## Pattern Formation on Growing Square Domains

## Adela Comanici

### Virginia Tech

Abstract:
Numerical simulations of reaction-diffusion systems with Neumann
boundary conditions (NBC) on growing domains by Maini et al. exhibit
square and 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 talk, I will discuss several of these
transitions between different types of squares and between squares
and rolls, using tools from bifurcation theory with symmetry and
dynamical systems. To understand these transitions, we will 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.

We have found that a generic continuous transition can occur between two
types of squares. Also, the transition between squares and rolls can
generically occur either via steady-states and time-periodic states
(standing waves), or via a jump. Moreover, interestingly, the symmetry
constraints induced by NBC are important in understanding which solutions
exist and which solutions are stable. Therefore, I will also point out
interesting differences between the NBC and PBC problems.

Our future goal is to use this research work to construct
reaction-diffusion systems on growing square domains that show prescribed
transitions between different patterns.