Hartman–Grobman 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 \colon \mathbb{R}^n \to \mathbb{R}^n. Now 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 neighbourhood N of the equilibrium u^* and a homeomorphism h \colon 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. A like result holds for iterated maps, and for fixed points of flows or maps on manifolds. A mere topological conjugacy does not provide geometric information about the behavior near the equilibrium. Indeed, neighborhoods of any two equilibria are topologically conjugate so long as the dimensions of the contracting directions (negative eigenvalues) match and the dimensions of the expanding directions (positive eigenvalues) match. But the topological conjugacy in this context does provide the full geometric picture. In effect, the nonlinear phase portrait near the equilibrium is a thumbnail of the phase portrait of the linearized system. This is the meaning of the following regularity results, and it is illustrated by the saddle equilibrium in the example below. 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 exponent arbitrarily close to 1. Moreover, on a surface, i.e., in dimension 2, the linearizing homeomorphism and its inverse are continuously differentiable (with, as in the example below, the differential at the equilibrium being the identity) And in any dimension, if f has Hölder continuous derivative, then the linearizing homeomorphism is differentiable at the equilibrium and its differential at the equilibrium is the identity. 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 \frac{dy}{dt} = -3y + yz \quad\text{and}\quad \frac{dz}{dt} = z+y^2. By direct computation it can be seen that the only equilibrium of this system lies at the origin, that is u^* = 0. The coordinate transform, u = h^{-1}(U) where U = (Y,Z), given by \begin{align} y & \approx Y + YZ + \tfrac{1}{42} Y^3 + \tfrac{1}{2} Y Z^2 \\[5pt] z & \approx Z - \tfrac{1}{7} Y^2 - \tfrac{1}{3} Y^2 Z \end{align} 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 linearization \frac{dY}{dt} = -3Y \quad\text{and}\quad \frac{dZ}{dt} = Z. That is, a distorted version of the linearization gives the original dynamics in some finite neighbourhood. ==See also==