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September  2008, 10(2&3, September): 377-400. doi: 10.3934/dcdsb.2008.10.377

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, Belgium
2. 

Hasselt University, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590 Diepenbeek, Belgium

Received  September 2006 Revised  May 2007 Published  June 2008

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.
Keywords: Normal Forms, resonance, Gevrey series, summation..
Mathematics Subject Classification: Primary: 37C15, 34C2.
Citation: Patrick Bonckaert, P. De Maesschalck. Gevrey and analytic local models for families of vector fields. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 377-400. doi: 10.3934/dcdsb.2008.10.377
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