Dynamics of a Discrete Brusselator Model: Escape to Infinity and Julia Set

We consider a discrete version of the Brusselator Model of the famous Belousov-Zhabotinsky reaction in chemistry. The original model is a reaction-diffusion equation and its discrete version is a coupled map lattice. We study the dynamics of the local map, which is a smooth map of the plane. We discuss the set of trajectories that escape to infinity as well as analyze the set of bounded trajectories – the Julia set of the system. 1. The Brusselator model for the Belousov-Zhabotinsky reaction The Brusselator model is a famous model of chemical reactions with oscillations. It was proposed by Prigogine and Lefever in 1968 and the name was coined by Tyson (see [6]). In the middle of the last century Belousov and Zhabotinsky discovered chemical systems exhibiting oscillations. More precisely, they observed that cerium(III) and cerium(IV) were the cycling species: in a mix of potassium bromate, cerium(IV) sulfate, and citric acid in dilute sulfuric acid, the ratio of concentration of the Ce(IV ) and Ce(III) ions oscillated. While for most chemical reaction a state of homogeneity and equilibrium is quickly reached, the BelousovZhabotinsky reaction is a remarkable chemical reaction that maintains a prolonged state of non-equilibrium leading to macroscopic temporal oscillations and spatial pattern formation that is very life-like. The simplified mechanism for the Belousov-Zhabotinsky reaction is as follows (see [5]): Ce(III) → Ce(IV ) (1.1) Ce(IV ) + CHBr(COOH)2 → Ce(III) +Br− + other products (1.2) Equation (1.1) is autocatalyzed by BrO− 3 , and strongly inhibited by Br − ions. Therefore, as Ce(IV ) is produced in equation (1.1), the rate of equation (1.2) increases. This results in a high concentration of Br− which inhibits and slows equation (1.1). After the discovery of oscillating chemical reactions, in 1968 Prigogine proposed a virtual oscillating chemical reaction system – the Brusselator model. The net reaction is A+ B → D + E with transient appearance of intermediates X and Y. Here A and B are reactants and D and E are products. The reaction consists of four steps shown on Table 1. Step 3 is autocatalystic, since two X 1991 Mathematics Subject Classification. 37N25, 37N10, 35K57.