Stability and synchronization of complex networks composed of dynamical nodes with discontinuities

随着通信、计算和互联网的快速发展，使我们能够收集和分析大量的信息，近年来人们对复杂网络的兴趣越来越大。一个复杂的网络是一个被称为节点的相互连接的动态单元的大集合。每个节点都是一个具有特定行为的动态系统。有许多自然的和人为的复杂网络的例子，如基因调控网络、通信网络、交通网络、电网、社会关系网络和万维网等等。由于其耦合结构，复杂网络可能表现出与单个节点完全不同的复杂动态。因此，研究复杂动态网络的集体动力学行为，如稳定性和同步性，以及发生在复杂网络中的其他相关过程的性质，如疾病和信息的传播，是非常重要的。*******在本研究计划中，我们将主要关注那些由具有切换或跳跃不连续的动态单元组成的复杂网络。这样的动态单元包括切换系统、脉冲系统或更普遍的混合系统，它们通常用于模拟具有某些嵌入式智能的实际系统（有时称为智能系统）。我们的目标包括：(i)发展这种复杂网络的稳定性理论和鲁棒性；（ii）建立复杂网络的各种同步标准；（iii）开发有效的脉冲控制和混合控制方法，用于复杂网络的稳定和同步；（iv）为复杂网络的计算建模和模拟开发有效和可靠的计算算法；(v)将获得的方法应用于实际问题，例如疾病在社会网络上的传播和蔓延。本研究项目的成果将产生新的稳定性和鲁棒性结果，新的同步准则，有效的脉冲和混合控制方法，以及用于复杂动态网络建模和仿真的有效计算算法。*******研究成果将提高加拿大在前沿研究方面的声誉，并为国际合作提供机会。我们希望每年在国际知名期刊上发表几篇高质量的论文。这个研究项目也将为培养研究生和其他高素质人才提供一个极好的机会。

With the rapid development of communication, computation and the Internet, which allow us to gather and analyze large amount of information, there has been a growing interest in complex networks in recent years. A complex network is a large set of interconnected dynamical units called nodes. Each node is a*dynamical system with specific behavior. There are numerous examples of natural and man made complex networks such as genetic regulatory networks,*communication networks, transportation networks, electrical power grids, social relation networks, and the World Wide Web, among many others. A*complex network can exhibit complicated dynamics that may be totally different from those of a single node due to its coupling architecture. Thus it is very*important to study the collective dynamical behavior, such as stability and synchronization, of complex dynamical networks as well as properties of other*relevant processes that take place in complex networks such as the spreading of diseases and information.*******In this research program, we will mainly focus on those complex networks composed of dynamical units with switching or jump discontinuities. Such*dynamical units include switched systems, impulsive systems, or more generally hybrid systems, which are often used to model practical systems*having certain embedded intelligence (sometime called the smart systems). Our objectives include: (i) to develop stability theory and robustness of such*complex networks;(ii) to establish various synchronization criteria of complex networks; (iii) to develop effective impulsive control and hybrid control*methods for stabilization and synchronization of complex networks; (iv) to develop efficient and reliable computational algorithms for computational modeling and simulation of complex networks; (v) to apply the obtained methods to practical problems such as disease transmission and spreading over social networks.*The outcomes of this research program will result in new stability and robustness results, new synchronization criteria, effective impulsive and hybrid*control methods, efficient computational algorithms for modeling and simulation of complex dynamical networks.*******The outcomes will enhance Canada's reputation for leading edge research and facilitate opportunities for international collaboration. We expect to publish several top quality papers per year in prestigious international journals. This research program will also provide an excellent opportunity for training graduate students and other highly qualified personnel.