Versal deformations of vector field singularities
Abstract
When a singular point of a vector field passes through resonance, a formal invariant cone appears. In the seventies, Pyartli proved that for -resonance the cone is in fact analytic and is the degeneration of a family of invariant cylinders. In his thesis, Stolovitch established a new type of normal form and proved that for a simple resonance and under arithmetic conditions the cone is (the germ of) an analytic variety. In this paper, we prove a versal deformation theorem for analytic vector fields with an isolated singularity over Cantor sets. Our result implies that, under arithmetic conditions, the resonant cone is the degeneration of a set of invariant manifolds like in Pyartli's example. For the multi-Hopf bifurcation, that is for the -resonance, this implies the existence of vanishing tori carrying quasi-periodic motions generalising previous results of Chenciner and Li.
Cite
@article{arxiv.2011.06802,
title = {Versal deformations of vector field singularities},
author = {Mauricio Garay and Duco van Straten},
journal= {arXiv preprint arXiv:2011.06802},
year = {2020}
}