A. Comanici and M. Golubitsky

Patterns on growing square domains via mode interactions

Dynamical Systems. 23, No.2 (2008) 167-206.

Numerical simulations of reaction-diffusion systems with Neumann boundary conditions (NBC) on growing square domains by Maini et al. exhibit square and stripe (or roll) patterns that are usually associated with bifurcations from a trivial equilibrium on a square lattice. However, these patterns change as the domain grows. In this paper we discuss several of these transitions; namely, transitions between different types of squares and between squares and stripes (or rolls). We show that these transitions can be understood by tracing paths through the unfoldings of certain codimension two mode interactions. To understand these transitions we need to discuss two issues: the speed at which the domain size changes and the relations between NBC and periodic boundary conditions (PBC) on a square.

First, in the simulations, the domain growth takes place on a time scale that is longer than the one needed for pattern formation. Therefore, we can assume that the domain growth is identified with quasi-static variation of time; that is, the domain grows slowly enough that the PDE solution of the time-dependent system tracks equilibria of the reaction-diffusion systems posed on a fixed size domain.

Second, as is well-known, NBC problems on a square of side length l can be embedded in PBC problems on a square with side length 2l. The PBC problem has translation symmetries that are not present in the NBC problem. These additional symmetries are called hidden symmetries in the literature. Moreover, solutions to PBC that restrict to the smaller square and satisfy NBC are just those solutions that satisfy certain symmetry constraints. We show further that the transitions between different patterns can be understood by considering relevant mode interaction bifurcation problems on the larger square and then restricting to the smaller square.

We have found that a generic continuous transition can occur between two types of squares. Also, the transition between squares and stripes can generically occur either via steady-states and time-periodic states ( standing waves) or via a jump.

Interestingly, in mode interactions, the symmetry constraints induced by NBC are important in understanding which solutions exist and which solutions are stable. For example, diagonal stripes cannot occur as a primary branch in the NBC problem but do in the PBC problem. Also, patterns can be stable in the NBC problem that are not stable in the PBC problem. As a consequence, in the NBC problem we see standing wave time-periodic solutions as stable patterns leading to stable stripes, whereas in the PBC problem we see wavy rolls steady-states as stable patterns leading to stable stripes. In principle, a classification of all transitions in NBC mode interactions is possible. However, we concentrate only on those transitions that are relevant to the numerically observed transitions.