Note on chaos of a coupled lattice system related with the Belusov–Zhabotinskii reaction

Abstract

In García Guirao and Lampart (J Math Chem 48:159–164, 2010) considered a lattice dynamical system which is stated by Kaneko (Phys Rev Lett 65:1391–1394, 1990) and related to the Belusov–Zhabotinskii reaction. In this note, we introduce the new concepts of density one transitivity and density one weak mixing which are stronger forms of topological transitivity and explore the following more general lattice dynamical systems:

$$\begin{aligned} x_{n}^{m+1}=(1-\beta )w_{n}\left( x_{n}^{m}\right) +\frac{1}{2}\beta \left[ w_{n}\left( x_{n-1}^{m}\right) -w_{n}\left( x_{n+1}^{m}\right) \right] , \end{aligned}$$

where m is discrete time index, n is lattice side index with system size T, $\varepsilon \in I=[0, 1]$ is coupling constant and $w_{n}$ is a continuous map of I for every $n\in \{1, 2, \ldots , T\}$. In particular, we show that the for zero coupling constant, this CML (Coupled Map Lattice) system is density one transitive (resp. density one weakly mixing) if and only if so is $w_{n}$ for any $n\in \{1, 2, \ldots , T\}$.