时滞诱发的非线性耦合系统复杂动力学的理论和实验研究

Since time delay is common and ubiquitous in real coupled systems, the dynamics of nonlinear coupled systems with time delays has attracted considerable attention and become a frontier research topic. This project focuses on various dynamical behaviors of nonlinear coupled systems with time delays. The main themes and contributions of the project include: The dynamical models are presented based on the characteristics of real coupled systems. The simple stability criteria of coupled systems are proposed and the influences of multiple time delays on the stability of coupled systems are discussed. The conditions of different bifurcations when the coupled system loses its stability locally are derived. New qualitative and quantitate methods are developed to determine the dynamical characteristics of bifurcation points. The high codimensional bifurcations and complex behaviors due to time delays and couplings are explored. Synchronization and its transitions of coupled systems are investigated. The effects of time delays on the existence and switches of different synchronization phenomena are given. The platform of circuit experiments for the nonlinear coupled systems with time delays is designed and can be used to show various properties of the coupled systems under different parameters. The results of theoretical analysis and numerical simulations can be validated through circuit experiments. The aims and objectives of the research are to develop novel and effective theoretical methods, efficient and general numerical algorithms, and high precision real-time experimental technology, reveal the mechanisms of complex dynamical behaviors due to time delays, and promote the application and development of the nonlinear dynamics of time-delayed systems.

由于时滞普遍存在于实际耦合系统中，时滞非线性耦合系统动力学受到了国内外学术界的广泛关注，为前沿研究课题。本项目研究时滞非线性耦合系统的各类动力学行为，主要内容包括：针对实际耦合系统的特性进行动力学建模，研究耦合系统的简易稳定性判据，考察多时滞情形对于系统稳定性的影响；研究系统失稳后各类分岔的产生条件，发展新的定性和定量方法分析分岔点的动力学特性，重点揭示时滞和耦合作用诱发的高余维分岔及其导致的复杂力学行为；研究系统的同步和转迁动力学，考察时滞对于各类同步现象及其相互间切换的影响；构建时滞非线性耦合系统的实验平台，实时显示多种参数条件下的各类动力学特性，以期作为理论分析和数值计算的佐证。通过本项目的研究，旨在发展新颖有效的理论分析方法、高效通用的数值算法以及高精度实时显示的实验技术，揭示时滞诱发的非线性耦合系统的复杂动力学机制，推动时滞系统非线性动力学的发展和应用。

近年来，由于在人工智能、控制以及保密通信等众多领域的广泛应用，时滞非线性耦合系统已经成为国内外学术界关注的研究热点。本项目研究时滞非线性耦合系统的各类动力学行为的触发、转迁、同步以及调控机制。主要研究内容包括：考察了高维时滞非线性系统的稳定性和分岔，揭示了分岔诱导的多种复杂行为；分析了各类同步现象的产生条件，阐释了不同同步行为之间的转迁机制；设计了电路仿真实验系统，获得了与已有结果相吻合的实验现象。本项目的研究成果主要包括：分析了时滞、非线性以及耦合作用等因素对系统特性的影响机理，揭示了多周期振荡、概周期响应、混沌吸引子以及多稳态共存等复杂现象的触发和转迁机制，并通过电路仿真实验加以验证。通过本项目的研究，发展了高维时滞非线性系统的理论分析方法、数值算法和实验技术，为实际应用提供了重要支撑。