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Conjugacy of normally tangent diffeomorphisms : a tool for treating moduli of stability

Patrick Bonckaert — 1990

Annales de l'institut Fourier

We give sufficient conditions for the conjugacy of two diffeomorphisms coinciding on a common invariant submanifold V and with equal normal derivative; moreover we obtain that the homeomorphism h realizing this conjugacy satisfies additional inequalities. These inequalities, implying also the existence of the normal derivative of h along V, serve to extend this conjugacy towards regions where moduli of stability are present.

Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Patrick Bonckaert;  Freek Verstringe — 2012

Annales de l’institut Fourier

We explore the convergence/divergence of the normal form for a singularity of a vector field on ℂ n with nilpotent linear part. We show that a Gevrey- α vector field X with a nilpotent linear part can be reduced to a normal form of Gevrey- 1 + α type with the use of a Gevrey- 1 + α transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

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