Hilbert's 16th problem. I. When differential systems meet variational methods

TL;DR

This paper establishes a polynomial upper bound on the number of limit cycles in planar polynomial differential systems of a given degree by employing variational methods and Morse inequalities, advancing the solution to Hilbert's 16th problem.

Contribution

It introduces a novel approach combining variational and dynamical techniques to bound limit cycles, providing a degree-dependent polynomial upper bound.

Findings

Derived a degree four polynomial upper bound for limit cycles

Connected counting limit cycles to critical points of a functional

Applied Morse inequalities to establish the bound

Abstract

We provide an upper bound for the number of limit cycles that planar polynomial differential systems of a given degree may have. The bound turns out to be a polynomial of degree four in the degree of the system. The strategy brings together variational and dynamical system techniques by transforming the task of counting limit cycles into counting critical points for a certain smooth, non-negative functional, through Morse inequalities, for which limit cycles are global minimizers. We thus solve the second part of Hilbert's 16th problem providing a uniform upper bound for the number of limit cycles which only depends on the degree of the polynomial differential system.