Ultradiscrete two-variable Oregonator

Abstract

Ultradiscretization is a limiting procedure transforming a given differential/difference equation into a ultradiscrete equation. Ultradiscrete equations are expressed by addition, subtraction and/or max. The procedure is expected to preserve the essential properties of the original equations. As a method of ultradiscretization, there is "tropical discretization" proposed by M. Murata. In this paper, we shall modify it, and derive a ultradiscrete equation from the continuous model of the BZ reaction. The derived equation generates a cellular automaton by restricting the values of the parameters, which is equivalent to one of those introduced by D. Takahashi, A. Shida, and M. Usami. By setting appropriate initial values, we can obtain the typical patterns of the BZ reaction. Furthermore, we consider the equation without diffusion effect and derive the explicit solutions. As a result, the solutions corresponding to the limit cycle (oscillation) appearing in the continuous model will be found.