Invariant manifolds for a singular ordinary differential equation

We study the singular ordinary differential equation
dU/dt = F(U)/z(U) + G(U).
The equation is singular because z(U) can attain the value 0. We focus on
the solutions of the above equation that
belong to a small neighbourhood of a point V such that F(U)= G(U) = 0 and
z(U) = 0. We
investigate the existence of manifolds that are locally invariant for the
above equation and that contain orbits with
a prescribed asymptotic behaviour. Under suitable hypotheses on the set {U
: z(U) = 0}, we extend
to the case of the singular ODE the definitions of center manifold,
center-stable manifold and of
uniformly stable manifold. We prove that the solutions of the singular ODE
lying on each of these manifolds are regular: this is not trivial since we
provide examples showing that, in general, a solution of a singular ODE is
not continuously differentiable. Finally, we show a decomposition result
for a center-stable manifold and for the uniformly stable manifold. An
application of our analysis concerns the study of the viscous profiles
with small total variation for a class of mixed hyperbolic-parabolic
systems in one space variable. Such a class includes the compressible
Navier Stokes equation.