复杂时滞耦合系统的动态特性及其调控机制研究

In recent years, the investigations of delay-coupled systems have become focal subjects in research due to their extensive applications in many fields, such as artificial intelligence, cooperative control, secret communication, mechanical engineering, and so on. This project focuses on the emergence, transition, synchronization, and regulation of the nonlinear dynamical behaviors of complex delay-coupled systems. The main themes and contributions of the project include: The stability criteria and the conditions of the existence of different bifurcations are studied. Interesting phenomena are explored by analyzing the interactions of different bifurcation points. The complicated dynamical behaviors induced by the interactions of fast and slow variables are investigated by considering different time scales. Different patterns of synchronization and their transitions are shown and effective controllers are designed to reach expected synchronization states. The influences of delayed couplings, nonlinearity, and different time scales on the complicated dynamics of the system are excavated. High-performance circuit experiments are built to validate the obtained theoretical and numerical results and to discover new phenomena. Various dynamical behaviors and the process of their variations are observed in real time through circuit experiments. The aims and objectives of the project are to develop theoretical methods, numerical algorithms and experimental technology for complex delay-coupled systems and to provide powerful support for their practical applications.

近年来，由于在人工智能、协同控制、保密通信、机械工程等众多领域中的广泛应用，时滞耦合系统已经成为国内外学术界关注的研究热点。本项目研究复杂时滞耦合系统的非线性动力学行为的产生、转迁、同步以及调控机制，主要内容包括：研究系统的稳定性判据和各类分岔产生条件，揭示不同分岔相互作用诱导的有趣现象；考虑不同时间尺度效应，研究快变量和慢变量同时作用所引发的复杂动力学行为；研究各类同步现象及其相互间的转迁机制，设计控制器调控系统行为至期望同步状态；发掘时滞耦合、非线性以及尺度效应等对于各类动力学行为的影响机理；搭建高性能电路实验系统以验证前期研究成果和发现新现象，并可实时显示各类动力学行为及其动态变化过程。通过本项目的研究，旨在发展复杂时滞耦合系统的理论分析方法、数值算法和实验技术，为实际应用提供有力支撑。

近年来，时滞耦合系统已成为力学、数学、控制、神经科学以及信息科学等诸多学科关注的热点领域。本项目研究复杂时滞耦合系统的动力学行为的产生、转迁以及调控机制。主要研究内容和成果包括：开展了时滞耦合系统的动力学建模，建立了相应的非线性微分方程；考察了平衡点的数目和类型，分析了局部稳定性和全局稳定性；研究了系统失稳后的各类分岔，在参数平面内给出了静态分岔和动态分岔曲线，探讨了不同分岔之间的相互作用机制，得到了多种有趣的振荡模式；研究了网络内部节点和不同网络间节点的同步性，阐释了不同同步运动状态的产生和转迁机制；揭示了复杂动力学的演化机制，发现了多周期轨道、混沌、多稳态共存等现象。借助电子元器件的非线性和延时性，设计了节点电路、时滞电路等模块，组建了高性能的电路实验平台，有效验证了理论分析与数值仿真的结果。重点揭示了时滞、非线性、忆阻以及电磁辐射等因素对于系统性能的影响情况，提出了各类动力学特性的调控策略。