Hartman–Grobman theorem

Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be continuously differentiable. Let $\varphi$ be the flow of $x' = f(x)$, and let $x^* \in U$ be a hyperbolic equilibrium point. Let $A = Df(x^*)$. Then there are open neighbourhoods $V \subseteq U$ of $x^*$ and $W \subseteq \mathbb{R}^n$ of $0$, and a homeomorphism $h : V \to W$ with $h(x^*) = 0$, as follows. For each $x \in V$ and each $t \in \mathbb{R}$ such that $\varphi_s(x) \in V$ for all $s$ between $0$ and $t$, $h(\varphi_t(x)) = e^{tA} h(x)$.

Remarks

So near $x^*$ the flow is topologically conjugate to the flow of the linearisation $y' = A y$, given by the matrix exponential. The map $h$ sends trajectories to trajectories and keeps the direction of time. The map $h$ is in general only continuous, not differentiable. The hypothesis of hyperbolicity is necessary. For $x' = -x^3$ and $x' = x^3$ on $\mathbb{R}$, the linearisation at $0$ is the same, but the two phase portraits are not conjugate.

History

The theorem is named after Philip Hartman and David M. Grobman. It is also called the linearization theorem.

Explanations

L1 human

Near a hyperbolic equilibrium, a nonlinear system looks like its linear approximation, after a continuous change of coordinates. So the pattern of trajectories of the linear system, such as a saddle or a spiral, also appears in the nonlinear system.