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

Abstract

Consider an analytical function f : V ⊂ R 2 → 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.