Hartman–Grobman theorem

In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearization theorem is a theorem about the local behavior of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearization—our first resort in applications—is unreasonably effective in predicting qualitative patterns of behavior.

The theorem states that the behavior of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behavior of its linearization near this equilibrium point, provided that no eigenvalue of the linearization has its real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearization of the system to analyze its behavior around equilibria.

Main theorem

Consider a system evolving in time with state $u(t)\in\mathbb R^n$ that satisfies the differential equation $du/dt=f(u)$ for some smooth map $f: \mathbb{R}^n \to \mathbb{R}^n$. Suppose the map has a hyperbolic equilibrium state $u^*\in\mathbb R^n$: that is, $f(u^*)=0$ and the Jacobian matrix $A=[\partial f_i/\partial x_j]$ of $f$ at state $u^*$ has no eigenvalue with real part equal to zero. Then there exists a neighborhood $N$ of the equilibrium $u^*$ and a homeomorphism $h : N \to \mathbb{R}^n$, such that $h(u^*)=0$ and such that in the neighbourhood $N$ the flow of $du/dt=f(u)$ is topologically conjugate by the continuous map $U=h(u)$ to the flow of its linearization $dU/dt=AU$.

Even for infinitely differentiable maps $f$, the homeomorphism $h$ need not to be smooth, nor even locally Lipschitz. However, it turns out to be Hölder continuous, with an exponent depending on the constant of hyperbolicity of $A$.

The Hartman-Grobman theorem has been extended to infinite dimensional Banach spaces, non-autonomous systems $du/dt=f(u,t)$ (potentially stochastic), and to cater for the topological differences that occur when there are eigenvalues with zero or near-zero real-part.

Example

The algebra necessary for this example is easily carried out by a web service that computes normal form coordinate transforms of systems of differential equations, autonomous or non-autonomous, deterministic or stochastic.

Consider the 2D system in variables $u=(y,z)$ evolving according to the pair of coupled differential equations

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): dy/dt=-3y+yz\quad\text{and}\quad dz/dt=z+y^2.

This system has an equilibrium at the origin, that is $u^*=0$, among others not analysed here. The coordinate transform, $u=h^{-1}(U)$ where $U=(Y,Z)$, given by

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): y\approx Y+YZ+\dfrac1{42}Y^3+\dfrac12YZ^2

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): z\approx Z-\dfrac17Y^2-\dfrac13Y^2Z

is a smooth map between the original $u=(y,z)$ and new $U=(Y,Z)$ coordinates, at least near the equilibrium at the origin. In the new coordinates the dynamical system transforms to its linearisation

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): dY/dt=-3Y\quad\text{and}\quad dZ/dt=Z.

That is, a distorted version of the linearization gives the original dynamics in some finite neighbourhood.