Abstract :
In this paper, the chaotic properties of the following Belusov-Zhabotinskii’s reaction model is explored: a l k+1=(1-η)θ(a l k)+ 1/2 η[θ(a l-1 k)θ(a l+1 k)], where k is discrete time index, l is lattice side index with system size M, η∊ [0, 1) is coupling constant and θ is a continuous map on W=[-1, 1]. This kind of system is a generalization of the chemical reaction model which was presented by García Guirao and Lampart in [Chaos of a coupled lattice system related with the Belusov–Zhabotinskii reaction, J. Math. Chem. 48 (2010) 159-164] and stated by Kaneko in [Globally coupled chaos violates the law of large numbers but not the centrallimit theorem, Phys. Rev. Lett. 65 (1990) 1391-1394], and it is closely related to the Belusov-Zhabotinskii’s reaction. In particular, it is shown that for any coupling constant η ∊ [0, 1/2), any r ∊ {1, 2, ...} and θ=Qr, the topological entropy of this system is greater than or equal to rlog(2-2η), and that this system is Li-Yorke chaotic and distributionally chaotic, where the map Q is defined by Q(a)=1-|1-2a|, a ∊ [0, 1], and Q(a)=-Q(-a), a ∊ [-1, 0]. Moreover, we also show that for any c, d with 0≤c≤ d≤ 1, η=0 and θ=Q, this system is distributionally (c, d)-chaotic.
Keywords :
Coupled map lattice , Distributional chaos , Principal measure , Chaos in the sense of Li , Yorke , Topological entropy
