Bifurcations and Chaos - Complex eigenvalues

[Rubik]

Homework Statement

Identify the stable, unstable and center eigenspaces for

\dot{y} = the 3x3 matrix
row 1: 0, -3, 0
row 2: 3, 0, 0
row 3: 0, 0, 1

Homework Equations

The Attempt at a Solution

This is an example used from the lecture and I understand how to get the eigenvalues
λ_1,2 = ±3i and λ₃ = 1.

However, I do not see how to find E^C?

I understand how E^U = the vector (0, 0, 1)
But I do not see how to find E^C = {(1, 0, 0) and (0, 1, 0)}

Can anyone explain this for me?

 

[tiny-tim]

Hi Rubik!

http://en.wikipedia.org/wiki/Center_manifold is very clear on this , so i'll just quote it (with highlights) …​

In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium points of dynamical systems is to linearize the system.

The eigenvectors corresponding to eigenvalues with negative real part form the stable eigenspace, which gives rise to the stable manifold. The stable manifold attracts orbits close to it.

Similarly, eigenvalues with positive real part yield the unstable manifold, which repels orbits close to it.

This concludes the story if the equilibrium point is hyperbolic (i.e., all eigenvalues of the linearization have nonzero real part). However, if there are eigenvalues whose real part is zero, then these give rise to the center manifold.

The behavior near the center manifold is not determined by the linearization and thus more difficult to study.

Center manifolds play an important role in bifurcation theory because the interesting behavior takes place on the center manifold.