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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!

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If you have only stable subspace, the critical point is Lyapunov asymptotically stable (you can quite easily construct Lyapunov function here and use Lyapunov second method). If unstable subspace is non-empty, and there are other subspaces, the critical point is unstable (Chetaev theorem can be used for proof here, as far as I remember). The most interesting case is when you have only center and stable subspace. In that case all depends on dynamics of system on center manifold (see explanation in Shilnikov-Shilnikov-Turaev-Chua book around formula 5.0.6). The dynamics is essentially nonlinear on the center manifold, but things like Lyapunov second method still can be applied.

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