412 15 Center Manifold Reduction 2

[@Mudsleeper11]

This video (and the previous one) is fantastic. Thank you very much!

[@ankailiu3365]

Question: it is not obvious that the calculation via power series utilize the fact that it is a center manifold. In fact it looks like such argument hold as long as the graph of h is an invariant manifold. Is there anything that I missed here? Of course in this case any invariant manifold that is not a stable manifold is a center manifold. But what about systems with unstable manifold?

[@hermannriecke6565]

Indeed, this calculation could be done with any invariant manifold. If you do it for the stable manifold you will get an equation for the fast contraction to the fixed point, but not an equation for the bifurcation of interest, i.e. for the purpose at hand you are looking for a lower-dimensional equation that captures the creation of the two new fixed points in the case of a pitch-fork bifurcation, say. Those fixed points are on the center manifold of the suspended system and not on the stable manifold of that system.

[@ankailiu3365]

@@hermannriecke6565 I see. Thanks you!