Undefined (mathematics)

In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearization theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point.

Basically the theorem states that the behaviour of a dynamical system near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearization near this equilibrium point provided that no eigenvalue of the linearization has its real part equal to 0. Therefore when dealing with such fixed points one can use the simpler linearization of the system to analyze its behaviour. That is, linearisation is 'unreasonably' effective.

Main theorem

Let $f:{\mathbb{ℝ}}^n\to {\mathbb{ℝ}}^n$ be a smooth map of a dynamical system with differential equation $du/dt=f(u)$. Suppose the map has a hyperbolic equilibrium point $u_0$: that is, $f(u_0)=0$ and the Jacobian matrix $A=[\partial f_i/\partial x_j]$ of $f$ at point $u_0$ has no eigenvalue with real part equal to zero. Then there exists a neighborhood $N$ of the equilibrium $u_0$ and a homeomorphism $h:N\to {\mathbb{ℝ}}^n$, such that $h(u_0)=0$ and such that in the neighbourhood $N$ the flow of $du/dt=f(u)$ is topologically conjugate by the smooth map $U=h(u)$ to the flow of its linearization $dU/dt=AU$.

In general, 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 $A$.

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 [1].

Consider the 2D system in variables $u=(y,z)$ evolving according to

$dy/dt=-3y+yz\text{and}dz/dt=z+y^2.$

This system has an equilibrium at the origin, that is $u_0=0$, among others not analysed here. The coordinate transform, $u=h^{-1}(U)$ where $U=(Y,Z)$, given by

$y\approx Y+YZ+\frac{1}{42}{Y}^3+\frac{1}{2}Y{Z}^2$

$z\approx Z-\frac{1}{7}{Y}^2-\frac{1}{3}{Y}^{2}Z$

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

$dY/dt=-3Y\text{and}dZ/dt=Z.$

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

External links

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