Consider a system $\dot{x} = f(x) \in \mathbb{R}^N$ with an equilibrium at $x_0$ for which the Jacobian has a zero eigenvalue and all other eigenvalues have negative real part. By the Reduction Principle, in a neighborhood of $x_0$, the system is topologically equivalent to
$$ \begin{align*} \dot{x}_c &= f_c(x_c) \\ \dot{x}_s &= -x_s \\ \end{align*} $$
where $f_c$ describes the dynamics on the center manifold, which satisfy the center manifold equation.
In general, finding an exact center manifold is not possible, so one has to compute series approximations that satisfy the center manifold equation up to some desired order.
My question is, when is the projection of the dynamics onto the center eigenspace a valid approximation of the center manifold dynamics (up to second order or higher)? is there any information about the center manifold dynamics in the projected dynamics?