Proof of the Hartman-Grobman Theorem

The Hartman-Grobman theorem establishes that near a hyperbolic equilibrium, a nonlinear system is topologically equivalent to its linearization.

Statement

Theorem6.7Hartman-Grobman theorem

Let $\mathbf{x}_0$ be a hyperbolic equilibrium of $\mathbf{x}' = \mathbf{F}(\mathbf{x})$ (i.e., all eigenvalues of $A = D\mathbf{F}(\mathbf{x}_0)$ have nonzero real parts). Then there exists a homeomorphism $h$ defined in a neighborhood of $\mathbf{x}_0$ that maps trajectories of the nonlinear system to trajectories of the linear system $\mathbf{u}' = A\mathbf{u}$ , preserving the direction of time.

RemarkTopological conjugacy

The theorem says there is a continuous change of coordinates (not necessarily smooth!) that transforms the nonlinear flow into the linear flow. The types of equilibria (node, saddle, spiral) are preserved, but quantitative features (eigenvalue magnitudes, spiral rates) may differ.

Proof Sketch

Proof

We outline the key steps for the case where $\mathbf{x}_0 = \mathbf{0}$ and $\mathbf{F}(\mathbf{x}) = A\mathbf{x} + \mathbf{N}(\mathbf{x})$ with $\mathbf{N}(\mathbf{0}) = \mathbf{0}$ and $D\mathbf{N}(\mathbf{0}) = 0$ .

Step 1: Global modification. Modify $\mathbf{N}$ outside a small ball to make it Lipschitz with small Lipschitz constant $\varepsilon$ . This does not change the local dynamics near $\mathbf{0}$ .

Step 2: Formulation as a fixed-point problem. Seek a homeomorphism $h$ such that $h \circ \phi_t = \psi_t \circ h$ where $\phi_t$ is the nonlinear flow and $\psi_t = e^{At}$ is the linear flow. Setting $h = \text{id} + u$ , this becomes a functional equation for $u$ :

$u(\psi_t(\mathbf{x})) = e^{At}u(\mathbf{x}) + \int_0^t e^{A(t-s)}\mathbf{N}(\psi_s(\mathbf{x}) + u(\psi_s(\mathbf{x})))\,ds.$

Step 3: Contraction mapping. Using the hyperbolicity (exponential dichotomy) of $e^{At}$ , show this equation defines a contraction on a space of bounded continuous functions, provided $\varepsilon$ is small enough.

Step 4: Properties of the solution. The fixed point $u$ is continuous and $h = \text{id} + u$ is a homeomorphism (invertibility follows from a similar construction for $h^{-1}$ ). $\blacksquare$

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Limitations and Extensions

ExampleThe conjugacy need not be smooth

For $x' = -x$ , $y' = -2y + x^2$ , the linearization is $u' = -u$ , $v' = -2v$ . The conjugacy exists but involves a term like $h(x,y) = (x, y - x^2)$ which, while smooth here, in general the Hartman-Grobman conjugacy is only continuous.

RemarkWhat the theorem does not say

The conjugacy $h$ is generally only continuous, not differentiable. The Sternberg theorem provides smooth (even $C^\infty$ ) conjugacy under additional non-resonance conditions on eigenvalues.
The theorem does not apply to non-hyperbolic equilibria. Centers ( $\lambda = \pm i\beta$ ) require separate analysis: the linearization shows periodic orbits, but the nonlinear system may have a spiral (stable or unstable).
The theorem is local: it says nothing about the global phase portrait.

RemarkStable and unstable manifolds

Near a hyperbolic equilibrium, the stable manifold theorem guarantees smooth invariant manifolds $W^s$ and $W^u$ tangent to the stable and unstable eigenspaces of $A$ . These manifolds organize the global dynamics and are preserved by the Hartman-Grobman conjugacy.