Pinning impulsive directed coupled delayed dynamical network and its applications

Abstract

The main objective of the present paper is to further investigate pinning synchronisation of a complex delayed dynamical network with directionally coupling by a single impulsive controller. By developing the analysis procedure of pinning impulsive stability for undirected coupled dynamical network previously, some simple yet general criteria of pinning impulsive synchronisation for such directed coupled network are derived analytically. It is shown that a single impulsive controller can always pin a given directed coupled network to a desired homogenous solution, including an equilibrium point, a periodic orbit, or a chaotic orbit. Subsequently, the theoretical results are illustrated by a directed small-world complex network which is a cellular neural network (CNN) and a directed scale-free complex network with the well-known Hodgkin–Huxley neuron oscillators. Numerical simulations are finally given to demonstrate the effectiveness of the proposed control methodology.

Keywords:

• pinning impulsive synchronisation
• complex dynamical network
• directionally coupling
• CNN model
• Hodgkin–Huxley model
• time delays

Acknowledgements

The authors sincerely thank the anonymous reviewers for their valuable comments that have led to the present improved version of the original manuscript.

Additional information

Funding

This work is supported by the National Science Foundation of China [grant number 11272791] [grant number 61203006]; the Innovation Program of Shanghai Municipal Education Commission [grant number 14ZZ151] [grant number 10ZZ61].

Notes on contributors

Chunnan Lin

Chunnan Lin received his BS degree in applied mathematics from Shanghai University, Shanghai, China, in 2011. He is currently a master student at Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, China. His current research interests include analysis and applications of hybrid systems, and impulsive control.

Quanjun Wu

Quanjun Wu received his BS degree in applied mathematics from Qingdao University, Qingdao, China, in 2004, MS degree in applied mathematics from Hebei University of Technology, Tianjin, China, in 2007 and PhD degree in dynamics and control from Shanghai University, Shanghai, China, in 2010. He is currently a Associate Professor of the School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, China. His current research interests include coordinated control of networked multi-agent systems, dynamics and control of complex networks and complex systems.

Lan Xiang

Lan Xiang received her BS degree in physics from Southwest Normal University, Chongqing, China, in 1984, and MS degree in physics from Hebei University of Technology, Tianjin, China, in 2002. She is currently an Associate Professor of Department of Physics, School of Science, Shanghai University, Shanghai, China. Her research interests include theoretical physics, and theoretical analysis of complex systems and complex networks.

Jin Zhou

Jin Zhou received his BS degree in mathematics from Sichuan Normal College, Sichuan, China, in 1983, MS degree in applied mathematics from Chongqing University, Chongqing, China, in 1989 and Ph.D. degree in operations research and cybernetics from Shanghai University, Shanghai, in 2003, China. He is currently a Professor and Doctoral Advisor of Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, China. His research interests include control and synchronization of complex dynamical networks, cooperative control of networked multi-agent systems, and differential equations and system control.