Gevrey and analytic local models for families of vector fields

Patrick Bonckaert; P. De Maesschalck

doi:10.3934/dcdsb.2008.10.377

Article Contents

2008, Volume 10, Issue 2&3: 377-400. Doi: 10.3934/dcdsb.2008.10.377

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Gevrey and analytic local models for families of vector fields

Patrick Bonckaert^1, and
P. De Maesschalck^2,

1.
Hasselt University, Agoralaan, gebouw D, B-3590 Diepenbeek
2.
Hasselt University, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590 Diepenbeek

Received: August 31, 2006

Revised: April 30, 2007

Published: August 2008

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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 Poincaré 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.

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Mathematics Subject Classification: Primary: 37C15, 34C20.

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