Remark on topological entropy and ${\mathscr {P}}$-chaos of a coupled lattice system with non-zero coupling constant related with Belusov-Zhabotinskii reaction

Abstract

In this paper we study some chaotic properties of the following systems which is posed by Kaneko in (Phys Rev Lett, 65: 1391-1394, 1990) and is related to the Belusov-Zhabotinskii reaction:

$$\begin{aligned} u_{b}^{a+1}=(1-\alpha )r(u_{b}^{a})+ \frac{1}{2}\alpha [r(u_{b-1}^{a})+r(u_{b+1}^{a})], \end{aligned}$$

where a is discrete time index, b is lattice side index with system size T, $\alpha \in [0, 1]$ is coupling constant and r is a continuous selfmap on $J=[0, 1]$. It is proven that for each continuous selfmap r on J, the topological entropy of such a coupled lattice system with $\alpha =0$ is not less than the topological entropy of r, and that for each continuous selfmap on J with positive topological entropy, the above system with $\alpha =0$ is ${\mathscr {P}}$-chaotic, where ${\mathscr {P}}$ is one of the three properties: Li–Yorke chaos, distributional chaos, $\omega$-chaos. Moreover, we deduce that for each continuous selfmap r on J and any $\alpha \in [0, 1]$, if r is $\omega$-chaotic, then so is the above system.