Invariant manifolds for a singular ordinary differential equation

We study the singular ordinary differential equation dU dt = 1 ζ(U) φs(U) + φns(U), (0.1) where U ∈ R , the functions φs ∈ R N and φns ∈ R N are of class C and ζ is a real valued C function. The equation is singular in the sense that ζ(U) can attain the value 0. We focus on the solutions of (0.1) that belong to a small neighbourhood of a point Ū such that φs(Ū) = φns(Ū) = ~0, ζ(Ū) = 0. We investigate the existence of manifolds that are locally invariant for (0.1) and that contain orbits with a suitable prescribed asymptotic behaviour. Under suitable hypotheses on the set {U : ζ(U) = 0}, we extend to the case of the singular ODE (0.1) the definitions of center manifold, center stable manifold and of uniformly stable manifold. We prove that the solutions of (0.1) lying on each of these manifolds are regular: this is not trivial since we provide examples showing that, in general, a solution of (0.1) is not continuously differentiable. Finally, we show a decomposition result for a center stable 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.