时滞系统动力学近期研究进展与展望

综述了1999年以来时滞系统动力学在力学、机械工程、航空航天、生态学、生物学、神经网络、激光、电子和信息技术、保密通讯和经济学等领域的研究进展, 总结了其中的研究方法. 通过本文可以看出时滞系统普遍存在于自然和工程实际中, 即使对已经非常熟悉的简单振子,考虑到时滞的影响, 仍有许多问题有待作更深入的理论研究和新现象的发现.针对以往研究中出现的问题, 提出今后几年的发展方向、建议和展望, 同时指出了在理论上急需解决的一些科学问题,例如以时滞反馈控制为中心的控制策略、非线性因素和时滞联合作用的影响、时滞导致的多级分岔使系统呈现出复杂动力学行为、以时滞状态变量耦合为中心构成的网络系统计算模型对系统的影响等问题都是非线性动力学系统所遇到的科学基础问题.      

生物神经元系统同步转迁动力学问题

生物神经元系统中存在着丰富的同步模式, 不同同步模式的实现条件已经被广泛地研究. 然而, 不同同步模式之间的转迁是神经动力学研究领域的难点问题, 近年来在此方面开展了许多相应的研究工作. 本文主要阐述近年来在神经元系统同步转迁动力学方面的研究进展, 揭示神经元系统在耦合、时滞和网络拓扑等不同参数作用下呈现的复杂的同步转迁动力学行为及其可能的动力学机制. 最后总结研究进展的内容并提出对同步动力学今后研究的展望.     

动力学,振动与控制学科未来的发展趋势

对近年来动力学，振动与控制的研究进展作了简要回顾，概述了非线性动力学与振动主动控制这两个研究热点的现状，提出了世纪之初应关注的若干研究前沿，即高维非线性系统的全局摄动法，全局分岔和混沌动力学，高维强非线性系统分岔与混沌动力学的实验研究，非线性时滞系统的动力学，流体－弹性体－刚体耦合系统动力学与控制，碰撞与变结构系统动力学，微机电系统动力学。最后，对我国动力学，振动与控制的发展提出了一些建议。     

血细胞生成的系统动力学研究进展

造血系统是人体中复杂的调控系统, 包括复杂的造血器官和各种血液细胞的增生、分化、成熟、死亡等过程和对这些过程的反馈控制, 是典型的非线性时滞动力系统. 血细胞生成过程的失调可以引起很多动态血液病. 血细胞生成的系统动力学研究对于人们了解和治疗这些血液病有重大意义. 本文从造血系统的基本知识、常见动态血液病的临床表现及其动力学特征、理论模型的建立和对模型的动力学分析等方面综述血细胞生成的系统动力学研究进展.     

谐波齿轮系统的快慢振荡机制研究

谐波齿轮减速器是一种新型的传动装置, 因其具有诸多的优点, 因而得到了广泛应用. 谐波齿轮减速器涉及不同振荡尺度之间的耦合作用, 这通常会诱发复杂的快慢振荡, 严重影响了谐波齿轮系统的正常工作. 本文考虑涉及扭转刚度非线性因素的谐波齿轮系统, 旨在研究系统的快慢动力学, 揭示新型的快慢振荡机制. 首先, 构建了非线性扭转刚度下的谐波齿轮系统的快慢动力学模型. 然后, 通过改变扭转刚度系数, 得到了系统从常规振荡向快慢振荡的转迁过程. 接着, 简要地论述了有关快慢系统的基础理论. 在此基础上, 采用快慢分析法研究了快子系统的动力学特性, 揭示了快慢振荡的产生机制. 研究表明, 当系统参数改变时, 快子系统的平衡点曲线并未发生失稳或分岔; 然而, 在某一点附近, 平衡点曲线能够产生急剧量变, 其特征是平衡点在局部小范围内可以在正坐标值与负坐标值之间快速转迁. 在此基础上, 揭示了一种诱发快慢振荡的新型动力学机制, 比较了这种诱发机制与其他相关机制之间的区别. 本文丰富了系统通向快慢振荡的路径, 为实际谐波齿轮传动系统中的快慢振荡机理与控制研究提供参考.      

Van der Pol时滞弱耦合系统多尺度分析

对于非线性耦合项中带有时滞的van der Pol系统,采用多尺度法对该系统进行定性以及定量的分析.研究结果表明,对于van der Pol时滞耦合系统,时滞的存在影响了系统的稳定性,使系统的周期解发生了静态分岔和Hopf分岔.研究还发现,对于耦合强度较弱的情形,利用多尺度法对系统进行定嚣分析是合理可靠的.我们取不同的耦合强度作用了系统的时间历程图,相图和分岔图,分析了解析解与数值解之间产生误差的原因.本文所研究的系统来源于耦合的激光振荡器.     

