Monodromy and Dulac's problem for piecewise analytical planar vector fields

Abstract

Consider an analytical function f:V⊂R2→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± are analytical vector fields defined on Σ±={p∈V:±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.