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Dynamics in a discrete Nagumo equation: Spatial topological chaos. (English) Zbl 0840.34012

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.

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
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