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

TL;DR

This paper establishes a polynomial upper bound on the maximum number of limit cycles in polynomial planar systems of degree n, using a variational approach rooted in global analysis, applicable even without dynamical systems expertise.

Contribution

It introduces a novel variational method within global analysis to bound limit cycles, providing explicit polynomial bounds depending on the degree n.

Findings

Upper bound on limit cycles is a degree 4 polynomial in n.

Exact maximum for quadratic systems is 4 limit cycles.

Method is entirely variational and does not require dynamical systems expertise.

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.