Gevrey and analytic local models for families of vector fields

Abstract

We give sufficient conditions on the spectrum at the equilibrium point such that a Gevrey-$s$ family can be Gevrey-$s$ conjugated to a simplified form, for $0\le s\le 1$. Local analytic results (i.e. $s=0$) are obtained as a special case, including the classical Poincare theorems and the analytic stable and unstable manifold theorem. As another special case we show that certain center manifolds of analytic vector fields are of Gevrey-$1$ type. We finally study the asymptotic properties of the conjugacy on a polysector with opening angles smaller than $s\pi$ by considering a Borel-Laplace summation.