Topological entropy and ${\fancyscript{P}}$-chaos of a coupled lattice system with non-zero coupling constant related with Belusov–Zhabotinskii reaction

Abstract

This paper focuses on the chaotic properties of the following systems stated by Kaneko (Phys Rev Lett 65:1391–1394, 1990) which 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 $L$, $\varepsilon \in (0, 1]$ is coupling constant and $f$ is a continuous selfmap of $[0, 1]$. It is shown that for every continuous selfmap $f$ of the interval $[0, 1]$, the topological entropy of such a coupled lattice system is not less than the topological entropy of the map $f$, and that for every continuous selfmap of the interval $[0, 1]$ with positive topological entropy, such a system is ${\fancyscript{P}}$-chaotic, where ${\fancyscript{P}}$ denotes one of the three properties: Li–Yorke chaos, distributional chaos, $\omega$-chaos. These results extend the ones of Wu and Zhu (J Math Chem 50:1304–1308, 2012), (J Math Chem 50:2439–2445, 2012) and Li et al. (J Math Chem 51:1712–1719, 2013).