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Center Conditions and Cyclicity for Generic Planar Polynomial Vector Fields

Yovani Villanueva, Warwick Tucker

Abstract

We study the center-focus problem for planar polynomial vector fields, which can be viewed as a local version of Hilbert's 16th problem. Based on a Lyapunov function approach, we establish novel results regarding the center-focus conditions. More precisely, under generic conditions, and for any degree of a polynomial vector field, we find an upper bound on the size of the Bautin ideal generated by the Lyapunov constants. This also provides an upper bound on the cyclicity of the systems we consider.

Center Conditions and Cyclicity for Generic Planar Polynomial Vector Fields

Abstract

We study the center-focus problem for planar polynomial vector fields, which can be viewed as a local version of Hilbert's 16th problem. Based on a Lyapunov function approach, we establish novel results regarding the center-focus conditions. More precisely, under generic conditions, and for any degree of a polynomial vector field, we find an upper bound on the size of the Bautin ideal generated by the Lyapunov constants. This also provides an upper bound on the cyclicity of the systems we consider.
Paper Structure (7 sections, 6 theorems, 59 equations, 1 table)

This paper contains 7 sections, 6 theorems, 59 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Theory on homogeneous differential systems
  4. Main theorem for generic homogeneous differential systems
  5. Generalization to non-homogeneous differential systems
  6. Conclusions
  7. Acknowledgments

Key Result

Proposition 1

Given a homogeneous system eq:homogeneous_system of degree $n \ge 2$, the homogeneous terms $V_2, V_3,\dots$ of the Lyapunov function $V$ appearing in eq:lyapunov_formula satisfy Here each $P_k$ is a vector of polynomials in the parameters $(f,g)$. Furthermore, the Lyapunov constants $L_1,L_2,\dots$ appearing in eq:lyapunov_formula satisfy Again, each $q_j$ and $r_j$ is a polynomial in the param

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Corollary 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • proof : Proof of Theorem \ref{['theo1']}:
  • ...and 8 more