation of limit cycle. In the non-autonomous case, the dynamics of system with periodic excitation will be discussed. Concretely, the periodic excitation is taken as the bifurcation parameter of fast subsystem. Non-smooth bifurcations by modulation of the periodic excitation which lead to a hysteresis loop will be considered here in order to identify the type of hysteresis loop. Furthermore, the bifurcations for emergence and termination of the spiking state will be analyzed to reveal the inherent dynamical nature and topological types of bursting patterns. The influence of the specific form of periodic excitation on bursting is also considered.

实际系统中广泛存在的各种非光滑因素引起了人们的高度关注，已成为当前非线性动力学研究的热点和难点之一。本项目拟围绕非光滑连续系统，考虑其中各状态量变化速率间的差异，基于非光滑分岔理论，分自治和非自治两种情形探究系统的各种非光滑簇发振荡模式及其诱发机制。在自治条件下，分析快子系统在不同运动状态转迁过程中所涉及的不同吸引子的非光滑分岔，着重关注向量场不同分界面引起的平衡点非光滑分岔之间的相互作用以及极限环的余维二分岔，给出自治非光滑连续系统多种簇发振荡模式的分岔转迁机制及其拓扑类型；在非自治条件下，主要探讨周期激励调制下系统经由滞后环的簇发振荡。分析周期激励导致滞后环产生的非光滑分岔机制和类型，进而探讨快子系统在静息态和激发态之间转迁的非光滑分岔模式，揭示系统经由不同类型滞后环的多种簇发振荡模式以及周期激励形式对簇发振荡行为的影响。上述研究将为非光滑系统理论发展和工程应用提供服务。

多时间尺度耦合的非光滑连续系统在工程实际中具有广泛的应用背景，该类系统的分岔机理和簇发转迁模式都较光滑系统有很大的不同。本项目紧密围绕一类典型的非光滑连续系统，将模型中各物理量按其变化速度的快慢进行分类，基于非光滑分岔理论，从数值计算和理论分析两方面探究了一定参数条件下该类系统簇发振荡的诱发机制及其拓扑类型。讨论了系统在向量场不同区域的运动形式及其动力学性质，分析了系统在向量场分界面处发生的非光滑分岔机理，指出了位于不同区域的吸引子之间相互转迁时相应分岔在其中的作用，从而给出了系统静息态、激发态两种不同运动模式相互转迁的机理；同时充分考虑各不同区域吸引子各自的动力学性质，并由此解释了相应多模态复合振荡模式下多个振荡频率产生的动力学机制以及周期簇发解随参数变化的演化机理；分析周期刺激对系统行为的影响，将周期刺激整体作为系统的调制参数，紧扣系统的非光滑特性，从参数演化与向量场分界面的相互关联出发，阐释了由于刺激量值缓慢而周期地穿越其与向量场非光滑性相关的分岔点，使得系统因非光滑分岔在各区域不同吸引子之间交替突发转迁，从而生成滞后环并形成经由滞后环的簇发振荡的动力学过程和机理，分别从频率、幅值等方面较为全面地揭示了慢变周期刺激对系统簇发行为进行调制的机理和本质。本项目研究工作为进一步了解非光滑动力系统的复杂行为，进而研究相应的控制策略提供了一定的基础。