分数阶时滞耦合网络的同步研究

In recent years,researches on structure and dynamic behaviors of complex networks have become a very important task,especially, synchronization problems of complex networks.In many concrete systems, synchronization has important practical applications.Ability and speed of synchronization of networks are closely related to stability and transmission efficiency of networks,and the structure of networks has inevitable effect on its synchronization.Since time-delay and fractional-order derivatives widely exist in many practical systems,this project mainly uses stability theory to discuss global synchronization of fractional-order coupled networks with time-delay and fractional-order singular coupled networks with time-delay.Combining with theory of spectral graph,we investigate global synchronization of complex networks on general graphs and the relation between eigenvalues of graphic matrices and synchronization,and seek the relation between structure and parameters of graphs and speed of synchronization of networks.This project can not only extend stability theory of fractional-order differential equations with time-delay,but be applied to many concret engineering systems such as Internet,image processing,communication systems,traffic systems and so on since the considered networks are general fractional-order systems.

近年来，复杂网络的结构和动力学行为的研究已成为一个非常重要的课题，其中最受关注的是复杂网络的同步问题。在很多实际系统中,同步已被发现具有重要的应用价值。复杂网络的同步能力和同步速度关系到网络的稳定性和传输效率等实际问题，而网络的结构性能对其同步有着必然的，甚至是本质的影响。由于时滞现象和分数阶导数广泛存在于各种实际系统中，且对系统的性态具有重要的影响，本项目主要运用稳定性理论研究分数阶时滞耦合网络和分数阶退化时滞耦合网络的全局同步；结合谱图理论知识，研究一般图上复杂网络的全局同步问题，建立图矩阵的特征值和网络同步的关系，寻求图的结构、参数和网络同步速度之间的联系。本项目的研究不仅在理论上可以拓广分数阶时滞微分方程的稳定性理论，而且由于所考虑的网络是一般的分数阶系统，所以本项目的研究将可以应用到现实世界中很多工程实际问题中，如因特网、图像处理、通信系统、交通系统等。

本项目主要研究分数阶时滞微分系统的稳定性和分数阶动力网络的同步问题等.充分考虑分数阶导数和时滞对系统的影响，所得的结果能够更精确地描述实际系统，丰富和发展了分数阶微分方程理论。一年来取得的成果有：建立了非线性分数阶系统的全局渐近稳定性；建立了非线性分数阶中立型系统的全局渐近稳定性结果；给出了分数阶线性系统的全局同步条件；研究了一类随机脉冲神经网络，根据不动点理论和分析技巧得到了网络指数稳定的一些充分条件；研究了分数阶线性和非线性差分系统的初值问题等。发表相关论文6篇，标注资助5篇，其中SCI收录5篇。