# Hartman-Grobman theorem

Consider the differential equation

 $x^{\prime}=f(x)$ (1)

where $f$ is a $C^{1}$ vector field. Assume that $x_{0}$ is a hyperbolic equilibrium of $f$. Denote $\Phi_{t}(x)$ the flow of (1) through $x$ at time $t$. Then there exists a homeomorphism $\varphi(x)=x+h(x)$ with $h$ bouded, such that

 $\varphi\circ e^{tDf(x_{0})}=\Phi_{t}\circ\varphi$

is a sufficiently small neighboorhood of $x_{0}$.

This fundamental theorem in the qualitative analysis of nonlinear differential equations states that, in a small neighborhood of $x_{0}$, the flow of the nonlinear equation (1) is qualitatively similar to that of the linear system $x^{\prime}=Df(x_{0})x$.

Title Hartman-Grobman theorem HartmanGrobmanTheorem 2013-03-22 13:18:37 2013-03-22 13:18:37 jarino (552) jarino (552) 4 jarino (552) Theorem msc 34C99