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

Abstract

Guirao and Lampart (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. Motivated by Guirao and Lampart (J Math Chem 48:159–164, 2010) and Li and Zhao (J Math Chem. doi: 10.1007/s10910-015-0538-y ), in this paper we study the following more general lattice dynamical systems: $$\begin{aligned} y_{i}^{j+1}=(1-\eta )h_{i}\left( y_{i}^{j}\right) +\frac{1}{2}\eta \left[ h_{i}\left( y_{i-1}^{j}\right) -h_{i}\left( y_{i+1}^{j}\right) \right] , \end{aligned}$$ where j is discrete time index, i is lattice side index with system size H, $$\eta \in J=[0, 1]$$ is coupling constant and $$h_{i}$$ is a continuous selfmap on J for any $$i\in \{1, 2, \ldots , H\}$$ . In particular, we obtain that the following hold: Moreover, we give two examples. These results extend the results of Guirao and Lampart (J Math Chem 48:159–164, 2010).