Versal deformations of vector field singularities

TL;DR

This paper proves a versal deformation theorem for analytic vector fields with isolated singularities over Cantor sets, extending classical results on invariant cones and manifolds in resonant vector fields.

Contribution

It introduces a new versal deformation theorem for resonant vector fields with isolated singularities over Cantor sets, generalizing previous work on invariant cones and tori.

Findings

Resonant cones are degenerations of invariant manifolds under arithmetic conditions.

Existence of vanishing tori with quasi-periodic motions in multi-Hopf bifurcations.

Extension of classical results to more general settings involving Cantor sets.

Abstract

When a singular point of a vector field passes through resonance, a formal invariant cone appears. In the seventies, Pyartli proved that for $(-1,1)$-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 $(-1,1)^d$-resonance, this implies the existence of vanishing tori carrying quasi-periodic motions generalising previous results of Chenciner and Li.