Hilbert's 16th problem. II. Pfaffian equations and variational methods

TL;DR

This paper explores the use of variational methods applied to Pfaffian equations to analyze Hilbert's 16th problem, focusing on limit cycles and extending techniques to higher dimensions.

Contribution

It introduces a perspective on variational methods for Pfaffian equations, aiming to detect limit cycles and extend the approach beyond planar systems.

Findings

Initial calculations in dimension 3 show promising results similar to the 2D case.

The variational approach can potentially be applied to higher-dimensional systems.

The paper suggests a broader applicability of the method for solving parts of Hilbert's 16th problem.

Abstract

Starting from a Pfaffian equation in dimension $N$ and focusing on compact solutions for it, we place in perspective the variational method used in [29] to solve Hilbert's 16th problem. In addition to exploring how this viewpoint can help in detecting and finding approximations for limit cycles of planar systems, we recall some of the initial important facts of the full program developed in [29] to motivate that the same proposal could eventually be used in other situations. In particular, we make some initial interesting calculations in dimension $N=3$ that lead to some similar initial conclusions as with the case $N=2$.