topological entropy‎, ‎distributional chaos and the principal‎‎ measure of a class of belusov−zhabotinskii’s reaction models presented by garcía guirao and lampart ‎

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.‎