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A variational approach to Hilbert's 16th problem within the framework of global analysis

Pablo Pedregal

Abstract

We focus on the second part of Hilbert's 16th problem and provide an upper bound on the number of limit cycles that a polynomial, differential, planar system may have, depending exclusively on the degree $n$ of the system. Such a bound turns out to be a polynomial of degree $4$ in $n$. More specifically, if $H(n)$ indicates the maximum number of limit cycles among planar, differential, polynomial systems of degree $n$, then \begin{gather} H(n)\le \dfrac52 n^4-\dfrac{23}2 n^3+ \dfrac{43}2n^2-\dfrac{37}2n+7\,\,\,\, \mbox{if $n$ is even, and} \nonumber H(n)\le \dfrac52 n^4-\dfrac{23}2 n^3+ \dfrac{41}2n^2-\dfrac{33}2n+6\,\,\,\, \mbox{if $n$ is odd}.\nonumber \end{gather} For quadratic systems, we find $H(2)=4$. Our proof is entirely variational and utilizes in a fundamental way tools and facts from global analysis to the point that no particular expertise in dynamical systems is necessary or required.

A variational approach to Hilbert's 16th problem within the framework of global analysis

Abstract

We focus on the second part of Hilbert's 16th problem and provide an upper bound on the number of limit cycles that a polynomial, differential, planar system may have, depending exclusively on the degree of the system. Such a bound turns out to be a polynomial of degree in . More specifically, if indicates the maximum number of limit cycles among planar, differential, polynomial systems of degree , then \begin{gather} H(n)\le \dfrac52 n^4-\dfrac{23}2 n^3+ \dfrac{43}2n^2-\dfrac{37}2n+7\,\,\,\, \mbox{if is even, and} \nonumber H(n)\le \dfrac52 n^4-\dfrac{23}2 n^3+ \dfrac{41}2n^2-\dfrac{33}2n+6\,\,\,\, \mbox{if is odd}.\nonumber \end{gather} For quadratic systems, we find . Our proof is entirely variational and utilizes in a fundamental way tools and facts from global analysis to the point that no particular expertise in dynamical systems is necessary or required.

Paper Structure

This paper contains 46 sections, 54 theorems, 466 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. Main corollaries of central theorem
  3. Driving idea for the proof of Theorem \ref{['principall']}
  4. Final comments
  5. Strategy and organization
  6. The abstract setting
  7. Initial difficulties
  8. Troubles persist
  9. Our guiding principle
  10. The general setting
  11. The concrete setting, and the full program
  12. Abstract results
  13. Morse inequalities
  14. Main concepts
  15. Perturbation
  16. ...and 31 more sections

Key Result

Theorem 1.1

Consider the polynomial differential system e1 of degree $n>1$. Assume that: Then an upper bound for the number $H(n)$ of limit cycles that such a differential system e1 may have is

Figures (1)

  • Figure 1: Situation around a contact point: top, a multiple contact point; bottom, a single contact point.

Theorems & Definitions (92)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Bezout Theorem, Fu
  • Theorem 1.4: Harnack Theorem, Gu
  • proof : Proof of Theorem \ref{['t5']} based on Theorem \ref{['principall']}
  • Corollary 1.5
  • proof
  • Corollary 1.6
  • proof
  • ...and 82 more