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

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.