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

2021/03/12 by Pablo Pedregal, Pedregal, Pablo
Computer Science · Engineering · Mathematics · #Advanced Differential Equations and Dynamical Systems #Control and Dynamics of Mobile Robots #Dynamical Systems (math.DS) #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2103.07193

openalex publication_date 2021/03/12 · openalex created_date 2022/08/29 · openalex updated_date 2026/07/28

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 H(n)≤ \dfrac52 n⁴-\dfrac232 n³+ \dfrac432n²-\dfrac372n+7 if n is even, and H(n)≤ \dfrac52 n⁴-\dfrac232 n³+ \dfrac412n²-\dfrac332n+6 if n is odd.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.

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