Get the answer to "How do stable, unstable, and centre subspaces correspond..."

College
Calculus and Analysis
Differential Calculus
Critical Point

Karley Castillo

Answered question

2022-11-12

How do stable, unstable, and centre subspaces correspond to stability of a critical point?

Answer & Explanation

Haylie Park

Beginner2022-11-13Added 14 answers

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). 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). The dynamics is essentially nonlinear on the center manifold, but things like Lyapunov second method still can be applied.

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