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On $k$-summable normal forms of vector fields with one zero eigenvalue

Peter De Maesschalck, Kristian Uldall Kristiansen

TL;DR

This paper extends the theory of analytic normal forms for saddle-nodes with a single zero eigenvalue to arbitrary Poincaré rank $k$ by introducing a Banach convolutional algebra in the $k$-order Borel plane and a generalized Laplace transform of order $k$. It develops a full functional-analytic framework, including $k$-summability and $k$-dependent spaces, to solve the conjugacy problem and obtain $k$-sums $\phi$ and $g$ that reduce the system to a pre-normal form even when nontrivial Jordan blocks are present. The main result proves the existence of these $k$-sums on a domain $\omega_k(\nu,\theta,\alpha)\times B^n(R)$, with a resonant set chosen to minimize removals, and provides detailed proofs via a Borel-plane fixed-point approach, including semi-simple and non-semi-simple cases. The work also demonstrates an application to zero-Hopf singularities and discusses directions for future generalizations and the potential to attain true convergence by removing Borel-plane singularities.

Abstract

In this paper, we study normal forms of analytic saddle-nodes in $\mathbb C^{n+1}$ with any Poincaré rank $k\in \mathbb N$. The approach and the results generalize those of Bonckaert and De Maesschalck from 2008 that considered $k=1$. In particular, we introduce a Banach convolutional algebra that is tailored to study differential equations in the Borel plane of order $k$. One of the subtleties that we take care of in this paper, is that nontrivial Jordan blocks are allowed in the linear part of the vector field. We anticipate that our approach can stimulate new research and be used to study different normal forms in future work.

On $k$-summable normal forms of vector fields with one zero eigenvalue

TL;DR

This paper extends the theory of analytic normal forms for saddle-nodes with a single zero eigenvalue to arbitrary Poincaré rank by introducing a Banach convolutional algebra in the -order Borel plane and a generalized Laplace transform of order . It develops a full functional-analytic framework, including -summability and -dependent spaces, to solve the conjugacy problem and obtain -sums and that reduce the system to a pre-normal form even when nontrivial Jordan blocks are present. The main result proves the existence of these -sums on a domain , with a resonant set chosen to minimize removals, and provides detailed proofs via a Borel-plane fixed-point approach, including semi-simple and non-semi-simple cases. The work also demonstrates an application to zero-Hopf singularities and discusses directions for future generalizations and the potential to attain true convergence by removing Borel-plane singularities.

Abstract

In this paper, we study normal forms of analytic saddle-nodes in with any Poincaré rank . The approach and the results generalize those of Bonckaert and De Maesschalck from 2008 that considered . In particular, we introduce a Banach convolutional algebra that is tailored to study differential equations in the Borel plane of order . One of the subtleties that we take care of in this paper, is that nontrivial Jordan blocks are allowed in the linear part of the vector field. We anticipate that our approach can stimulate new research and be used to study different normal forms in future work.

Paper Structure

This paper contains 12 sections, 16 theorems, 168 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and statement of the main result
  3. Main result
  4. Proof of Theorem \ref{['theorem:main1']}
  5. The Borel plane and Laplace transforms
  6. Definition of the space Gkn{z}
  7. The conjugacy equation in the Borel plane
  8. Completing the proof of Theorem \ref{['theorem:main1']}
  9. Proof of Lemma \ref{['lemma:LHSinv']} in the semi-simple case
  10. Proof of Lemma \ref{['lemma:LHSinv']} for $\Xi\ne 0$
  11. Application to the zero-Hopf
  12. Discussion

Key Result

Theorem 2.1

Consider (eq:prenormalform), satisfying Assumption assumption:Adiag with $f$ being analytic near the origin, and any $C>0$. We then define $\mathcal{R}=\mathcal{R}(C)$ by Moreover, let $\theta$ be a complex direction, so that the following holds for some $\xi>0$ small enough: Then there exist $R>0$, $\nu>0$ and an $\alpha>0$, all small enough, and two analytic functions $\phi=\phi(x,z)$ and $g=g

Figures (2)

  • Figure 1: Illustrations of the different domains $\Omega$ and $\omega_k$. Fig. (a) is in the "Borel plane" $w\in \mathbb C$, whereas Fig. (b) is in the $x$-domain. Here $\omega_k$ is the complex domain upon which the Laplace transform of $\mathcal{G}_k$-functions on $\Omega$ are defined, see Lemma \ref{['lemma:laplace']}.
  • Figure 2: For any $N\in \mathbb N$ there is a $C>0$ small enough so that the projection $\mathcal{R}_1$ of $\mathcal{R}$ onto $i=1$ is a subset of the purple region. The case $i=2$ is obtained by reflecting the $i=1$ case around the bisector.

Theorems & Definitions (39)

  • Remark 1
  • Definition 1
  • Definition 2
  • Theorem 2.1
  • Theorem 2.1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 3.1
  • proof
  • ...and 29 more