We derive a necessary and sufficient condition for Turing instabilities to
occur in two-component systems of reaction-diffusion equations with Neumann
boundary conditions. We apply this condition to reaction-diffusion systems
built from vector fields with one fixed point and a supercritical Hopf
bifurcation. For the Brusselator and the Ginzburg-Landau reaction-diffusion
equations, we obtain the bifurcation diagrams associated with the transition
between time periodic solutions and asymptotically stable solutions (Turing
patterns). In two-component systems of reaction-diffusion equations, we show
that the existence of Turing instabilities is neither necessary nor sufficient
for the existence of Turing pattern type solutions. Turing patterns can exist
on both sides of the Hopf bifurcation associated to the local vector field,
and, depending on the initial conditions, time periodic and stable solutions
can coexist.