I am starting studying bifurcations, and I have encountered the term invariant manifold. I have a little confusion about what this is.
What I have understood is that if I consider for example a manifold $M$, it is invariant if any trajectory on the manifold remains in the manifold, but I am not sure that it is so simple since i am just stranslating the concept of invariant subspaces into manifolds.
I have seen that the invariant manifolds can be stable, unstable and center and so this makes me think that there is something more I should know.
Also, to give some context and to make clear my question, I am approching this topic since I am going to study center manifold and bifurcations (in 2D).
So, my question is: What is an invariant manifold?
I don't know if this will answer your questions , but if the term invariant manifold is not clear some references such nonlinear systems by Hassan Khalil use the term invariant set instead .
I hope this example clear out some of issues about stable, unstable and center manifolds . Consider a vector field $\dot{x}=f(x)$ defined on the surface of the sphere $S^2$ with a stable node at the north pole and unstable node at the south pole . The surface of the sphere is an invariant set (manifold) which is both the stable invariant set (manifold) -not including the unstable node - of stable node and unstable invariant set (manifold) -not including the stable node - of unstable node .