# Ultradiscrete two-variable Oregonator

## TL;DR

This paper introduces a modified ultradiscretization method to derive a cellular automaton model from the continuous Belousov-Zhabotinsky (BZ) reaction, capturing oscillatory behaviors and pattern formations.

## Contribution

It proposes a new ultradiscretization approach, deriving a cellular automaton from the BZ reaction model and analyzing explicit solutions for oscillations.

## Key findings

- Derived a cellular automaton equivalent to the BZ reaction model
- Obtained explicit solutions representing oscillations and limit cycles
- Demonstrated pattern formation consistent with continuous BZ dynamics

## 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.