Remark on positive entropy of a coupled lattice system related with Belusov–Zhabotinskii reaction

Abstract

In Garcia Guirao and Lampart (J Math Chem 48:159–164, 2010) presented a lattice dynamical system (LDS) stated by Kaneko (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this paper, we consider the following more general LDS: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f_{n}\left( x_{n}^{m}\right) +\frac{1}{2}\varepsilon \left[ f_{n}\left( x_{n-1}^{m}\right) -f_{n}\left( x_{n+1}^{m}\right) \right] , \end{aligned}$$ where m is discrete time index, n is lattice side index with system size L, \(\varepsilon \in I=[0, 1]\) is coupling constant and \(f_{n}\) is a continuous selfmap on I for every \(n\in \{1, 2, \ldots , L\}\). In particular, we prove that for zero coupling constant, if there is \(n\in \{1, 2, \ldots , L\}\) such that \(f_{n}\) has positive topological entropy, then so does this coupled map lattice system. This result extends the existing one.