考虑时滞效应时耦合化学反应系统的动力学行为

在耦合自催化反应系统中,采用数值分析方法研究了考虑时滞效应和流速扰动时子系统的动力学行为.与原系统相比,该系统呈现出更加丰富的动力学现象.反应过程中出现了结构复杂的混沌吸引子和由在周期解邻域内振荡而产生的概周期运动,并且存在混沌由倍周期分岔演变为新的混沌吸引子的过程.这些结果对于解释耦合化学反应系统中的复杂现象、揭示其反应机理具有一定的指导意义.     

一类平面自治非线性时滞控制系统的多稳态和混沌

众所周知，平面自治系统即使具有光滑非线性存在，系统也不会出现复杂的动力学行为。本文研究这样的系统存在时滞时，时滞量对系统的动力学行为的影响。通过对一个平面自治非线性系统引入时滞反馈，得到数学模型。利用泛函分析和平均法建立系统平衡态随时滞量变化的失稳机理，研究表明：时滞量平面自治系统动力学行为的影响是本质的．时滞量不但可以使系统出现Hopf分岔，产生周期振动。而且还可以使系统出现多稳态的周期运动或周期吸引子，这些共存的吸引子相碰是导致系统复杂的动力学行为，包括概周期和混沌运动。     

时滞动力系统的稳定性与分岔:从理论走向应用

本文综述了近年来时滞动力系统稳定性与分岔方面的研究进展, 重点阐述了作者及其团队在稳定性分析、Hopf分岔计算、利用时滞改善系统稳定性等方面的一些理论和方法研究结果, 介绍了时滞对颤振主动控制系统、不稳定系统镇定、网络系统的影响等方面的研究. 基于研究体会, 对进一步的研究提出了若干展望.     

非线性时滞动力系统的研究进展

具有时滞的动力系统广泛存在于各工程领域．本文从动力学角度对时滞动力系统的研究进展作一综述,内容包括时滞动力系统的特点、研究方法、动力学热点问题的研究进展等．由于时滞动力系统的演化趋势不仅依赖于系统的当前状态,还依赖于系统过去某一时刻或若干时刻的状态,其运动方程要用泛国微分方程来描述,解空间是无穷维的．即使系统中的时滞非常小,在许多情况下也不能忽略不计．对于非线性时滞常微分方程,目前的研究思路基本上与常微分方程系统理论相平行．主要研究方法可分为时域法和频域法,前者包括Ｔａｙｌｏｒ级数法,中心流形法,Ｐｏｉｎｃａｒｅ映射法等,后者包括Ｎｙｑｕｉｓｔ法等．目前对这类系统的动力学研究主要集中在稳定性、Ｈｏｐｆ分岔、混沌等方面．研究表明：时滞动力系统具有非常丰富和复杂的动力学行为,如单变量的一维非线性时滞动力系统可发生混沌现象,与用常微分方程描述的系统有本质性差别．另一方面,人们可巧妙地利用时滞来控制动力系统的行为,如时滞反馈控制是控制混饨的主要方法之一．最后,本文展望了存在的一些问题以及近期值得关注的研究．     

Dynamics of firing patterns, synchronization and resonances in neuronal electrical activities： experiments and analysis

Recent advances in the experimental and theoretical study of dynamics of neuronal electrical firing activities are reviewed. Firstly, some experimental phenomena of neuronal irregular firing patterns, especially chaotic and stochastic firing patterns, are presented, and practical nonlinear time analysis methods are introduced to distinguish deterministic and stochastic mechanism in time series. Secondly, the dynamics of electrical firing activities in a single neuron is concerned, namely, fast-slow dynamics analysis for classification and mechanism of various bursting patterns, one- or two-parameter bifurcation analysis for transitions of firing patterns, and stochastic dynamics of firing activities （stochastic and coherence resonances, integer multiple and other firing patterns induced by noise, etc.）. Thirdly, different types of synchronization of coupled neurons with electrical and chemical synapses are discussed. As noise and time delay are inevitable in nervous systems, it is found that noise and time delay may induce or enhance synchronization and change firing patterns of coupled neurons. Noise-induced resonance and spatiotemporal patterns in coupled neuronal networks are also demonstrated. Finally, some prospects are presented for future research. In consequence, the idea and methods of nonlinear dynamics are of great significance in exploration of dynamic processes and physiological functions of nervous systems.     

COMPLICATED BURSTING BEHAVIORS AS WELL AS THE MECHANISM OF A THREE DIMENSIONAL NONLINEAR SYSTEM 1)

由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.     

Dynamics of a multiplex neural network with delayed couplings

Multiplex networks have drawn much attention since they have been observed in many systems,e.g.,brain,transport,and social relationships.In this paper,the nonlinear dynamics of a multiplex network with three neural groups and delayed interactions is studied.The stability and bifurcation of the network equilibrium are discussed,and interesting neural activities of the network are explored.Based on the neuron circuit,transfer function circuit,and time delay circuit,a circuit platform of the network is constructed.It is shown that delayed couplings play crucial roles in the network dynamics,e.g.,the enhancement and suppression of the stability,the patterns of the synchronization between networks,and the generation of complicated attractors and multi-stability coexistence.     

