Synchronization of complex networks with time‐varying inner coupling and outer coupling matrices

Abstract

Synchronization of complex networks with time-varying coupling matrices is studied in this paper. Two kinds of timevarying coupling are taken into account. One is the time-varying inner coupling in the node state space and the other is the time-varying outer coupling in the network topology space. By respectively setting linear controllers and adaptive controllers, time-varying complex networks can be synchronized to a desired state. Meanwhile, different influences of the control parameters of linear controllers and adaptive controllers on the synchronization have also been investigated. Based on the Lyapunov function theory, we construct appropriate positive-definite functions, and several sufficient synchronization criteria are obtained. Numerical simulations further illustrate the effectiveness of conclusions. IntroductionOver the last few decades, the dynamics research of complex networks has aroused extensive concern among researchers in all disciplines domestic and overseas. A complex network is a structural system that contains a great deal of nodes that are connected by a great deal of edges. In a network model, nodes represent individual elements and edges represent relationships between individuals. Owing to the universality of the application and the simplicity of description, complex networks have been widely used in various fields, examples of which include biochemical reaction networks, multi-agent systems, traffic networks, neural networks, financial networks, and social networks [1][2][3][4][5][6].Controlling complex networks has been a central goal for the study of complex networks, whose theories and methods are derived from classical control theory, while there remain many differences between the classical control theory and the control of complex networks. The classical control theory emphasizes individual intrinsic dynamics, and the control of complex networks concerns both intrinsic dynamics of the node itself and the dynamics among nodes [7,8]. The study of controlling complex networks is of great significance for the deep understanding of nonlinear dynamics mechanism. In particular, rapidly growing interests have focused on the synchronization problem.Synchronization is one of the most important nonlinear phenomena in the real world, and it is also an important part of the complex network control area [9,10]. Synchronization can be described as a process, in which all the nodes seek to adjust a certain property of their motion to a common behavior in the limit as time tends to infinity. Over the past 10 years, many network models (for example, static network model [11,12], time-varying network model [13,14], delayed network model [15][16][17], etc.) and synchronization patterns (for example, projective synchronization [18,19], exponential synchronization [20], cluster synchronization [21,22], lag synchronization [23], etc.) had been proposed, which greatly enriched the study of nonlinear dynamics. Actually, however, complex networks could not often synchronize by themselves, and a large am...

Synchronization for fractional‐order multi‐linked complex network with two kinds of topological structure via periodically intermittent control

“…One of the major topics drawing considerable concerns is synchronization of CNs. Currently, many results on synchronization analysis of CNs have been achieved for the strong demands from diverse sorts of fields in parallel image processing, pattern recognition, and secure communication . Besides, fractional‐order calculus can be seen as a generalization of integer‐order differentiation and integration to arbitrary non‐integer order.…”

“…Currently, many results on synchronization analysis of CNs have been achieved for the strong demands from diverse sorts of fields in parallel image processing, pattern recognition, and secure communication. [7][8][9] Besides, fractional-order calculus can be seen as a generalization of integer-order differentiation and integration to arbitrary non-integer order. Actually, many processes with memory and hereditary properties generally or most likely are fractional-order CNs, rather than integer-order CNs.…”

Synchronization of Uncertain Complex Networks with Time‐Varying Node Delay and Multiple Time‐Varying Coupling Delays

“…Recently, synchronization of all dynamical nodes in a network, as an interesting and significant topic, has drawn special attention. Many efforts have also been to address the synchronization problem of complex networks [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

“…In addition, since complex networks cannot often synchronize by themselves, a large amount of control technologies such as slide control [17], impulsive control [18], pinning control [19], adaptive control [20], and intermittent control [21] have been designed to force the network to synchronize.…”

“…Recently, synchronization of all dynamical nodes in a network, as an interesting and significant topic, has drawn special attention. Many efforts have also been to address the synchronization problem of complex networks . Specifically, some synchronization patterns, such as cluster synchronization , bounded synchronization , projective synchronization , phase synchronization , guaranteed cost synchronization , and finite‐time synchronization , as well as some other hybrid synchronization patterns have been proposed.…”