Abstract:
This talk deals with the characterisation of initial conditions (IC) leading
to asymptotic non-homogeneous solutions in space,  for a two dimensional
reaction-diffusion (RD) system of FitzHugh-Nagumo (FHN) oscillatory type.
We focus here, on the case where the inhomogeneities are only carried by
the IC. We will first give a review of the origin of the FHN ODE equation and
its qualitative properties. Then, we will go ahead with the RD equation.
Numerical simulations indicate that "most" of IC lead to homogeneous
solutions in space. We will provide some theorems, preventing this
homogeneous asymptotic behavior, and illustrate these theoretical results
numerically. Finally,  we will present some patterns emerging from some IC
with particular stochastic distribution.