Traveling wave solutions for discrete-Time Model of delayed Cellular Neural Networks

The aim of this work is to study the existence and stability of traveling wave solutions for discrete-time model of delayed cellular neural networks distributed in the one-dimensional integer lattice ℤ1. Since the dynamics of each given cell depends on its left and right neighboring cells, it is not easy to construct the traveling wave solutions. Using the method of step along with positive characteristic roots of the equations, we successfully prove the existence of traveling wave solutions. Moreover, we show that all the traveling wave solutions are unstable. We also provide some numerical results to support our results, and point out the different structures of traveling wave solutions between the continuous-time and discrete-time models.

[1]  TRAVELING WAVE SOLUTIONS IN CELLULAR NEURAL NETWORKS WITH MULTIPLE TIME DELAYS , 2005 .

[2]  L.O. Chua,et al.  Cellular neural networks , 1993, 1988., IEEE International Symposium on Circuits and Systems.

[3]  O. Diekmann,et al.  On the bounded solutions of a nonlinear convolution equation , 1978 .

[4]  Leon O. Chua,et al.  Pattern formation properties of autonomous Cellular Neural Networks , 1995 .

[5]  Suh-Yuh Yang,et al.  On camel-like traveling wave solutions in cellular neural networks , 2004 .

[6]  Lin-Bao Yang,et al.  Cellular neural networks: theory , 1988 .

[7]  Wenxian Shen,et al.  Traveling waves in cellular neural networks , 1999 .

[8]  R. Yuan,et al.  Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications , 2010 .

[9]  Leon O. Chua,et al.  Cellular neural networks: applications , 1988 .

[10]  Suh-Yuh Yang,et al.  Diversity of Traveling Wave Solutions in Delayed Cellular Neural Networks , 2008, Int. J. Bifurc. Chaos.

[11]  Suh-Yuh Yang,et al.  Structure of a class of traveling waves in delayed cellular neural networks , 2005 .

[12]  Wan-Tong Li,et al.  Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays , 2006 .

[13]  Jonq Juang,et al.  Cellular Neural Networks: Mosaic Pattern and Spatial Chaos , 2000, SIAM J. Appl. Math..

[14]  Vitaly Volpert,et al.  Traveling Wave Solutions of Parabolic Systems , 1994 .

[15]  Cheng-Hsiung Hsu.  Smale Horseshoe of Cellular Neural Networks , 2000, Int. J. Bifurc. Chaos.

[16]  Dr. Gabriele Manganaro,et al.  Cellular Neural Networks , 1999, Springer Series in Advanced Microelectronics.

[17]  Patrick Thiran,et al.  Dynamics and self-organization of locally coupled neural networks , 1996 .

[18]  Leon O. Chua,et al.  The CNN paradigm , 1993 .

[19]  Klaus W. Schaaf.  Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations , 1987 .

[20]  Xiao-Qiang Zhao,et al.  Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models , 2003 .

[21]  Peixuan Weng,et al.  Existence of traveling wave solutions in nonlinear delayed cellular neural networks , 2009 .

[22]  Luigi Fortuna,et al.  The CNN Paradigm for Complexity , 2008 .

[23]  Peixuan Weng,et al.  Deformation of Traveling Waves in Delayed Cellular Neural Networks , 2003, Int. J. Bifurc. Chaos.

[24]  Rong Yuan,et al.  Traveling waves of delayed reaction–diffusion systems with applications , 2011 .

[25]  Shiwang Ma,et al.  Traveling Wavefronts for Delayed Reaction-Diffusion Systems via a Fixed Point Theorem , 2001 .

[26]  Xingfu Zou,et al.  Asymptotic and Periodic Boundary Value Problems of Mixed FDEs and Wave Solutions of Lattice Differential Equations , 1997 .

[27]  Bingtuan Li,et al.  Existence of traveling waves for integral recursions with nonmonotone growth functions , 2009, Journal of mathematical biology.

[28]  Chih-Wen Shih,et al.  Complete Stability for Standard Cellular Neural Networks , 1999 .

[29]  Cheng-Hsiung Hsu,et al.  Existence and multiplicity of traveling waves in a lattice dynamical system , 2000 .

[30]  Chunhua Ou,et al.  Persistence of wavefronts in delayed nonlocal reaction-diffusion equations , 2007 .

[31]  Leon O. Chua,et al.  The analogic cellular neural network as a bionic eye , 1995, Int. J. Circuit Theory Appl..