若干分数阶复杂网络的同步分析及计算

发布时间：2018-04-20 20:10

本文选题：分数阶导数 + 复杂网络　；参考：《兰州大学》2016年博士论文

【摘要】：著名物理学家霍金称:“二十一世纪是复杂性科学的世纪”.复杂网络是复杂性科学中的新兴学科,它广泛存在于各种不同的领域,如生物、物理、社会、计算机、工程等.同步普遍存在于各类复杂网络中,是复杂网络上非常典型的集体行为,也是复杂网络中最重要的动力学特征之一.复杂网络的同步与控制有助于理解和解决自然和社会中的许多问题.最近十几年,人们将复杂网络推广到分数阶情形,一方面分数阶复杂网络可以更好地刻画模型所具有的记忆和遗传性质,另一方面分数阶复杂网络通过分数阶导数的阶数增加了一个自由度,极大地丰富了动力学行为.因此,分数阶复杂网络的同步与控制有着更加广阔的应用空间.本文研究了分数阶复杂网络的同步问题,重点关注分数阶模糊神经网络、分数阶时滞复杂网络、分数阶Takagi-Sugeno(T-S)模糊复杂网络和一般的分数阶复杂网络等,理论分析了自适应控制、牵制控制、脉冲控制、牵制脉冲控制等方法的合理性,数值试验表明了这些方法的有效性.全文共分为六章.第一章主要介绍了整数阶复杂网络和分数阶复杂网络同步的背景和研究意义,对目前的研究现状进行了综述,并概述了本文的主要工作和创新点.在第二章,为了动态增加耦合强度,引入了模糊算子和相互作用,得到了带有相互作用的分数阶模糊神经网络模型.通过压缩映像原理和分数阶导数的性质证明了平衡点的唯一性、网络的有界性.根据模糊理论和分数阶非线性系统的Lyapunov定理,在自适应控制器的作用下,给出了分数阶相互作用模糊神经网络的同步标准.最后通过数值试验证实了所得结果的有效性.本章的研究深入洞察了分数阶神经网络在不同耦合状态下的同步状况.第三章研究了带有时滞、非线性耦合、外部扰动分数阶复杂动力网络的外部牵制同步.建立了一般的分数阶时滞系统的渐近稳定性定理,提出了一组新的控制器,结合分数阶导数理论和矩阵不等式,实现了分数阶复杂网络所有节点的完全同步.在上述的框架下,耦合配置矩阵和内部耦合矩阵可以是对称的或非对称的,网络可以是时滞的或非时滞的,节点内部耦合可以是线性的或非线性的.数值仿真结果证实了该控制方法的有效性.第四章围绕着分数阶T-S模糊复杂网络的脉冲同步.建立了分数阶脉冲系统比较原理,可用于一般的分数阶脉冲问题解的大小比较.本章采用分数阶T-S模糊系统作为复杂网络动力系统,具有局部线性等优点.基于比较原理和分数阶GronwallBellman不等式,得到了在脉冲控制作用下分数阶T-S模糊复杂网络的同步标准.最后,通过相应的数值例子来验证所提出标准的准确性和有效性.第五章研究了分数阶复杂动力网络的牵制脉冲控制同步.将牵制控制和脉冲控制结合,充分发挥两者的优势,得到了分数阶网络的靶向牵制脉冲控制方法,初步解决了牵制控制哪些节点的问题.基于分数阶脉冲比较原理和分数阶系统的Volterra积分等价形式,建立了一般分数阶复杂网络的同步标准.数值仿真进一步表明了所给出理论结果的正确性和提出方法的有效性.第六章对全文工作进行了总结并对未来的工作进行了展望。
[Abstract]:The famous physicist Hocking said, "twenty-first Century is the century of complexity science." complex networks are emerging disciplines in complexity science. It exists widely in various fields, such as biology, physics, society, computers, engineering, etc., which are common in various complex networks, and are a very typical collective behavior on complex networks, It is also one of the most important dynamic characteristics in complex networks. Synchronization and control of complex networks can help to understand and solve many problems in nature and society. In the last decade, complex networks have been extended to fractional order. On the one hand, the fractional complex network can better describe the memory and genetic properties of the model, and the other is a better description of the memory and genetic properties of the model. On the one hand, the fractional complex network increases a degree of freedom by the order of the fractional derivative, which greatly enriches the dynamic behavior. Therefore, the synchronization and control of the fractional complex networks have a wider application space. This paper studies the synchronization problem of fractional complex networks, focusing on fractional fuzzy neural networks, and the fractional order fuzzy neural networks. The order time delay complex network, the fractional Takagi-Sugeno (T-S) fuzzy complex network and the general fractional complex network, etc., the rationality of adaptive control, control, pulse control, pulse control and other methods are analyzed theoretically. Numerical experiments show the effectiveness of these methods. The full text is divided into six chapters. The first chapter mainly introduces the integer. The background and research significance of the synchronization of order complex network and fractional complex network are summarized, and the main work and innovation of this paper are summarized. In the second chapter, in order to dynamically increase the coupling strength, the fuzzy operator and interaction are introduced, and a fractional order fuzzy neural network model with interaction is obtained. The uniqueness of the equilibrium point and the boundedness of the network are proved by the principle of the compression image and the properties of the fractional derivative. According to the Lyapunov theorem of the fuzzy theory and the fractional order nonlinear system, the synchronization standard of the fractional order interacted fuzzy neural network is given under the action of the adaptive controller. Finally, the numerical experiment is proved. The study of the results is effective. The study of this chapter deeply insights into the synchronization of fractional neural networks under different coupling states. The third chapter studies the external synchronization of fractional complex dynamical networks with time-delay, nonlinear coupling and external disturbances. The asymptotic stability theorems for the general fractional order time-delay systems are established. A new set of controllers, combined with fractional derivative theory and matrix inequality, realizes complete synchronization of all nodes in a fractional complex network. Under the above framework, the coupling configuration matrix and the internal coupling matrix can be symmetric or asymmetric, and the network can be time-delay or non time-delay, and the internal coupling of nodes can be linear. The numerical simulation results confirm the effectiveness of the control method. The fourth chapter focuses on the pulse synchronization of the fractional T-S fuzzy complex network. The comparison principle of the fractional order pulse system is established, which can be used to compare the size of the general fractional order pulse problem. In this chapter, the fractional order T-S fuzzy system is used as a complex network power system. Based on the comparison principle and the fractional GronwallBellman inequality, the synchronization standard of the fractional order T-S fuzzy complex networks under the impulse control is obtained. Finally, the accuracy and effectiveness of the proposed standard are verified by the corresponding numerical examples. The fifth chapter studies the fractional complex dynamic network. With the combination of control and pulse control, the control and pulse control are combined to give full play to the advantages of the two. The target controlled pulse control method of the fractional order network is obtained. The problem of which nodes is controlled is solved preliminarily. Based on the fractional pulse comparison principle and the Volterra integral equivalent form of the fractional order system, the general fraction is established. The numerical simulation further shows the correctness of the theoretical results and the effectiveness of the proposed method. The sixth chapter summarizes the full text work and looks forward to the future work.

【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O157.5

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