STABLE PATTERN AND STANDING-WAVE FORMATION IN A SIMPLE ISOTHERMAL CUBIC AUTOCATALYTIC REACTION SCHEME

The formation of stable patterns is considered in a reaction-diffusion system based on the cubic autocatalator, A + 2B --> 3B, B --> C, with the reaction taking place within a closed region, the reactant A bein g replenished by the slow decay of precursor P via the reaction P --> A. The linear stability of the spatially uniform steady state (a, b) = (mu(-1),mu), where a and b are the dimensionless concentrations of re actant A and autocatalyst B and mu is a parameter representing the ini tial concentration of the precursor P, is discussed first. It is shown that a necessary condition for the bifurcation of this steady state t o stable, spatially non-uniform, solutions (patterns) is that the para meter D < 3 - 2 root 2 where D = D-b/D-a (D-a and D-b are the diffusio n coefficients of chemical species A and B respectively). The values o f mu at which these bifurcations occur are derived in terms of D and l ambda (a parameter reflecting the size of the system). Further informa tion about the nature of the spatially non-uniform solutions close to their bifurcation points is obtained from a weakly nonlinear analysis. This reveals that both supercritical and subcritical bifurcations are possible. The bifurcation branches are then followed numerically usin g a path-following method, with mu as the bifurcation parameter, for r epresentative values of D and lambda. It is found that the stable patt erns can lose stability through supercritical Hopf bifurcations and th ese stable, temporally periodic, spatially non-uniform solutions are a lso followed numerically.