DIFFERENTIAL-FLOW-INDUCED INSTABILITY IN A CUBIC AUTOCATALATOR SYSTEM

The formation of spatio-temporal stable patterns is considered for a r eaction-diffusion-convection system based upon the cubic autocatalator , A + 2B --> 3B, B --> C, with the reactant A being replenished by the slow decay of some precursor P via the simple step P --> A. The react ion is considered in a differential-flow reactor in the form of a ring . It is assumed that the reactant A is immobilised within the reactor and the autocatalyst B is allowed to flow through the reactor with a c onstant velocity as well as being able to diffuse. The linear stabilit y of the spatially uniform steady state (a, b) = (mu(-1), mu), where a and b are the dimensionless concentrations of the reactant A and auto catalyst B, and mu is a parameter reflecting 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, spatiall y non-uniform, flow-generated patterns is that the flow parameter phi > phi(c)(mu, lambda) where phi(c)(mu, lambda) is a (strictly positive) critical value of phi and lambda is the dimensionless diffusion coeff icient of the species B and also reflects the size of the system. Valu es of phi(c) at which these bifurcations occur are derived in terms of mu and lambda. Further information about the nature of the bifurcatin g branches (close to their bifurcation points) is obtained from a weak ly nonlinear analysis. This reveals that both supercritical and subcri tical Hopf bifurcations are possible. The bifurcating branches are the n followed numerically by means of a path-following method, with the p arameter phi as a bifurcation parameter, for representatives values of mu and lambda. It is found that multiple stable patterns can exist an d that it, is also possible that any of these can lose stability throu gh secondary Hopf bifurcations. This typically gives rise to spatio-te mporal quasiperiodic transients through which the system is ultimately attracted to one of the remaining available stable patterns.