Global asymptotic and exponential synchronization of ring neural network with reaction–diffusion term and unbounded delay

Swati Tyagi1, Syed Abbas1, Mokhtar Kirane2
1School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, India
2LaSIE, Faculté des Sciences et Technologies, Université de La Rochelle, La Rochelle, France

Tóm tắt

In this paper, we consider a ring neural network of coupled neurons with distributed and discrete time-varying delays along with the reaction–diffusion terms. We derive sufficient conditions that ensure the existence and uniqueness of the equilibrium point, synchronized asymptotic stability and exponential synchronization by using the theory of topological degree, properties of M-matrix, Lyapunov functional and analytic methods. The obtained results remove the assumption on the boundedness of activation functions. At the end, we give two examples to show the validity of our analysis.

Từ khóa


Tài liệu tham khảo

Abbas S (2012) Existence and attractivity of k-pseudo almost automorphic sequence solution of a model of bidirectional neural networks. Acta Appl Math 119(1):57–74 Abbas S (2009) Pseudo almost periodic sequence solutions of discrete time cellular neural networks. Nonlinear Anal Model Control 14(3):283–301 Balasubramaniam P, Vembarasan V, Rakkiyappan R (2011) Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays. Commun Nonlinear Sci Numer Simul 16(4):2109–2129 Baldi P, Atiya AF (1994) How delays affect neural dynamics and learning. IEEE Trans Neural Netw 5(4):612–621 Bungay SD, Campbell SA (2007) Patterns of oscillation in a ring of identical cells with delayed coupling. Int J Bifurc Chaos 17(09):3109–3125 Campbell SA, Ruan S, Wolkowicz G, Wu J (1999) Stability and bifurcation of a simple neural network with multiple time delays. Fields Inst Commun 21(4):65–79 Campbell SA, Yuan Y, Bungay SD (2005) Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling. Nonlinearity 18(6):2827 Chen S, Cao J (2012) Projective synchronization of neural networks with mixed time-varying delays and parameter mismatch. Nonlinear Dyn 67(2):1397–1406 Feng C, Plamondon R (2012) An oscillatory criterion for a time delayed neural ring network model. Neural Netw 29:70–79 Gana Q, Liub T, Chang Liua TL (2016) Synchronization for a class of generalized neural networks with interval time-varying delays and reaction–diffusion terms. Nonlinear Anal Model Control 21(3):379–399 Guo D (1985) Nonlinear functional analysis. Shundong Sci. Tech. Press, Jinan Hale JK, Lunel SMV (1993) Introduction to functional differential equations. Applied Mathematical Sciences, vol 99. Springer-Verlag, New York, x+447 pp. ISBN: 0-387-94076-6 Hu C, Jiang H, Teng Z (2010) Impulsive control and synchronization for delayed neural networks with reaction–diffusion terms. IEEE Trans Neural Netw 21(1):67–81 Li R, Cao J (2016) Stability analysis of reaction–diffusion uncertain memristive neural networks with time-varying delays and leakage term. Appl Math Comput 278:54–69 Li X, Shen J (2010) LMI approach for stationary oscillation of interval neural networks with discrete and distributed time-varying delays under impulsive perturbations. IEEE Trans Neural Netw 21(10):1555–1563 Liao X, Fu Y, Gao J, Zhao X (2000) Stability of Hopfield neural networks with reaction–diffusion terms. Acta Electron Sin 28(1):78–80 Lou XY, Cui BT (2006) Asymptotic synchronization of a class of neural networks with reaction–diffusion terms and time-varying delays. Comput Math Appl 52(6):897–904 Lu JG, Lu LJ (2009) Global exponential stability and periodicity of reaction–diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions. Chaos Solitons Fractals 39(4):1538–1549 Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64(8):821 Phat VN, Trinh H (2010) Exponential stabilization of neural networks with various activation functions and mixed time-varying delays. IEEE Trans Neural Netw 21(7):1180–1184 Sheng L, Yang H (2008) Exponential synchronization of a class of neural networks with mixed time-varying delays and impulsive effects. Neurocomputing 71(16):3666–3674 Sheng L, Yang H, Lou X (2009) Adaptive exponential synchronization of delayed neural networks with reaction–diffusion terms. Chaos Solitons Fractals 40(2):930–939 Song Q (2009) Design of controller on synchronization of chaotic neural networks with mixed time-varying delays. Neurocomputing 72(13):3288–3295 Song Q, Cao J (2011) Synchronization of nonidentical chaotic neural networks with leakage delay and mixed time-varying delays. Adv Differ Equ 2011(1):1–17 Song Y, Han Y, Peng Y (2013) Stability and Hopf bifurcation in an unidirectional ring of \(n\) neurons with distributed delays. Neurocomputing 121:442–452 Tyagi S, Abbas S, Ray RK (2015) Stability analysis of an integro differential equation model of ring neural network with delay. Springer Proc Math Stat 143:37–49 Wang Y, Cao J (2007) Synchronization of a class of delayed neural networks with reaction–diffusion terms. Phys Lett A 369(3):201–211 Wang L, Zhang R, Wang Y (2009) Global exponential stability of reaction–diffusion cellular neural networks with S-type distributed time delays. Nonlinear Anal Real World Appl 10(2):1101–1113 Wang L, Zhao H, Cao J (2016) Synchronized bifurcation and stability in a ring of diffusively coupled neurons with time delay. Neural Netw 75:32–46 Wang Z, Zhang H (2010) Global asymptotic stability of reaction–diffusion Cohen–Grossberg neural networks with continuously distributed delays. IEEE Trans Neural Netw 21(1):39–49 Wei PC, Wang JL, Huang YL, Xu BB, Ren SY (2016) Impulsive control for the synchronization of coupled neural networks with reaction–diffusion terms. Neurocomputing 207:539–547 Wei-Yuan Z, Jun-Min L (2011) Global exponential stability of reaction–diffusion neural networks with discrete and distributed time-varying delays. Chin Phys B 20(3):030701 Yuan K, Cao J, Li HX (2006) Robust stability of switched Cohen–Grossberg neural networks with mixed time-varying delays. IEEE Trans Syst Man Cybern Part B Cybern 36(6):1356–1363 Yuan Y, Campbell SA (2004) Stability and synchronization of a ring of identical cells with delayed coupling. J Dyn Differ Equ 16(3):709–744 Zhang CK, He Y, Wu M (2010) Exponential synchronization of neural networks with time-varying mixed delays and sampled-data. Neurocomputing 74(1):265–273 Zhu Q, Cao J (2011) Exponential stability analysis of stochastic reaction–diffusion Cohen–Grossberg neural networks with mixed delays. Neurocomputing 74(17):3084–3091