Limit cycles for some families of smooth and non-smooth planar systems

TL;DR

This paper investigates the number of limit cycles in certain planar systems, including non-smooth cases, using averaging methods and Brouwer degree, and proposes a variant of Hilbert's 16th problem focusing on monomials.

Contribution

It introduces a novel approach combining averaging and Brouwer degree to analyze limit cycles in both smooth and non-smooth systems, and presents a new perspective on Hilbert's 16th problem.

Findings

Provides lower bounds for the number of limit cycles.

Extends analysis to systems with non-smoothness on specific sets.

Proposes a variant of Hilbert's 16th problem based on monomials.

Abstract

In this paper, we apply the averaging method via Brouwer degree in a class of planar systems given by a linear center perturbed by a sum of continuous homogeneous vector fields, to study lower bounds for their number of limit cycles. Our results can be applied to models where the smoothness is lost on the set $\Sigma=\{xy=0\}$. We also apply them to present a variant of Hilbert 16th problem, where the goal is to bound the number of limit cycles in terms of the number of monomials of a family of polynomial vector fields, instead of doing this in terms of their degrees.