多时间尺度耦合效应下分数阶系统的复杂动力学

The fractional-order and the multiple time-scale systems are both important in the theoretical and engineering aspect, which are focus in the dynamics filed. For the past few years, the fractional-order system with multiple time scales has gradually attracted the attention from the scholars at home and abroad. In this project we will investigate the fractional-order system containing multiple time scales, and the complicated dynamical behaviors under the slow-fast effect will be explored. Firstly, harmonic balance method will be used to investigate the analytical solution, which can reveal the generation mechanism of the dynamical phenomenon. The analytical solution based on the method of variation of parameter will be given for the system with periodic excitation. Based on nonlinear dynamical theory, the bifurcation and chaos as well as their critical conditions will be discussed. Secondly, the different typical bursting oscillations as well as the bifurcation mechanism will be studied, and various methods such as slow-fast analysis, enveloping analysis and transition portrait are used to reveal the switch between the quiescent states and spiking states. The bursting as well as its bifurcation mechanism in high dimensional fractional-order system will be discussed, especially when high co-dimensional bifurcation is involved. Thirdly, we will focus on the parameters depending on the practical physical conditions, where the difference and relation of the dynamics behaviors between the fractional-order and integer-order systems are considered. To some extent, the project not only may enrich the dynamical theory about the fractional-order system with different timescales, but also can be used to serve for many aspects related with the experiment and design of engineering practice.

具有广泛工程背景的分数阶系统和多时间尺度耦合系统目前都是动力学领域研究的热点和前沿课题之一，本项目拟从不同角度深入探讨多时间尺度耦合效应下分数阶系统的各种动力学特征。关注在不同快慢子结构下系统分岔、混沌及各种振荡，重点研究簇发振荡行为，利用快慢分析、包络分析、转换相图等方法给出激发态与沉寂态的转迁模式，探讨高维分数阶系统中簇发振荡行为，考察在不同慢变量个数下的高余维簇发振荡及其诱导机制；通过谐波平衡法及其改进形式，研究复杂微分项的数学处理，解决分数阶微分项的平均处理问题，基于常数变易法的思想，给出周期扰动导致的多尺度耦合分数阶系统近似解析解的计算方法；研究分数阶系统与整数阶系统多尺度效应的区别与联系，找出适合不同时间尺度耦合分数阶系统中某些动力学行为的研究方法，分析与实际自然物理条件密切相关的敏感参数对系统动力学行为尤其是对簇发振荡行为的影响规律，为实际工程问题提供有效的理论服务。

具有广泛工程背景的分数阶系统和多时间尺度耦合系统都是目前动力学领域研究的热点和前沿课题之一，本项目深入研究了分数阶van der Pol-Rayleigh、Duffing、Brusselator多尺度耦合系统的动力学行为。研究发现，在整数阶多尺度耦合系统中用于研究快慢效应的快慢分析法同样适用于分数阶多尺度系统，可以定性揭示分数阶系统簇发振荡行为的分岔机理。通过利用分数阶稳定性理论，得出了分岔临界参数条件，给出了几类系统双参分岔图，确定了在不同参数范围内的系统稳定性与吸引子类型。发现了几类系统中存在的不同簇发振荡类型，这些簇发振荡的产生机理涉及到平衡点的Fold分岔、Hopf分岔、叉形分岔和极限环分岔。簇发振荡会包含一种、两种甚至三种分岔类型，包含的分岔类型越多行为就越复杂。研究了不同参数对簇发振荡的影响规律，在一定的参数范围内，周期激励作用的慢变过程和其他快变过程共同影响整个系统的响应类型，幅值不同时，快慢过程对整个系统响应的影响作用也不同，且两者之间存在有趣的博弈现象。分数阶阶次影响系统的分岔类型、吸引子类型、参数区间长度等，进而会影响簇发振荡类型和形状等。给出了适用于分数阶非线性系统的平均法、谐波平衡法、KBM法、多尺度等定量研究方法，并分析了分数阶van der Pol、Van der Pol-Rayleigh、Duffing-van der Pol、Mathieu-Duffing系统的主共振、超谐共振、亚谐共振，得到了系统共振的近似程度较高的解析解。利用Melnikov方法，给出了分数阶导数的Duffing系统的混沌阈值。磁流变阻尼器具有明显的分数阶特性，本项目对其进行了分数阶建模，研究其非线性动力学特性。本研究成果在一定程度上丰富了非线性动力学理论，为相关工程领域的振动控制等提供了理论指导。