I would like the simplest example of the failure of an ODE to be locally diffeomorphic to its linearization, despite being locally homeomorphic to it. More precisely, consider x' = f(x) with f(0) = 0 in R^n. Let A = f'(0) so that the local linearization is x' = Ax. Suppose the eigenvalues of A all have nonzero real part (i.e., 0 is a hyperbolic critical point).

The Hartman-Grobman theorem tells us that there is a homeomorphism of a neighborhood of 0 which conjugates the system x'=f(x) to its linearization x'=Ax. If one reads the elementary `differential equations from the dynamical systems point of view' literature, however, you will gain the false impression that the is a diffeomorphism h : U --> U of a nhood of 0 which does this, and that further, one can even do this with h'(0) = I, the identity matrix. The point of this is to ensure that the trajectories of the nonlinear system are tangent to the trajectories of the linearization: if $h'(0)\neq I$ then this may be false.

Smale's stable manifold theorem gives partial information in this direction, saying that the stable manifolds of the system and its linearization are tangent, and similarly for the unstable manifolds. In 2D, at a saddle, this is sufficient to imply that the separatrices of the original system are tangent to those of the linearization. For a node in 2D, or in higher dimensions, I am under the impression this need not hold. I even think I had worked out an example many years ago, which I no longer recall.

Any enlightenment on this issue would be much appreciated. I am not at all expert in these matters, so welcome any corrections, if I have distorted the facts. I am hoping for a 2 dimensional example.