Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme

Abstract

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 being replenished by the slow decay of precursor P via the reaction P → A. The linear stability of the spatially uniform Steady state % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaabmGabaGaamyyaiaacYcacaWGIbaacaGLOaGaayzkaaGaeyyp% a0ZaaeWaceaacqaH8oqBdaahaaWcbeqaaiabgkHiTiaaigdaaaGcca% GGSaGaeqiVd0gacaGLOaGaayzkaaaaaa!48C3!\[\left( {a,b} \right) = \left( {\mu ^{ - 1} ,\mu } \right)\], where a and b are the dimensionless concentrations of reactant A and autocatalyst B and μ is a parameter representing the initial concentration of the precursor P, is discussed first. It is shown that a necessary condition for the bifurcation of this steady state to stable, spatially non-uniform, solutions (patterns) is that the parameter % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadseacqGH8aapcaaIZaGaeyOeI0IaaGOmamaakaaabaGaaGOm% aaWcbeaaaaa!4139!\[D < 3 - 2\sqrt 2 \] where D=D _b/D_a (D _a and D _b are the diffusion coefficients of chemical species A and B respectively). The values of μ at which these bifurcations occur are derived in terms of D and λ (a parameter reflecting the size of the system). Further information 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 using a path-following method, with μ as the bifurcation parameter, for representative values of D and λ. It is found that the stable patterns can lose stability through supercritical Hopf bifurcations and these stable, temporally periodic, spatially non-uniform solutions are also followed numerically.