Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice Z1. For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space (γ,δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.

[1]  Gábor Fáth,et al.  Propagation failure of traveling waves in a discrete bistable medium , 1997, patt-sol/9711003.

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

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

[4]  Suh-Yuh Yang,et al.  Existence of Monotonic Traveling Waves in Lattice Dynamical Systems , 2005, Int. J. Bifurc. Chaos.

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

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

[7]  John Mallet-Paret,et al.  The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems , 1999 .

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

[9]  Shui-Nee Chow,et al.  Traveling Waves in Lattice Dynamical Systems , 1998 .

[10]  John Mallet-Paret,et al.  The Fredholm Alternative for Functional Differential Equations of Mixed Type , 1999 .

[11]  B. Zinner,et al.  Traveling wavefronts for the discrete Fisher's equation , 1993 .

[12]  James P. Keener,et al.  Propagation and its failure in coupled systems of discrete excitable cells , 1987 .

[13]  Leon O. Chua,et al.  Rtd-Based Cellular Neural Networks with Multiple steady States , 2001, Int. J. Bifurc. Chaos.

[14]  Thomas Erneux,et al.  Propagating waves in discrete bistable reaction-diffusion systems , 1993 .

[15]  B. Zinner,et al.  Existence of traveling wavefront solutions for the discrete Nagumo equation , 1992 .

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

[17]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

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