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Analytic normal forms and inverse problems for unfoldings of 2-dimensional saddle-nodes with analytic center manifold. (arXiv:1810.04890v1 [math.DS])
来源于:arXiv
We give normal forms for generic k-dimensional parametric families
$(Z_\varepsilon)_\varepsilon$ of germs of holomorphic vector fields near
$0\in\mathbb{C}^2$ unfolding a saddle-node singularity $Z_0$, under the
condition that there exists a family of invariant analytic curves unfolding the
weak separatrix of $Z_0$. 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. 查看全文>>