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Let $x'(t)=Ax(t)$ be a linear ODE system.

Then the span of the eigenvectors belonging to eigenvalues with negative real-part is called the stable subspace, the subspace spanned by the eigenvectors of eigenvalues of positive real-part is called unstable subspace and the subspace spanned by the eigenvectors belonging to eigenvalues with zero real-part is called centre subspace.

  1. What kind of stability is the stable subspace of?

  2. What kind of unstability (as negation of what kind of stability) is the unstable subspace of?

  3. Is the centre subspace stable or unstable? Are there situation in which it is stable (and which kind of stability then)?

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  1. Globally asymptotically stable: solutions with initial values anywhere in this subspace converge to $0$.

  2. Negation of any kind of stability: the solutions with initial values in this subspace are unbounded, eventually leaving every compact set.

  3. Center is (globally) Lyapunov stable, not asymptotically stable.

All of this follows by considering the compression of $A$ to the relevant subspace, since the orbits beginning there stay in it. Reference: G. Teschl, ODE and Dynamical Systems.

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  • $\begingroup$ Just a comment to @Normal Human's definition: Asymptotic stability means: a solution starting close remains close and eventually converge. Thus, for the centre case: solutions starting close remain close but not necessarily converges $\endgroup$ – Cybernetician Nov 11 '15 at 21:49
  • $\begingroup$ @Humberto starting close to what? Close to the stable subspace? $\endgroup$ – M. Meyer Nov 21 '15 at 18:25
  • $\begingroup$ For example, let the stable subspace be spanned by only one eigenvector. What, if we start on that subspace? I guess then we converge to the equilibrium, or? And what if we start close to this subspace (in this case a line)? Do we then eventually converge to the origin by "going" along near that line? $\endgroup$ – M. Meyer Nov 21 '15 at 18:52

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