On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

TL;DR

This paper investigates the maximum number of limit cycles in piecewise linear vector fields with algebraic discontinuity sets, showing that this maximum grows quadratically with the degree of the algebraic curve, improving previous bounds.

Contribution

It introduces a second order Melnikov method for nonsmooth systems, providing new lower bounds for limit cycles that grow as fast as n^2, surpassing previous estimates for n≥4.

Findings

Limit cycles grow as fast as n^2 for piecewise linear systems.

New lower bounds improve previous estimates for n≥4.

Method uses a second order Melnikov approach for nonsmooth systems.

Abstract

The second part of the Hilbert's sixteenth problem consists in determining the upper bound $\mathcal{H}(n)$ for the number of limit cycles that planar polynomial vector fields of degree $n$ can have. For $n\geq2$, it is still unknown whether $\mathcal{H}(n)$ is finite or not. The main achievements obtained so far establish lower bounds for $\mathcal{H}(n)$. Regarding asymptotic behavior, the best result says that $\mathcal{H}(n)$ grows as fast as $n^2\log(n)$. Better lower bounds for small values of $n$ are known in the research literature. In the recent paper "Some open problems in low dimensional dynamical systems" by A. Gasull, Problem 18 proposes another Hilbert's sixteenth type problem, namely improving the lower bounds for $\mathcal{L}(n)$, $n\in\mathbb{N}$, which is defined as the maximum number of limit cycles that planar piecewise linear differential systems with two zones separated by a branch of an algebraic curve of degree $n$ can have. So far, $\mathcal{L}(n)\geq [n/2],$ $n\in\mathbb{N}$, is the best known general lower bound. Again, better lower bounds for small values of $n$ are known in the research literature. Here, by using a recently developed second order Melnikov method for nonsmooth systems with nonlinear discontinuity manifold, it is shown that $\mathcal{L}(n)$ grows as fast as $n^2.$ This will be achieved by providing lower bounds for $\mathcal{L}(n)$, which improves every previous estimates for $n\geq 4$.