Global Asymptotic and Exponential Stability of Tri-Cell Networks with Different Time Delays

Document Type: بنیادی - نظری


Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran


In this paper‎, ‎a bidirectional ring network with three cells and different time delays is presented‎. ‎To propose this model which is a good extension of three-unit neural networks‎, ‎coupled cell network theory and neural network theory are applied‎. ‎In this model‎, ‎every cell has self-connections without delay but different time delays are assumed in other connections‎. ‎A suitable Lyapunov function is presented for this model which helps to get sufficient conditions to guarantee asymptotic and exponential stability of the model‎. ‎Also‎, ‎these conditions are independent of time delays‎. ‎Finally‎, ‎analytical results are confirmed by numerical examples which are stated.


Main Subjects


‎Ahmadkhanlou F.‎, ‎Adeli H‎. ‎(2005)‎. ‎``Optimum cost design of reinforced concrete slabs using neural dynamics model"‎, ‎Engineering Applications of Artificial Intelligence‎, ‎18‎, ‎65--72‎.

‎bibitem{R2} Amari S.I.‎, ‎Cichocki A‎. ‎(1998)‎. ‎``Adaptive blind signal processing-neural network approaches"‎, ‎Proceedings of the IEEE‎, ‎86‎, ‎2026--2048‎.

‎bibitem{ZD} Dadi Z‎. ‎(2017)‎. ‎``Dynamics of two-cell systems with discrete delays"‎, ‎Advances in Computational Mathematics‎, ‎43:3‎, ‎653--676‎.

‎bibitem{G2} Golubitsky M.‎, ‎Pivato M.‎, ‎Stewart I‎. ‎(2004)‎. ‎``Interior symmetry and local bifurcation in coupled cell networks"‎, ‎Dynamical Systems‎, ‎19:4‎, ‎389--407‎.

‎bibitem{G3} Golubitsky M.‎, ‎Stewart I‎. ‎(2006)‎. ‎``Nonlinear dynamics of networks‎: ‎the groupoid formalism"‎, ‎Bulletin of the American Mathematical Society‎, ‎43‎, ‎305--364‎.

‎bibitem{G} Golubitsky M.‎, ‎Stewart I.‎, ‎Buono P.L.‎, ‎Collins J.J‎. ‎(1998)‎. ‎``A modular network for legged locomotion"‎, ‎Physica D‎, ‎115‎, ‎56--72‎.

‎bibitem{G4} Golubitsky M.‎, ‎Stewart I.‎, ‎T"{o}r"{o}k A‎. ‎(2005)‎. ‎``Patterns of synchrony in coupled cell networks with multiple arrows"‎, ‎SIAM Journal on Applied Dynamical Systems‎, ‎4:1‎, ‎78--100‎.

‎bibitem{R3} Guo S.‎, ‎Huang L‎. ‎(2003)‎. ‎``Hopf bifurcating periodic orbits in a ring of neurons with delays"‎, ‎Physica D‎, ‎183‎, ‎19--44‎.

‎bibitem{dj} Javidmanesh E.‎, ‎Dadi Z.‎, ‎Afsharnezhad Z.‎, ‎Effati S‎. ‎(2014)‎. ‎``Global stability analysis and existence of periodic solutions in an eight-neuron BAM neural network model with delays"‎, ‎Journal of Intelligent and Fuzzy Systems‎, ‎27:1‎, ‎391--406‎.

‎bibitem{R4} Jiang F.‎, ‎Shen J.‎, ‎Li X‎. ‎(2013) ``The LMI method for stationary oscillation of interval neural networks with three neuron activations under impulsive effects"‎, ‎Nonlinear Analysis‎: ‎Real World Applications‎, ‎14:3‎, ‎1404--1416‎.

‎bibitem{R5} Kwon O.M.‎, ‎Kwon J.W.‎, ‎Kim S.H‎. ‎(2011)‎. ‎``New results on stability criteria for neural networks with time-varying delays"‎, ‎Chinese Physics B‎, ‎20:5‎, ‎050505‎.

‎bibitem{R6} Kwon O.M.‎, ‎Park M.‎, ‎Lee S.M.‎, ‎Park J.H.‎, ‎Cha E.J‎. ‎(2013)‎. ‎``Stability for neural networks with time-varying delay via some new approaches"‎,

‎IEEE Transactions on Neural Networks and Learning Systems‎, ‎24:2‎, ‎181--193‎.

‎bibitem{R7} Li H.‎, ‎Liao X.‎, ‎Li C.‎, ‎Huang H.‎, ‎Li C‎. ‎(2011)‎. ‎``Edge detection of noisy images based on cellular neural networks"‎, ‎Communications in Nonlinear Science and Numerical Simulations‎, ‎16:9‎, ‎3746--3759‎.

