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

• DOI: 10.1007/s10910-015-0482-x
• Publication Date: 2015-02-20T12:56:39Z
• Authors: Jinxing Liu; Tianxiu Lu; Risong Li
• Keywords: Topological Entropy; Lattice (music); Coupling constant; Crystal system; Continuous map

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).