Analytic normal forms and inverse problems for unfoldings of 2-dimensional saddle-nodes with analytic center manifold

2018/10/11 by Rousseau, C., Teyssier, Loïc Jean Dit
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.04890

Abstract

We give normal forms for generic k-dimensional parametric families (Z_ε)_ε of germs of holomorphic vector fields near 0∈ℂ² unfolding a saddle-node singularity Z₀, under the condition that there exists a family of invariant analytic curves unfolding the weak separatrix of Z₀. These normal forms provide a moduli space for these parametric families. In our former 2008 paper, a modulus of a family was given as the unfolding of the Martinet-Ramis modulus, but the realization part was missing. We solve the realization problem in that partial case and show the equivalence between the two presentations of the moduli space. Finally, we completely characterize the families which have a modulus depending analytically on the parameter. We provide an application of the result in the field of non-linear, parameterized differential Galois theory.

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