一类三维非线性系统的复杂簇发振荡行为及其机理

由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.      

Multiple synchronization transitions due to periodic coupling strength in delayed Newman–Watts networks of chaotic bursting neurons

In this paper, we numerically study the effect of time-periodic coupling strength on the synchronization of firing activity in delayed Newman–Watts networks of chaotic bursting neurons. We first examine how the firing synchronization transitions induced by time delay under fixed coupling strength changes in the presence of time-periodic coupling strength, and then focus on how time-periodic coupling strength induces synchronization transitions in the networks. It is found that time delay can induce more synchronization transitions in the presence of time-periodic coupling strength compared to fixed coupling strength. As the frequency of time-periodic coupling strength is varied, the firing exhibits multiple synchronization transitions between spiking antiphase synchronization and in-phase synchronization of various firing behaviors including bursting, spiking, and both bursting and spiking, depending on the values of time delay. These results show that time-periodic coupling strength can increase the synchronization transitions by time delay and can induce multiple synchronization transitions of various firing behaviors in the neuronal networks. This means that time-periodic coupling strength plays an important role in the information processing and transmission in neural systems.     

Anticipatory, complete and lag synchronization of chaos and hyperchaos in a nonlinear delay-coupled time-delayed system

We study the synchronization of chaos and hyperchaos in first-order time-delayed systems that are coupled using the nonlinear time-delay excitatory coupling. We assign two characteristic time delays: the system delay that is same for both the systems, and the coupling delay associated with the coupling path. We show that depending upon the relative values of the system delay and the coupling delay the coupled systems show anticipatory, complete, and lag synchronization. We derive a general stability condition for all the synchronization processes using the Krasovskii–Lyapunov theory. Numerical simulations are carried out to corroborate the analytical results. We compute a quantitative measure to ensure the occurrence of different synchronization phenomena. Finally, we set up an experiment in electronic circuit to verify all the synchronization scenario. It is observed that the experimental results are in good agreement with our analytical results and numerical observations.     

Synchronization error bound of chaotic delayed neural networks

Synchronization of master–slave chaotic neural networks are well studied through asymptotic and exponential stability of error dynamics. Besides qualitative properties of error dynamics, there is a need to quantify the error in real-time experiments especially in secure communication system. In this article, we focused on quantitative analysis of error dynamics by finding the exact analytical error bound for the synchronization of delayed neural networks. Using the Halanay inequality, the error bound is going to be obtained in terms of exponential of given system parameters and delay. The time-varying coupling delay has been considered in the neural networks which does not require any restrictive condition on the derivative of the delay. The proposed method can also be applied to find error bound for state estimation problem. The analytical synchronization bound has been corroborated by two examples.     

Projective-dual synchronization in delay dynamical systems with time-varying coupling delay

The existence of projective-dual-anticipating, projective-dual, and projective-dual-lag synchronization in a coupled time-delayed systems with modulated delay time is investigated via nonlinear observer design approach. Transition from projective-dual-anticipating to projective-dual synchronization and from projective-dual to projective-dual-lag synchronization as a function of variable coupling delay τ _p (t) is discussed. Using Krasovskii–Lyapunov stability theory, a general condition for projective-dual synchronization is derived. Numerical simulations on the chaotic Ikeda and Mackey–Glass systems are given to demonstrate the effectiveness of the theoretical results.     

Stability and bifurcation for a coupled nonlinear relative rotation system with multi-time delay feedbacks

In this paper, we investigate the stability and bifurcation of a class of coupled nonlinear relative rotation system with multi-time delay feedbacks. Using dissipative system Lagrange equation, the dynamics equation of coupled nonlinear relative rotation system with three masses is established. The dynamical behaviors of the system under multi-time delay feedbacks, with two state variables, are discussed. First, characteristic roots and the stable regions of time delay are determined by direct method. The relation between two time delays ratio or time delay feedbacks gains and the stable regions of time delay is analyzed. Second, the direction and stability of Hopf bifurcation are decided by normal form theorem and center manifold argument. Finally, numerical simulation can confirm the validity of the conclusion.     

Robust synchronization of uncertain fractional-order chaotic systems with time-varying delay

This paper presents a new technique using a recurrent non-singleton type-2 sequential fuzzy neural network (RNT2SFNN) for synchronization of the fractional-order chaotic systems with time-varying delay and uncertain dynamics. The consequent parameters of the proposed RNT2SFNN are learned based on the Lyapunov–Krasovskii stability analysis. The proposed control method is used to synchronize two non-identical and identical fractional-order chaotic systems, with time-varying delay. Also, to demonstrate the performance of the proposed control method, in the other practical applications, the proposed controller is applied to synchronize the master–slave bilateral teleoperation problem with time-varying delay. Simulation results show that the proposed control scenario results in good performance in the presence of external disturbance, unknown functions in the dynamics of the system and also time-varying delay in the control signal and the dynamics of system. Finally, the effectiveness of proposed RNT2SFNN is verified by a nonlinear identification problem and its performance is compared with other well-known neural networks.