[Paper Review] Generic 1-parameter pertubations of a vector field with a singular point of codimension k

Arnaud Chéritat, Chrisitane Rousseau|arXiv (Cornell University)|Jan 12, 2017

Advanced Differential Equations and Dynamical Systems3 citations

TL;DR

This paper classifies generic 1-parameter perturbations of complex vector fields with a singular point of multiplicity $k+1$ via holomorphic conjugacy in coordinates and parameter. It introduces an eigenvalue function $\lambda(\delta)$, where $\delta = \epsilon^{1/(k+1)}$, and proves that two such families are equivalent under coordinate and parameter change if and only if their eigenvalue functions are equivalent up to rotation of order dividing $k+1$. The key result is a canonical normal form for the perturbed vector field: $\dot{z} = (z^{k+1} - \epsilon)\sigma(z)$, with $\sigma(0) = 1$ and no terms of degree $m(k+1)$ in its power series, ensuring almost uniqueness.

ABSTRACT

We describe the equivalence classes of germs of generic 1-parameter families of complex vector fields z dot = omega_epsilon(z) on C unfolding a singular point of multiplicity k+1: omega_0 = z^{k+1} + o(z^{k+1}). The equivalence is under conjugacy by holomorphic change of coordinate and parameter. We provide a description of the modulus space and (almost) unique normal forms. As a preparatory step, we present the complete bifurcation diagram of the family of vector fields z dot = z^{k+1} - epsilon, over CP1.

Motivation & Objective

• To classify generic 1-parameter families of complex vector fields unfolding a singular point of multiplicity $k+1$ under holomorphic conjugacy in both coordinates and parameter.
• To describe the modulus space of such families and construct (almost) unique normal forms.
• To analyze the bifurcation diagram of the model family $\dot{z} = z^{k+1} - \epsilon$ on $\mathbb{C}\mathbb{P}^1$ and its restriction to a disk.
• To solve two equivalence problems: one fixing the parameter, and one allowing parameter change, using eigenvalue functions as invariants.

Proposed method

• Introduce the eigenvalue function $\lambda(\delta)$, where $\delta = \epsilon^{1/(k+1)}$, encoding the eigenvalues at the $k+1$ singular points of the perturbed system.
• Use the complex time flow (rectifying coordinate) to analyze the dynamics and derive conjugacy invariants.
• Apply the method of infinite descent in the ring $\mathbb{C}\{\epsilon\}$ to prove uniqueness of the normal form up to canonical parameter choice.
• Reduce the conjugacy problem to equivalence of eigenvalue functions under right-composition by rotations of order dividing $k+1$, using geometric and analytic techniques inspired by Douady and Sentenac.
• Show that any generic 1-parameter perturbation can be conjugated to the form $\dot{z} = (z^{k+1} - \epsilon)\sigma(z)$ with $\sigma(0) \neq 0$, and further normalized so $\sigma(0) = 1$ and no terms of degree $m(k+1)$ in its expansion.
• Prove that the eigenvalue function $\lambda(\delta)$ vanishes to order exactly $k$ at $\delta = 0$, and that any such function arises as the eigenvalue function of some family.

Experimental results

Research questions

• RQ1 What are the possible phase portraits and bifurcations in a generic 1-parameter unfolding of a complex vector field with a singular point of multiplicity $k+1$?
• RQ2 How can two such perturbations be classified as equivalent under holomorphic change of coordinates and parameter?
• RQ3 What is the structure of the modulus space of such 1-parameter families?
• RQ4 Can a canonical normal form be constructed for these families, and if so, what are its defining properties?
• RQ5 How does the eigenvalue function $\lambda(\delta)$ encode the dynamical data of the singular points and serve as a complete invariant?

Key findings

• Any generic 1-parameter perturbation of a vector field with a codimension-$k$ singularity is conjugate to the form $\dot{z} = (z^{k+1} - \epsilon)\sigma(z)$ with $\sigma(0) \neq 0$, via a holomorphic change of coordinates preserving the parameter.
• The eigenvalue function $\lambda(\delta)$, defined for $\delta = \epsilon^{1/(k+1)}$, vanishes to order exactly $k$ at $\delta = 0$, and any such function arises from some family.
• Two such families are equivalent under coordinate and parameter change if and only if their eigenvalue functions are equivalent under right-composition by a rotation of order dividing $k+1$.
• The eigenvalue function can be normalized to the form $\lambda(\delta) = (k+1)\delta^k \sigma(\delta)$ with $\sigma(0) = 1$ and no terms of degree $m(k+1)$ in its power series, yielding an almost unique normal form.
• The bifurcation diagram of $\dot{z} = z^{k+1} - \epsilon$ on $\mathbb{C}\mathbb{P}^1$ is fully described, and its restriction to a disk containing the singularities is derived from it.
• The proof of uniqueness of the normal form uses an infinite descent argument in the ring $\mathbb{C}\{\epsilon\}$, showing that all coefficients of the conjugating map and parameter transformation vanish to arbitrary order, hence are identically zero.