Local Analytic Models For Families Of Hyperbolic Vector Fields

Abstract

We look for analytic models near hyperbolic singularities of families of real analytic vector fields Xe. The interesting case deals with saddles, since for sources or sinks we have the results of Poincar´e [1, 2]. For families we cannot use the Siegel theorem since the condition on the small divisors is fragile. Even on the formal level (i.e. power series) the number of resonances between the eigenvalues is infinite for a family: for instance in the case of a planar saddle this comes to the density of the rationals in R. One option is to use a Ck (k < 1) normal form for the family [5]. Here we want to remain within the analytic category, and have to allow a less simplified form. A first standard simplification is to use stable and unstable manifolds, and to ’straighten’ them, i.e. to write the vector field such that these are linear subspaces. The fact that these invariant manifolds are analytic and depend analytically on the parameter will also follow from the results in this paper. The normal form we aim at will be moreover be ’as flat as desired’ along these invariant manifolds if there are no low order resonances for X0. This approach can already be found in [3, 11, 12] and we extend the results in [12], on which our methods are inspired. Even though in this paper we confine ourselves to the case of a family of vector fields, we can prove similar results for a family of diffeomorphisms [7]. We shall also prove that possible symmetries are preserved in our local analytic model and by the changes of variables.