Clasificación topológica de una familia de campos vectoriales lineales a trozos simétricos en el plano

摘要: The contents of this memoir deal with Qualitative Theory of Differential Equations. The aim of this work is to describe the bifurcation set and all the phase portraits of a family of planar and symmetric piecewise linear vector fields that we call fundamental systems. The memoir is divided in four chapters. In Chapter 1 we give a brief exposition of basic facts on Qualitative Theory of planar differential equations. We will touch only the aspects of the theory that we will use in the rest of the memoir, and we restrict our attention to autonomous ordinary differential equations. In Section 1.9 we formalize some ideas about the Poincare compactification of flows. This allows us to prove a characterization of the vector fields which can be compactified. In Chapter 2 we introduce the fundamental systems x= Ax+ ϕ (k T x) b,(1) where A is a 2× 2 matrix with real coefficients, x∈ R 2, k, b∈ R 2 {0} and ϕ (σ)=