Bi-periodicity in an isothermal autocatalytic reaction-diffusion system

Abstract

We have examined an autocatalytic process under isothermal conditions, the so-called Gray-Scott model, with diffusion in one spatial dimension. We have found, that under symmetric conditions we may have stable stationary solutions or stable periodic solutions, but no bi-periodic solutions despite the presence of two pairs of complex conjugated eigenvalues with positive real parts. Bi-periodic solutions are seen, when we break the symmetry by making the boundary conditions unequal. In a two-parameter plane the region of bi-periodicity is bounded by a curve of Hopf bifurcation points with a point of self-intersection. This point is deformed into a cusp point as a third parameter approaches a critical value. This codimension 3 event can be formulated as a zero point problem, and we describe two different methods of formulating such a zero point problem. The computational effort was made small by using an orthogonal collocation method to discretise the PDEs into a low-dimensional system of ODEs.