kato’s chaos and p−chaos of a coupled lattice system given by garcía guirao and lampart which is related with belusov−zhabotinskii reaction

In (j. math. chem., 48: 66-71, 2010) and (j. math. chem., 48: 159- 164, 2010) garcía guirao and lampart presented the following lattice dynamical system stated by kaneko in (phys rev lett, 65: 1391-1394, 1990) which is related to the belusov-zhabotinskii reaction: z ^v+1 v = (1 − η)θ(z^u v) + 1/2η [θ(z^u v-1) − θ(z^u v+1)], where u is discrete time index, v is lattice side index with system size m, ηε [0,1] is coupling constant and θ is a continuous selfmap on h. they proved that for the tent map θ defined as θ(z) = 1 − |1-2z| for any zεh, the above system with η=0 has positive topological entropy and that such a system is li-yorke chaotic and devaney chaotic. in this article, we further consider the above system. in particular, we give a sufficient condition under which the above system is kato chaotic for η=0 and a necessary condition for the above system to be kato chaotic for η=0. moreover, it is deduced that for η=0, if θ is p-chaotic then so is this system, where a continuous map θ from a compact metric space z to itself is said to be p-chaotic if it has the pseudo-orbit-tracing property and the closure of the set of all periodic points for θ is the space z. also, an example and three open problems are presented.