Monodromy and Dulac's Problem for Piecewise analytical planar vector fields

TL;DR

This paper studies the behavior of piecewise analytical planar vector fields near singular points, characterizing monodromic points and conditions for the absence of limit cycles, contributing to the understanding of Dulac's problem.

Contribution

It provides a characterization of monodromic singular points in piecewise analytical vector fields and establishes conditions for limit cycle absence near these points.

Findings

Characterization of $ ext{Sigma}$-monodromic singular points.

Conditions under which neighborhoods are free of limit cycles.

Insights into Dulac's problem for piecewise systems.

Abstract

Consider an analytical function $f:V\subset\mathbb R^2\rightarrow\mathbb R$ having $0$ as its regular value, a switching manifold $\Sigma=f^{-1}(0)$ and a piecewise analytical vector field $X=(X^+,X^-)$, i.e. $X^\pm$ are analytical vector fields defined on $\Sigma^\pm=\{p\in V: \pm f(p)>0\}$. We characterize when the vector field $X$ has a monodromic singular point in $\Sigma$, called $\Sigma$-monodromic singular point. Moreover, under certain conditions, we show that a $\Sigma$-monodromic singular point of $X$ has a neighborhood free of limit cycles.