几类时滞复杂振子网络的动力学与控制

发布时间：2018-08-30 14:03

【摘要】：复杂网络广泛存在于自然界和人类社会中,如互联网、交通网、电力网、蛋白质相互作用网、人际关系网等等.可以说任何由相同或不同个体构成的复杂系统,当我们把这些个体抽象为节点,个体之间的相互作用抽象为边时,都可以用复杂网络来表征.近年来,复杂网络的研究已经在数学、物理、计算机科学、力学、生命科学及信息科学等多个学科领域内展开,取得了令人瞩目的成就.随着复杂网络在流行病的防控、互联网的拥塞控制、交通系统的优化、电力系统级联故障的分析及生态系统的演化等领域的普遍应用,复杂网络的动力学与控制受到了学术界众多科研工作者的广泛关注和深入研究.研究表明,复杂网络的大多数复杂动力学行为都是伴随着网络拓扑结构和节点动力学性质的改变而产生的.特别地,具有时滞和非线性因素的复杂网络通常会出现诸如稳定、不稳定、同步、振荡、分岔和混沌等特殊的行为,对这些动力学行为的研究为复杂网络在各领域的实际应用提供了非常重要的理论基础和依据.然而,具有时滞的复杂网络系统的特征方程是包含指数函数的超越方程,无法精确解出它的无穷多个特征根,给网络动力学的分析带来了一定的难度.因此,关于时滞复杂网络动力学的研究极具挑战性,同时对人类社会的发展具有重要的科学意义和实际应用价值.目前,研究复杂网络的动力学行为主要为实现以下几个目的:一是理解网络的拓扑结构如何影响网络的动力学行为;二是理解网络上的动力学行为如何决定网络的拓扑结构;三是如何采用合适的控制策略来控制网络达到预期的动力学行为.本文以具有时滞的耦合振子网络为对象,开展复杂网络的动力学行为与控制的研究.论文的主要研究内容和创新点如下:(1)研究一类具有激励或抑制长连接的时滞小世界振子网络的动力学行为.根据矩阵的扰动理论,分别给出网络连接强度矩阵的最大和最小特征值的上界和下界.基于时滞系统的稳定性基本理论,考察了系统特征方程的特征根的分布,研究了网络的稳定性和不稳定性,给出网络系统完全稳定和完全不稳定的区域.讨论了小世界网络的鲁棒稳定性,并将本文给出的稳定性判定条件与平均场理论给出的稳定性条件进行比较分析.虽然文中给出的稳定性条件是保守的,但能确保在绝大多数情形下系统稳定,特别是在激励和抑制连接同时存在的条件下.给出网络系统发生叉型分岔和Hopf分岔的充分条件.最后通过数值模拟讨论了连接强度矩阵的特征值离开稳定性区域的先后顺序和方向.(2)研究一类具有时滞的复杂振子网络的牵制控制.首先利用正交变换,将复杂网络系统映射成为一个简单的具有相同动力学性质的等价系统.通过对转换后系统的特征方程进行分析,发现所研究系统的平衡点对任何时滞都是不稳定的.为镇定不稳定的网络系统,本文提出了一种具有时滞状态反馈的有效控制策略-牵制控制,即通过控制网络的一少部分节点达到控制整个网络的目的.通过对受控网络系统进行分析并利用正交变换的逆变换,给出了复杂网络平衡点的局部稳定性、Hopf分岔及余维2分岔的存在条件.利用中心流形定理和规范形理论研究了Hopf分岔的方向和分岔周期解的稳定性.最后通过数值模拟考察了网络系统通向混沌的路径.(3)研究一类具有时滞的双层耦合复杂振子网络的稳定性和复杂时空动力学.首先,应用分而治之算法,分析了双层网络与每个单层网络的邻接矩阵的特征值之间的关系,进而给出系统的平衡点局部稳定的判定条件.发现双层网络的稳定性完全可以通过单层网络矩阵的最大和最小特征值来确定,因此仅需要通过研究单层网络即可以了解整个网络的动力学行为.当网络的节点数量较大时,大大降低了理论分析的难度并简化了计算过程.应用中心流形定理分析了系统的周期性,表明双层振子之间相互作用可以产生复杂的时空动力学行为,如反射波,镜面波等.最后,数值模拟验证了理论分析的结果.(4)研究一类具有随机长连接强度的时滞小世界振子网络的稳定性和不稳定性.应用随机矩阵理论和矩阵扰动理论,分析了网络连接强度矩阵最大特征值的概率分布和最小特征值的下界,进而通过对时滞系统特征方程的特征根进行分析给出系统的平衡点局部完全稳定和完全不稳定的判定条件.对于给定的系统参数,给出了计算系统稳定的概率公式.并讨论长连接概率和随机长连接强度的均值、方差对网络稳定性的影响.
[Abstract]:Complex networks exist widely in nature and human society, such as the Internet, transportation networks, power networks, protein-protein interaction networks, interpersonal networks, etc. It can be said that any complex system composed of the same or different individuals, when we abstract these individuals as nodes, the interaction between individuals as edges, we can use complexity. In recent years, research on complex networks has been carried out in many fields, such as mathematics, physics, computer science, mechanics, life science and information science, and has made remarkable achievements. The dynamics and control of complex networks have been extensively and deeply studied by many researchers in the academia. The results show that most of the complex dynamic behaviors of complex networks are accompanied by the changes of topological structure and dynamic properties of nodes. In general, complex networks with time-delay and nonlinearity usually exhibit special behaviors such as stability, instability, synchronization, oscillation, bifurcation and chaos. The study of these dynamic behaviors provides a very important theoretical basis and basis for the practical application of complex networks in various fields. Characteristic equation is a transcendental equation containing exponential function, which can not accurately solve its infinite number of characteristic roots. It brings a certain degree of difficulty to the analysis of network dynamics. To study the dynamic behavior of complex networks is to understand how the topological structure of the network affects the dynamic behavior of the network; to understand how the dynamic behavior of the network determines the topological structure of the network; and to adopt appropriate control strategies to control the network to achieve the desired dynamic behavior. The main contents and innovations of this paper are as