Synchronization Of Complex Dynamical Networks With Switching Topology

Complex network is one of most challenging and high-profile science subjects in the research on complex system and complexity science, which has comprehensive practical background and theoretical research significance. At present, the study on complex dynamic network has penetrated into many fields, such as mathematics, physics, biology, social science, etc. Synchronization of complex dynamical networks has been paid more and more attention as an important research branch. Network topology and node dynamics plays a key role in the process of the complex dynamic network synchronization. When a case that the network with switching topology structure or a network with non identical nodes, the dynamic behavior of complex dynamical network system will become very complex, and a lot of synchronization analysis and synchronization control problem to be solved. This dissertation studies the synchronization problem of complex networks with switching topology by switched systems theory and Lyapunov theory. The main contributions are as follows:(1) Exponential synchronization problem of switched complex network are studied under any given dwell time switching. The studied complex dynamic network has the following characteristics:All sub-networks can achieve synchronization; Complex dynamical networks have switching topology; The outer coupling matrix is asymmetric matrices and the sum of matrix row elements equal to zero, that is, the network can be directed connected. The outer connection matrices can be simultaneous triangularization. By constructing a time-varying piecewise continuous Lyapunov function, sufficient conditions are proposed in term of linear matrix inequality, which ensure the global synchronization of switched complex networks under any given dwell time switching. Compared with the existing results, the dwell time in this paper is any given dwell time rather than the calculated time, and multiple-Lyapunov function makes the corresponding Lyapunov function value is monotone decreasing at the switching times, which avoids solving the growth rate "μ" of two adjacent Lyapunov functions at the switching time.(2) Based on dwell time method, synchronization problem of complex dynamical networks with identical nodes are investigated. The complex network systems have the following characteristics: all sub-networks can not achieve synchronization; Complex dynamical networks have switching topology; The outer coupling matrix is asymmetric matrices and the sum of matrix row elements equal to zero; The outer coupling matrices can be simultaneous triangularization. The dwell time is viewed as the input, which is designed to achieve the synchronization. The synchronization criterion is obtianed. Different from the existing methods, when all of the sub-networks are not synchronized, the switched complex networks can achieve synchronization by designing switching signal in the frame of dwell time. So, this problem belongs to synchronous switching signal design problem.(3) Synchronization of switched complex networks with non-identical nodes is studied. The complex network systems have the following characteristics:The complex nerwoks have non-identical nodes and no common equilibrium solution is assumed to exist; Complex dynamical networks have switching topology; The outer coupling matrices can be simultaneous triangularization. We choose the average state of all nodes as the target of synchronization, establish the dynamical error equation, and then simplify the error system by similarity transformation and coordinate transformation. By the common Lyapunov function method and the final boundedness theorem, a common Lyapunov function is constructed by those of several lower dimensional dynamic systems, and a global synchronization criterion in the sense of boundedness is proposed under arbitrary switching topology. When all of the sub-networks can not achieve synchronization, synchronization is achieved by designing switching law.(4) Pinning synchronization of switched complex dynamical networks is studied. Two basic problems are considered: one is pinning synchronization of switched complex networks under arbitrary switching; the other is pinning synchronization of switched complex networks by design of switching when synchronization cannot achieved by using any individual connection topology alone. For the two problems, common Lyapunov function method and single Lyapunov function method are used respectively, some global synchronization criteria are proposed and the designed switching law is given. This synchronization condition is relatively simple, and adapt to the non switching system, so it can be seen as further expansion of existing results.(5) Synchronization problems between two switched complex networks are studied. The considered synchronization problems belong to outer synchronization, and the synchronization scheme consists of identical drive and response complex networks. The drive system is coupled with the response system by the controller. The designed controllers have two forms: one is the distributed controller, compared with the ordinary feedback controller, distributed controller can better reflect the distribution characteristics of the network, because it does not depend on a node, so it has better robustness and fault-tolerant, so it is very suitable for engineering applications; the other is pinning controller, which can reduce the number of controlled nodes. For the two kinds of control strategy, single Lyapunov function method are applied respectively, some criteria are proposed and switching signals are designed which guarantee the synchronization between drive and response complex system. The switched complex network can get synchronization by design switching when none of the sub-networks can achieve synchronization.Finally, the results of the dissertation are summarized and further research problems are point out.