Gevrey and analytic local models for families of vector fields
Patrick Bonckaert P. De Maesschalck
Discrete & Continuous Dynamical Systems - B 2008, 10(2&3, September): 377-400 doi: 10.3934/dcdsb.2008.10.377
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.
On dynamical systems close to a product of $m$ rotations
Patrick Bonckaert Timoteo Carletti Ernest Fontich
Discrete & Continuous Dynamical Systems - A 2009, 24(2): 349-366 doi: 10.3934/dcds.2009.24.349
We consider one parameter families of analytic vector fields and diffeomorphisms, including for a parameter value, say $\varepsilon = 0$, the product of rotations in $\R^{2m}\times \R^n$ such that for positive values of the parameter the origin is a hyperbolic point of saddle type. We address the question of determining the limit stable invariant manifold when $\varepsilon$ goes to zero as a subcenter invariant manifold when $\varepsilon = 0$.
keywords: subcenter invariant manifolds bifurcations. Perturbations of rotations

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