Bifurcation and Stability Analysis of a One-Dimensional Diffusion-Autocatalytic Reaction System

Abstract

Results of a numerical analysis of a set of one-dimensional reaction -diffusion equations are presented. The basis of these equations is a model scheme of chemical reactions, involving auto-and cross-catalytic steps (“Brusselator”). The steady state problem is solved numerically, fully exploiting the properties of recently developed continuation codes. Bifurcation diagrams are constructed for zero flux boundary conditions. For a relatively large diffusivity of initial species the Brusselator displays a huge number of dissipative steady state structures. At low system lengths a mechanism of perturbed bifurcation may be percieved. Bifurcations coincide with turning points of asymmetric solution branches. Completely isolated solutions prove to exist as well. For the problem without limited diffusion of the initial species, a careful bifurcation analysis show s the existence of a number of higher order bifurcations. At some of these points asymmetric profiles emanate from other asymmetric structures. Bifurcation points and limit points do not necessarily coincide. Stability analysis shows that relatively few steady states are stable. Especially symmetric solutions are found to be stable.