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Surfaces with central configuration and Dulac's problem for a three dimensional isolated Hopf singularity

Nuria Corral, María Martín Vega, Fernando Sanz Sánchez

Abstract

Let $ξ$ be a real analytic vector field with an elementary isolated singularity at $0\in \mathbb{R}^3$ and eigenvalues $\pm bi,c$ with $b,c\in \mathbb{R}$ and $b\neq 0$. We prove that all cycles of $ξ$ in a sufficiently small neighborhood of $0$, if they exist, are contained in a finite number of subanalytic invariant surfaces entirely composed by a continuum of cycles. In particular, we solve Dulac's problem, i.e. finiteness of limit cycles, for such vector fields.

Surfaces with central configuration and Dulac's problem for a three dimensional isolated Hopf singularity

Abstract

Let be a real analytic vector field with an elementary isolated singularity at and eigenvalues with and . We prove that all cycles of in a sufficiently small neighborhood of , if they exist, are contained in a finite number of subanalytic invariant surfaces entirely composed by a continuum of cycles. In particular, we solve Dulac's problem, i.e. finiteness of limit cycles, for such vector fields.
Paper Structure (19 sections, 19 theorems, 105 equations, 5 figures)

This paper contains 19 sections, 19 theorems, 105 equations, 5 figures.

Table of Contents

  1. Introduction and statements
  2. The semi-hyperbolic case
  3. Admissible blowing-ups and adapted reduction of singularities
  4. Formal normal form and truncated normal forms
  5. The first blowing-up
  6. Characteristic cycles and successive blowing-ups
  7. Adapted reduction of singularities.
  8. Analytic approximations of normal forms
  9. Characteristic cycles as limit sets
  10. Analysis of final adapted simple singularities
  11. Infinitely near points of the rotational axis
  12. Simple corner characteristic cycles
  13. Simple non-corner characteristic cycles
  14. Invariant formal surface along $\gamma_I$
  15. Poincaré first-return map associated to $\gamma_I$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\xi\in\mathcal{H}^3$ with isolated singularity. Then, there is some neighborhood $U$ of $0\in \mathbb{R}^3$ where a representative of $\xi$ is defined for which exactly one of the following possibilities holds:

Figures (5)

  • Figure 1: Illustration of case $(ii)$.
  • Figure 2: Definition of the Poincaré map $P_\gamma$
  • Figure 3: Several sequences of admissible blowing-ups.
  • Figure 4: Cross-section of the neighborhoods $\widetilde{W}_{I_j}\subset W_{I_j}$ and of $\tilde{U}$.
  • Figure 5: $\widetilde{\Sigma}_{N,C_1,\delta_1}(S_1)$ and $\varphi^{-1}(\widetilde{\Sigma}_{N,C_2,\delta_2}(S_2))$

Theorems & Definitions (48)

  • Theorem 1.1: Structure of cycle-locus
  • Corollary 1.2
  • Example 1.3
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Remark 3.6
  • ...and 38 more