Chow, Shui-Nee; Shen, Wenxian Dynamics in a discrete Nagumo equation: Spatial topological chaos. (English) Zbl 0840.34012 SIAM J. Appl. Math. 55, No. 6, 1764-1781 (1995). A coupled map lattice \[ u_j(n+ 1)= u_j(n)+ k(u_{j- 1}(n)+ u_{j+ 1}(n)- 2u_j(n))+ \alpha f(u_j(n)),\tag{1} \] where \(u_j(n)\in \mathbb{R}\), \(f(u)= u(u- a)(1- u)\), \(0< a< 1\), is considered. \(j\in \mathbb{Z}\) is treated as a spatial coordinate, \(n\in \mathbb{R}^+\) as time. Equation (1) is a discrete analog of the Nagumo equation \[ {\partial u\over \partial t}= D {\partial^2 u\over \partial x^2}+ f(u). \] The problem of the existence of standing waves, traveling waves and of spatial topological chaos is under investigation. A standing wave is a time-independent solution, a traveling wave is a solution with the form \(u_j(n)= g(j+ cn)\), where \(g\) is a scalar function. Spatial topological chaos occurs when the translation dynamical system generated by shift maps of the set of the stable standing waves behaves stochastically. Regions of parameters \(\alpha\), \(k\) corresponding to existence of spatial topological chaos and to non-existence of standing and traveling waves, are established. Reviewer: Yu.N.Bibikov (St.Peterburg) Cited in 65 Documents MSC: 34A35 Ordinary differential equations of infinite order 35K57 Reaction-diffusion equations 37C75 Stability theory for smooth dynamical systems 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior Keywords:discrete analog of the Nagumo equation; standing waves; traveling waves; spatial topological chaos PDFBibTeX XMLCite \textit{S.-N. Chow} and \textit{W. Shen}, SIAM J. Appl. Math. 55, No. 6, 1764--1781 (1995; Zbl 0840.34012) Full Text: DOI