时滞耦合系统非线性动力学的研究进展

时滞耦合系统非线性动力学的研究进展

张舒;徐鉴

【摘要】随着对自然界客观规律的深入认识,工程系统设计的精细化和复杂性要求也与日剧增.在许多耦合的动态系统设计过程中要考虑由耦合过程的时滞所引发的动力学行为,该时滞来自于与传感系统、作动系统和控制系统耦合的过程.耦合时滞也广泛存在于交通、系统生物学、电子通讯、神经和信息网络等技术中.本文首先从耦合时滞出发,在以时滞为中心的耦合系统复杂动力学机制、时滞镇定耦合系统的实验基础和实现、快慢变时滞耦合系统动力学和时滞神经网络同步和去同步4个方面,对耦合时滞诱发的动力学研究进展进行综述.着重介绍了时滞耦合系统中耦合时滞诱发的高余维分岔奇异性及新的定量分析方法、中立型时滞微分方程的规范型计算、具有耦合时滞的非线性系统中耦合时滞和非线性参数的辨识方法与实验实现、快慢变时滞耦合系统的张弛振荡、耦合时滞诱发的网络系统的同步模式切换等问题的研究进展;然后在应用方面重点介绍了车床磨削加工过程中耦合时滞诱发的颤振及其机理、具有惯性项和耦合时滞的神经网络系统中耦合时滞诱发的高余维分岔和复杂动力学、时滞动力吸振器与隔振装置的设计与实验实现.最后,从耦合时滞系统的一般性理论和工程应用两个方面展望了近期值得关注的一些问题.%With the deep understanding towards the objective laws of nature, requirements on refinement and complex-ity in engineering system design are increasing. Many coupled dynamic system designs need to take into account the dynamics induced by the time delay existing in the coupling process. Such coupling time delay may come from the process of coupling with the sensing system, the actuation system and the control system. Coupling delays also exten-sively exist in the fields such as transportation

system, system biology, electronic communication, neural and information networks and etc. Firstly, based on the concept of coupling delay, this paper reviews the recent research progresses on dynamics induced by such delay from the following four aspects:(1) the delay-centered mechanism of complex dynamics in coupled systems;(2) experimental foundation and realization of stabilizing coupled systems by utilizing time delay;(3) dynamics of fast-slow coupled system with time delay; and (4) synchronization and desynchronization of delayed neural networks. Some advances in the general theory of systems with coupling delay are highlighted including the coupling-delay-induced bifurcation and singularity with high codimention and the novel quantitative method of analysis, normal form computation for neutral delay differential equations, identification of time delay and nonlinear parameters in nonlinear systems with coupling delay and the relevant experiment, relaxation oscillation in the fast-slow system with coupling delay, and transition of modes of synchronization induced by coupling delay in network systems. Secondly, as for the application, some new results are presented in details such as the coupling-delay-induced chatter in grinding pro-cess and its mechanism, bifurcation with high codimension and complex dynamics induced by coupling delay in neural networks with inertial terms, and design and experiments of vibration absorber and isolator using coupling delay. Finally, some problems which are worthy of attention in near future are highlighted from perspectives of the general theory of systems with coupling delay and the potential applications.