Assuming some knowledge of center manifold theory, I would like more details on the following statement found on the Wikipedia page https://en.wikipedia.org/wiki/Center_manifold.
"The center manifold emergence theorem then says that the neighborhood may be chosen so that all solutions of the system staying in the neighborhood tend exponentially quickly to some solution $y(t)$ on the center manifold. That is, $x(t) = y(t) + O( e^{−\beta t} )$ as $t \to \infty$ for some rate $\beta$."
I have never seen a statement of the center manifold theorem where the proof of this result is explicit. The attractivity part of the Theorem in Ioos and Adelmeyer (referenced by Wikipedia) is stated without proof. In Kuznetsov's book it states that a necessary condition for a center manifold to be attractive is that the unstable manifold vanishes, but not that this condition is sufficient. Guckenheimer + Holmes only state the Theorem of Henry and Carr that says if the unstable manifold vanishes and the origin is a stable solution of the reduced equation, then it is a stable solution of the whole system.
In summary, I am looking for a proof of the intuitive fact that, if the unstable manifold is empty, for large time any trajectory starting in the neighborhood of the equilibrium point (taken w.l.o.g. to be the origin) approaches a solution of the reduced equation on the center manifold, regardless the nature of the origin as an equilibrium of the reduced equation.
Motivating example is the system \begin{equation*} \dot{x} = A x + f(x,y), \quad \dot{y} = By +g(x,y)\end{equation*} where $A \in \Bbb R^{2 \times 2}$ has an imaginary pair of eigenvalues and all eigenvalues of $B \in \Bbb R^{n \times n}$ have negative real part. Suppose the origin is a center of the reduced equation $\dot{x}=Ax+f(x,h(x))$ on the center manifold $y= h(x)$. Then my intuition says solutions approach a periodic solution of reduced equation that is realized as a limit cycle of the total system. When is this the case? What happens if the reduced equation admits a family of periodic orbits?