# Lyapunov theorem (dynamical system) in linear differential equation

I have a question about center spaces. Taking into account a linear differential equation we know Lyapunov theorem states: if all the eigenvalues of the matrix A have strictly negative real part then c is a stable equilibrium point for the equation. If A has at least one eigenvalue with strictly positive real part then c is an unstable equilibrium point for the equation. But the theorem does non say anything about center space. For example if I have all the eigenvalue with real part equal to zero or all negative with at least one eigenvalue with real part equal to 0 can I say something about center space or theorem simply does not apply ? Moreover If I can say something about center space how can I say if center spaces are stable or unstable ?

Thank you !

• The Lyapunov linearization theorem does not deal with linear problems (and if the there is/are an eigenvalue/eigenvalues for the linearization with zero real part and the remaining eigenvalues have negative real parts then nothing can be said). For linear equations, one can say much more in such a case, but that is another theorem (as far as I know, not named after anyone, certainly not after Lyapunov). – user539887 Jun 22 '18 at 19:09
• See, e.g., here math.stackexchange.com/a/2240282/310971 – Dmitry Jun 24 '18 at 8:43
• which Lyapunov theorem you are talking about (i.e. direct or indirect )? – CroCo Jul 8 '18 at 10:15