Turbulence, erratic property and horseshoes in a coupled lattice system related with Belusov-Zhabotinsky reaction

Abstract

In this paper we continue to study the chaotic properties of the following lattice dynamical system: bji+1= a1 g(bji)+ a2 g(bj-1i)+ a3 g(bj+1i), where i is discrete time index, j is lattice side index with system size L, g is a selfmap on [0, 1] and a1+a2+a3 ∊ [0, 1] with a1+a2+a3=1 are coupling constants. In particular, it is shown that if g is turbulent (resp. erratic) then so is the above system, and that if there exists a g-connected family G with respect to disjointed compact subsets D1, D2, …, Dm, then there is a compact invariant set K'⊆D' such that F |K' is semi-conjugate to m-shift for any coupling constants a1+a2+a3 ∊ [0, 1] with  a1+a2+a3=1, where D' ⊆ IL is nonempty and compact. Moreover, an example and two problems are given.