follows: (1) The dynamic behavior of a class of small-world oscillator networks with time-delay and excitation or suppression of long connections is studied. According to the perturbation theory of matrices, the network connections are given respectively. The upper and lower bounds of the maximum and minimum eigenvalues of the strength matrix are given. Based on the stability theory of time-delay systems, the distribution of eigenvalues of the eigenvalues of the eigenvalues of the systems is investigated, the stability and instability of the networks are studied, and the regions of complete stability and complete instability of the networks are given. The stability criterion given in this paper is compared with the stability criterion given in the mean field theory. Although the stability criterion given in this paper is conservative, it can ensure the stability of the system in most cases, especially under the condition that both excitation and restraint connections exist simultaneously. Finally, the order and direction of the eigenvalues of the connection strength matrix leaving the stability region are discussed by numerical simulation. (2) The pinning control of a class of complex oscillator networks with time delay is studied. By analyzing the characteristic equations of the transformed system, it is found that the equilibrium point of the studied system is unstable for any time delay. To stabilize the unstable network system, an effective control strategy with time-delay state feedback, i.e. pinned control, is proposed. By analyzing the controlled network system and using the inverse transformation of orthogonal transformation, the local stability of equilibrium point, the existence conditions of Hopf bifurcation and codimension 2 bifurcation are given. The direction of Hopf bifurcation and the stability of bifurcation periodic solution are studied by using the central manifold theorem and the normal form theory. The path to chaos is investigated by numerical simulation. (3) The stability and complex space-time dynamics of a class of two-layer coupled complex oscillator networks with time-delay are studied. Firstly, the relationship between the eigenvalues of the adjacency matrix of the two-layer network and each single-layer network is analyzed by using the divide-and-conquer algorithm, and then the equilibrium of the system is given. It is found that the stability of a two-layer network can be determined by the maximum and minimum eigenvalues of the single-layer network matrix, so the dynamic behavior of the whole network can be understood only by studying the single-layer network. The periodicity of the system is analyzed by using the central manifold theorem. It is shown that the interaction between two-layer oscillators can produce complex spatio-temporal dynamic behaviors, such as reflected waves, mirror waves, etc. Finally, the results of theoretical analysis are verified by numerical simulation. (4) A class of small-world oscillator networks with stochastic long connection strength is studied. Stability and instability. By using random matrix theory and matrix perturbation theory, the probability distribution of the maximum eigenvalue and the lower bound of the minimum eigenvalue of the network connection strength matrix are analyzed. For a given system parameter, the probabilistic formula for calculating the stability of the system is given. The effects of the mean and variance of the long connection probability and the stochastic long connection strength on the stability of the network are discussed.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O157.5

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