A note on Li–Yorke chaos in a coupled lattice system related with Belusov–Zhabotinskii reaction

Abstract

This paper is concerned with the following system which comes from a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) and is related to the Belusov–Zhabotinskii reaction: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f(x_{n}^{m})+\frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f(x_{n+1}^{m})\right] , \end{aligned}$$ where \(m\) is discrete time index, \(n\) is lattice side index with system size \(L\) (i.e., \(n=1, 2, \ldots , L\)), \(\varepsilon \) is coupling constant, and \(f(x)\) is the unimodal map on \(I\) (i.e., \(f(0)=f(1)=0\) and \(f\) has unique critical point \(c\) with \(0<c<1\) and \(f(c)=1\)). It is proved that for coupling constant \(\varepsilon =1\), this CML (Coupled Map Lattice) system is chaotic in the sense of Li–Yorke for each unimodal selfmap on the interval \(I=[0, 1]\).