The monodromic singularity to piecewise linear vector fields

Abstract

Abstract: Consider in R2 the semi-planes N = {y > 0} and S = {y < 0} having as common boundary the straight line D = {y = 0}. In N and S are defined linear vector fields X and Y , respectively, leading to a discontinuous polynomial vector field Z = (X, Y ). If the vector fields X and Y satisfy suitable conditions, they produce a transition flow from a segment of the splitting line to another segment and this produces a generalized singular point on the line. This point can be a focus or a center. In this paper we give necessary and sufficient conditions to a D-singular point be a monodromic.