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

Abstract

Consider an analytical function $f:V\subset {\mathbb{R}}^2\to \mathbb{R}$ 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.

24 pages, 7 figures