The problem I have is that I have a (very general) system of three ODEs which I need to analyse the critical points of. I have found that the system has a critical point at the origin, with eigenvalues $0,0,-c$ for some $c > 0$.
I would think this critical point is stable.
But because of the presence of the zero eigenvalues, there is a centre subspace. Because of the $-c$ eigenvalue, so there is a stable subspace also. So my difficulty is in understanding how the subspaces correspond to the stability of the point. I know that any solution starting in a subspace remains in it, but what about points starting outside the union of the subspaces?
Also, what happens if we have eigenvalues $0,a,-b$ with $a,b > 0$ of the Jacobian at the critical point? We have all three subspaces non-empty, but how do they correspond to stability of the critical point?
Thanks in advance!