El teorema de Hartman-Grobman

Abstract

The stability theory of di erential equations studies variations in the solutions of Cauchy Problems under small perturbations of initial conditions. The objective of this document is to study a classic result about of stability of differential equations. The Hartman-Grobman Theorem. We give a proof of the theorem for autonomous di erential equation. The theorem states that the behavior of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behavior of its linearization near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. It is a important result because through the theorem, the stability of many autonomous systems is characterized for the stability of the simpler linearization of the systems.