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 N , the functions φs ∈ R N and φns ∈ R N are of class C 2 and ζ is a real valued C 2 function. The equation is singular because ζ(U) can attain the value 0. We focus on the solutions of (0.1) that belong to a small neighbourhood of a point U¯ such that φs(U¯ ) = φns(U¯ ) = ~0 and ζ(U¯ ) = 0. We investigate the existence of manifolds that are locally invariant for (0.1) and that contain orbits with a 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 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.