In this talk we shall present some results regarding the attractors of
certain systems of parabolic equations. These include the FitzHugh-Nagumo
equations and the Olmstead et al. model for the flow of a non-Newtonian fluid
in the presence of memory.
The emphasys will be on conditions for the existence of a Lyapunov
functional, the existence of periodic behaviour, and the stability of
stationary solutions. Regarding the latter, we introduce a scalar parabolic
equation with the same set of stationary solutions as the original system,
and with the property that the dimension of the unstable manifold for a
given stationary solution for the system is always greater than or equal
to that of the corresponding solution of the reduced equation. Conditions
giving equality of the two dimensions are also presented.