Remark on topological entropy and $${\mathscr {P}}$$ 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.