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 ${\mathrm{\Phi }}_t(x)$ the flow of (1) through $x$ at time $t$. Then there exists a homeomorphism $\phi (x)=x+h(x)$ with $h$ bouded, such that

$$\phi \circ e^{tDf(x_0)}={\mathrm{\Phi }}_t\circ \phi$$

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 $x^{\prime }=Df(x_0)x$.