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.