论文标题
矢量场奇点的多种变形
Versal deformations of vector field singularities
论文作者
论文摘要
当矢量场的单一点通过共振，就会出现形式的不变锥。在七十年代，Pyartli证明，对于$（1,1）$ - 共振，该锥体实际上是分析性的，并且是一个不变的圆柱体的退化。 Stolovitch在他的论文中建立了一种新型的正常形式，并证明，对于简单的共鸣，在算术条件下，该锥体是分析品种的（细菌）。在本文中，我们证明了一种具有孤立奇异性的分析矢量场的多种变形定理。我们的结果意味着，在算术条件下，谐振锥是一组不变的歧管的变性，例如Pyartli的例子。对于多-HOPF分叉，即$（-1,1）^d $ - 谐音，这意味着存在携带准周期性动作的消失的托里，从而推广了Chenciner和Li的先前结果。
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.
