A note on the principal measure and distributional $({\varvec{p, q}})$-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction

Abstract

García Guirao and Lampart in (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this paper, we prove that for any non-zero coupling constant $\varepsilon \in (0, 1)$, this coupled map lattice system is distributionally $(p, q)$-chaotic for any pair $0\le p\le q\le 1$, and that its principal measure is not less than $(1-\varepsilon )\mu _{p}(f)$. Consequently, the principal measure of this system is not less than

$$\begin{aligned} (1-\varepsilon )\left( \frac{2}{3}+\sum \limits _{n=2}^{\infty }\frac{1}{n}\frac{2^{n-1}}{(2^{n}+1) (2^{n-1}+1)}\right) \end{aligned}$$

for any non-zero coupling constant $\varepsilon \in (0, 1)$ and the tent map $\Lambda$ defined by

$$\begin{aligned} \Lambda (x)=1-|1-2x|,\quad x\in [0, 1]. \end{aligned}$$