‎bibitem{R9} Luo R.‎, ‎Xu H.‎, ‎Wang W.S.‎, ‎Sun J.‎, ‎Xu W‎. ‎(2016)‎. ‎``A weak condition for global stability of delayed neural networks"‎, ‎Journal of Industrial and Management Optimization‎, ‎12:2‎, ‎505--514‎.

‎bibitem{R10} Shen Y.‎, ‎Wang J‎. ‎(2012)‎. ‎``Robustness analysis of global exponential stability of recurrent networks in the presence of time delays and random disturbances"‎, ‎IEEE Transactions on Neural Networks and Learning Systems‎, ‎23:1‎, ‎87--96‎.

‎bibitem{R11} Shu Y.‎, ‎Liu X.‎, ‎Liu Y‎. ‎(2016)‎. ‎``Stability and passivity analysis for uncertain discrete-time neural networks with time-varying delay"‎, ‎Neurocomputing‎, ‎173:3‎, ‎1706--1714‎.

‎bibitem{R12} Song Y.‎, ‎Han M.‎, ‎Wei J‎. ‎(2005)‎. ‎``Stability and Hopf bifurcation on a simplified BAM neural network with delays"‎, ‎Physica D‎: ‎Nonlinear Phenomena‎, ‎200‎, ‎185--204‎.

‎bibitem{R13} Tian J.‎, ‎Xie X‎. ‎(2010)‎. ‎``New asymptotic stability criteria for neural networks with time varying delay"‎, ‎Physics Letters A‎, ‎374:7‎, ‎938--943‎.

‎bibitem{S} Stewart I.‎, ‎Golubitsky M.‎, ‎Pivato M‎. ‎(2003)‎. ‎``Symmetry groupoids and patterns of synchrony in coupled cell networks"‎, ‎SIAM Journal on Applied Dynamical Systems‎, ‎2:4‎, ‎606--646‎.

‎bibitem{R15} Wei J.‎, ‎Li M.Y‎. ‎(2004)‎. ‎``Global existence of periodic solutions in a tri-neuron network model with delays"‎, ‎Physica D‎, ‎198‎, ‎106--119‎.

‎bibitem{R16} Wei J.J.‎, ‎Velarde M.G‎. ‎(2004)‎. ‎``Bifurcation analysis and existence of periodic solutions in a simple neural network with delays"‎, ‎Chaos‎, ‎143‎, ‎940--953‎.

‎bibitem{R17}Yan X.P‎. ‎(2006)‎. ‎``Hopf bifurcation and stability for a delayed tri-neuron network model"‎, ‎Journal of Computational and Applied Mathematics‎, ‎196‎, ‎579--595‎.

‎bibitem{R8} Yiping L.‎, ‎Lemmert R.‎, ‎Volkmann P‎. ‎(2001)‎. ‎``Bifurcation of periodic solution in a three-unit neural network with delay"‎, ‎Acta Mathematicae Applicatae Sinica‎, ‎17:3‎, ‎375--381‎.

‎bibitem{R18}Yuan Y‎. ‎(2007)‎. ‎``Dynamics in a delayed-neural network"‎, ‎Chaos‎, ‎Solitons and Fractals‎, ‎33:2‎, ‎443--454‎.

‎bibitem{R19} Zeng Z.‎, ‎Huang D.S.‎, ‎Wang Z‎. ‎(2008)‎. ‎``Pattern memory analysis based on stability theory of cellular neural networks"‎, ‎Applied Mathematical Modelling‎, ‎32:1‎, ‎112--121‎.

‎bibitem{R20} Zeng Z.‎, ‎Wang J‎. ‎(2007)‎. ‎``Analysis and design of associative memories based on recurrent neural networks with linear saturation activation functions and time-varying delays"‎, ‎Neural computation‎, ‎19:8‎, ‎2149--2182‎.

‎bibitem{R21} Zheng M.‎, ‎Mao Z.‎, ‎Li K.‎, ‎Fei M‎. ‎(2016)‎. ‎``Quadratic separation framework for stability analysis of a class of systems with time delays"‎,

‎Neurocomputing‎, ‎174‎, ‎466--474‎.

‎bibitem{R22} Zheng C.D.‎, ‎Shan QH‎, ‎Wang Z‎. ‎(2012)‎. ‎``Novel stability criterion for cellular neural networks‎: ‎an improved Gu's discretized LKF approach"‎, ‎Journal of the Franklin Institute‎, ‎349:1‎, ‎25--41‎.

‎bibitem{R23} Zou S.‎, ‎Huang L.‎, ‎Wang Y‎. ‎(2010)‎. ‎``Bifurcation of a three-unit neural network"‎, ‎Applied Mathematics and Computation‎, ‎217:2‎, ‎904--917‎.