João Carlos Medrado | Searcharxiv

arXiv2023

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

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

math.DS↗

arXiv2017

On the existence of limit cycles and invariant surfaces of sewing piecewise linear differential systems on $\mathbb{R}^3$

We consider a class of discontinuous piecewise linear differential systems in $\mathbb{R}^3$ with two pieces separated by a plane. In this class we show that there exist differential systems having: a unique limit cycle, a unique one-parameter family of periodic orbits, scrolls, invariant cylinders foliated by orbits which can be periodic or no.

math.DS↗