Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A universal method to approach the Poincaré center problem

Isaac A. García, Jaume Giné

Abstract

We address the classical (degenerate or non-degenerate) center problem posed by Poincaré in the 19th century for monodromic singularities of analytic families of planar vector fields $\mathcal{X}$. We prove that every analytic center admits a Laurent inverse integrating factor $V$ in weighted polar coordinates. Moreover, we show that when $\mathcal{X}$ has no local curves of zero angular speed, the Poincaré map is analytic, and if, in addition, $V$ has an essential singularity, then the singularity of $\mathcal{X}$ is a center. Based on this result, we derive a theoretical procedure to determine parameter constraints within the family that characterize any center of a polynomial vector field. Applications to nontrivial families that have resisted other methods are also provided.

A universal method to approach the Poincaré center problem

Abstract

We address the classical (degenerate or non-degenerate) center problem posed by Poincaré in the 19th century for monodromic singularities of analytic families of planar vector fields . We prove that every analytic center admits a Laurent inverse integrating factor in weighted polar coordinates. Moreover, we show that when has no local curves of zero angular speed, the Poincaré map is analytic, and if, in addition, has an essential singularity, then the singularity of is a center. Based on this result, we derive a theoretical procedure to determine parameter constraints within the family that characterize any center of a polynomial vector field. Applications to nontrivial families that have resisted other methods are also provided.
Paper Structure (16 sections, 10 theorems, 58 equations)

This paper contains 16 sections, 10 theorems, 58 equations.

Table of Contents

  1. Introduction and main results
  2. A procedure to solve generic center-focus problems
  3. Ascending expansions
  4. Descending expansions when $\mathcal{X}$ is a polynomial
  5. How the method works in a toy example
  6. Non-trivial examples
  7. Mañosa monodromic family I
  8. The (3,5)-semi-homogeneous family
  9. Proofs
  10. Proof of Proposition \ref{['prop-monpq']}
  11. Proof of Theorem \ref{['center-V']}
  12. Proof of Theorem \ref{['center-V-2']}
  13. Proof of Theorem \ref{['center-Pi']}
  14. Appendix
  15. Periodic solutions of linear periodic differential equations
  16. ...and 1 more sections

Key Result

Proposition 1

Any analytic center with $\Omega_{pq} = \emptyset$ admits an analytic inverse integrating factor $V(\varphi, \rho)$ of the form V-Laurent-a with $m \geq 0$.

Theorems & Definitions (19)

  • Proposition 1
  • Theorem 3
  • Theorem 4
  • Theorem 7
  • Proposition 12
  • proof
  • Proposition 13
  • proof
  • Proposition 14
  • proof
  • ...and 9 more