Synchronization of delayed coupled reaction-diffusion systems on networks

This paper mainly focus on the exponential stabilization problem of coupled systems on networks with mixed time-varying delays. Periodically intermittent control is used to control the coupled systems on networks with mixed time-varying delays.Moreover, based on the graph theory and Lyapunov method, two different kinds of stabilization criteria are derived, which are in the form of Lyapunov-type theorem and coefficients-type criterion, respectively. These laws reveal that the stability has a close relationship with the topology structure of the networks. In addition, as a subsequent result, a decision theorem is also presented. It is straightforward to show the stability of original system can be determined by that of modified system with added absolute value into the coupling weighted-value matrix. Finally, the feasibility and validity of the obtained results are demonstrated by several numerical simulation figures.

Section: Lyapunov-type Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A graph‐theoretic method to stabilize the delayed coupled systems on networks based on periodically intermittent control

Guo

Xiao

Zhang

2017

“…Synchronization has become an active research subject in nonlinear science. The current problems of synchronization are very interesting, non-traditional, and indeed very challenging [4][5][6][7]. Many scientists who are interested in this field have struggled to achieve the synchronization of different fractional-order chaotic systems [8], mainly due to its potential applications in secure communication and cryptography [9,10].…”

Section: Introductionmentioning

confidence: 99%

A robust method for new fractional hybrid chaos synchronization

Ouannas

Azar

Vaidyanathan

2016

101

In this paper, a robust mathematical method is proposed to study a new hybrid synchronization type, which is a combining generalized synchronization and inverse generalized synchronization. The method is based on Laplace transformation, Lyapunov stability theory of integer-order systems and stability theory of linear fractional systems. Sufficient conditions are derived to demonstrate the coexistence of generalized synchronization and inverse generalized synchronization between different dimensional incommensurate fractional chaotic systems. Numerical test of the method is used.

“…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.)…”

mentioning

confidence: 99%

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

Zhang

Wang

2017

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...