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

2011/10/17 by Bonckaert, P., Verstringe, F.
#34C20 #37C10 #37G05 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1110.3810

Abstract

We explore the convergence/divergence of the normal form for a singularity of a vector field on \Cⁿ 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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