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.