A study on ⟨(Q,S,R)-γ⟩-dissipative synchronisation of coupled reaction–diffusion neural networks with time-varying delays

ABSTRACT

This study examines the problem of dissipative synchronisation of coupled reaction–diffusion neural networks with time-varying delays. This paper proposes a complex dynamical network consisting of N linearly and diffusively coupled identical reaction–diffusion neural networks. By constructing a suitable Lyapunov–Krasovskii functional (LKF), utilisation of Jensen's inequality and reciprocally convex combination (RCC) approach, strictly $〈(\mathcal{Q},\mathcal{S},\mathcal{R})-\gamma 〉$-dissipative conditions of the addressed systems are derived. Finally, a numerical example is given to show the effectiveness of the theoretical results.

KEYWORDS:

• Complex dynamical networks
• $〈(\mathcal{Q},\mathcal{S},\mathcal{R})-\gamma 〉$-dissipativity
• reaction–diffusion
• synchronisation

Additional information

Funding

This work was jointly supported by Department of Science and Technology (DST) [project number SR/FTP/MS-039/2011]; the National Natural Science Foundation of China [61773217], [61374080]; the Natural Science Foundation of Jiangsu Province [BK20161552]; the Alexander von Humboldt Foundation of Germany [fellowship CHN/1163390]; Qing Lan Project of Jiangsu Province.

Notes on contributors

M. Syed Ali

M. Syed Ali graduated from the Department of Mathematics of Gobi Arts and Science College affiliated to Bharathiar University, Coimbatore, in 2002. He received his post-graduation in mathematics from Sri Ramakrishna Mission Vidyalaya College of Arts and Science affiliated to Bharathiar University, Coimbatore, Tamil Nadu, India, in 2005. He was awarded master of philosophy in 2006 in the field of mathematics with specialised area of numerical analysis from Gandhigram Rural University Gandhigram, India. He was conferred with doctor of philosophy in 2010 in the field of mathematics specialised in the area of fuzzy neural networks in Gandhigram Rural University, Gandhigram, India. He was selected as a post-doctoral fellow in the year 2010 for promoting his research in the field of mathematics at Bharathidasan University, Trichy, Tamil Nadu and also worked there from November 2010 to February 2011. Since March 2011, he is working as an assistant professor in the Department of Mathematics, Thiruvalluvar University, Vellore, Tamil Nadu, India. He was awarded young scientist award 2016 by The Academy of Sciences, Chennai. He has published more than 70 research papers in various SCI journals holding impact factors. He has also published research articles in national journals and international conference proceedings. He also serves as a reviewer for several SCI journals. His research interests include stochastic differential equations, dynamical systems, fuzzy neural networks, complex networks and cryptography.

Quanxin Zhu

Quanxin Zhu received the PhD degree from Sun Yatsen (Zhongshan) University, Guangzhou, China, in 2005. From July 2005 to May 2009, he was with the South China Normal University. From May 2009 to August 2012, he was with the Ningbo University. He is currently a professor of Nanjing Normal University, and he has obtained the Alexander von Humboldt Foundation of Germany. Prof. Zhu is an associate editor of the Journal of Mathematical Problems in Engineering, Journal of Applied Mathematics, Research in Applied Mathematics, Journal of Transnational Journal of Mathematical Analysis and Applications, Journal of Advances in Applied Mathematics, and he is a reviewer of Mathematical Reviews and Zentralblatt Math. Also, Prof. Zhu is a senior member of the IEEE and he is the lead guest editor of “Recent Developments on the Stability and Control of Stochastic Systems” in Mathematical Problems and its Applications in Engineering. Prof. Zhu is the lead editor of “Dynamical Analysis of Biological Systems Possessing Random Noises” in Frontiers in Applied Mathematics Statistics, and is the editor of “ICMMCMSE 2017” in Mathematics and Computers in Simulation. Prof. Zhu has obtained 2011 Annual Chinese “One Hundred The Most Influential International Academic Paper” award and has been one of most cited Chinese researchers in 2014–2016, Elsevier. Prof. Zhu is a reviewer of more than 40 other journals and he is the author or coauthor of more than 120 journal papers. His research interests include random processes, stochastic control, stochastic differential equations, stochastic partial differential equations, stochastic stability, nonlinear systems, Markovian jump systems and stochastic complex networks.

S. Pavithra

S. Pavithra was born in 1994. She received her BSc and MSc degrees in the field of mathematics from Auxilium College of Arts and Science affiliated to Thiruvalluvar University, Vellore, Tamil Nadu, India. Currently she is pursuing MPhil degree in the Department of Mathematics, Thiruvalluvar University, Tamil Nadu, India. Her research interests include complex dynamical networks.

N. Gunasekaran

N. Gunasekaran was born in 1987. He received his BSc degree in the field of mathematics during 2006–2009 from Mahendra Arts and Science College, Namakkal affiliated to Periyar University, Salem, Tamil Nadu, India. He received his post-graduation in mathematics from Jamal Mohamed College affiliated to Bharathidasan University, Trichy, Tamil Nadu, India, during 2010–2012. He was awarded Master of Philosophy in 2014 in the field of mathematics with specialized area of cryptography from Bharathidasan University, Trichy, India. He received Doctor of Philosophy in mathematics from Thiruvalluvar University, Tamil Nadu, India, in 2017. He was availed Junior Research Fellowship during 2014–2017 from Department of Science and Technology-Science and Engineering Research Board (DST-SERB), Government of India, New Delhi, India. Currently he is working as a postdoctoral research fellow, Research Center for Wind Energy Systems in Kunsan National University, South Korea. He serves as a reviewer for various SCI journals. His research interest is in the area of stability analysis, sampled-data control, complex dynamical networks, fuzzy neural networks and cryptography.