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Monodromic singularities of Brunella-Miari vector fields with two edges in the Newton diagram

Isaac A. García, Jaume Giné, Víctor Mañosa

Abstract

This work focuses on the study of monodromic singularities in planar analytic families of vector fields whose Newton diagram consists of exactly two edges. We begin by analyzing the desingularization scheme of a minimal model of polynomial vector fields, denoted by ${X}$, which includes only the monomials corresponding to the vertices of the Newton diagram. We then extend this minimal model to the so-called Brunella-Miari vector fields ${X} \subset {X}^{[1]}$, incorporating all monomials associated with points lying on the edges of the Newton diagram. As a second extension, we consider vector fields ${X}^{[1]} \subset {X}^{[2]}$ that include higher-order terms corresponding to points located above the polygonal line in the Newton diagram. The key point of our approach is to preserve the desingularization geometry at each extension step. We provide explicit desingularization procedures, which enable the computation of the linear part of the return map $Π$ in cases where the desingularized singularity is associated with a hyperbolic polycycle. Several nontrivial examples are included to illustrate the method.

Monodromic singularities of Brunella-Miari vector fields with two edges in the Newton diagram

Abstract

This work focuses on the study of monodromic singularities in planar analytic families of vector fields whose Newton diagram consists of exactly two edges. We begin by analyzing the desingularization scheme of a minimal model of polynomial vector fields, denoted by , which includes only the monomials corresponding to the vertices of the Newton diagram. We then extend this minimal model to the so-called Brunella-Miari vector fields , incorporating all monomials associated with points lying on the edges of the Newton diagram. As a second extension, we consider vector fields that include higher-order terms corresponding to points located above the polygonal line in the Newton diagram. The key point of our approach is to preserve the desingularization geometry at each extension step. We provide explicit desingularization procedures, which enable the computation of the linear part of the return map in cases where the desingularized singularity is associated with a hyperbolic polycycle. Several nontrivial examples are included to illustrate the method.
Paper Structure (37 sections, 18 theorems, 145 equations, 5 figures)

This paper contains 37 sections, 18 theorems, 145 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background and some notations
  3. The minimal model
  4. minimal models with hyperbolic monodromic polycycle
  5. The minimal model *
  6. The minimal model $\dag$
  7. The minimal model $\ddag$
  8. Minimal model with non-hyperbolic monodromic polycycle
  9. Linear term of the Poincaré map
  10. The first and second extensions of the minimal models
  11. Changes in the blow-up scheme for first extension fields
  12. Extensions of particular minimal models
  13. An example of extension of minimal models within the framework of Theorem \ref{['t:Teo-main1']}
  14. An example within the framework of Theorems \ref{['t:Teo-main2']} and \ref{['t:v1']}
  15. An example within the framework of Theorems \ref{['t:Teo-main2-new']} and \ref{['t:v2']}
  16. ...and 22 more sections

Key Result

Theorem 4

The origin is a monodromic singular point of the polynomial vector field e:minimal, and therefore is a center, if and only if its exponents belong to the set and its coefficients lie in the monodromic parameter space Moreover, restricting to $\mathcal{E} \cap \Lambda$, the conditions $\# W(\mathbf{N}(\mathcal{X})) = 2$ and $\mathcal{X} = \mathcal{X}_\Delta$ hold.

Figures (5)

  • Figure 1: Transition maps scheme for $\mathcal{A}$-type characteristic directions.
  • Figure 2: Transition maps in local coordinates for $\mathcal{A}$-type characteristic directions in the case $p$ and $q$ odd.
  • Figure 3: Transition maps scheme for $\mathcal{B}$-type $2$-tuples of characteristic directions.
  • Figure 4: Transition maps in local coordinates for $\theta_*\in\{0,\pi\}$ belonging to a $\mathcal{B}$-type $2$-tuple of characteristic directions in the case $p$ and $q$ odd.
  • Figure 5: Transition maps in local coordinates for $\theta_*\in\{\frac{\pi}{2},\frac{3\pi}{2}\}$ belonging to a $\mathcal{B}$-type $2$-tuple of characteristic directions in the case $p$ and $q$ odd. While the transition maps in the $(z,w)$-coordinates associated with systems \ref{['e:cordA']} depend on the parity of $p$ and $q$, the transition maps in the $(u,y)$-coordinates associated with systems \ref{['e:cordC']} are independent on the parity of $p$ and $q$.

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Theorem 4
  • Proposition 6
  • Proposition 7
  • Theorem 8
  • Theorem 10
  • Theorem 12
  • Theorem 13
  • Theorem 14
  • ...and 25 more