Hartman-Grobman theorem

Consider the differential equationMathworldPlanetmath

x=f(x) (1)

where f is a C1 vector field. Assume that x0 is a hyperbolic equilibrium of f. Denote Φt(x) the flow of (1) through x at time t. Then there exists a homeomorphism φ(x)=x+h(x) with h bouded, such that


is a sufficiently small neighboorhood of x0.

This fundamental theorem in the qualitative analysis of nonlinear differential equations states that, in a small neighborhoodMathworldPlanetmath of x0, the flow of the nonlinear equation (1) is qualitatively similar to that of the linear system x=Df(x0)x.

Title Hartman-Grobman theorem
Canonical name HartmanGrobmanTheorem
Date of creation 2013-03-22 13:18:37
Last modified on 2013-03-22 13:18:37
Owner jarino (552)
Last modified by jarino (552)
Numerical id 4
Author jarino (552)
Entry type Theorem
Classification msc 34C99