Considering a parameter-dependent ODE
$\dot{x} = Ax + f_A(x,y,z,\mu) \\ \dot{y} = By + f_B(x,y,z,\mu) \\ \dot{z} = Cz + f_C(x,y,z,\mu) $
where A has only eigenvalues with zero real part, B only eigenvalues with negative real part and C has only eigenvalues with positive real part and such that $(x,y,z,\mu) \in \mathbb{R}^{n_0} \times\mathbb{R}^{n_-} \times\mathbb{R}^{n_+} \times\mathbb{R}^{p}$. Suppose the system has an equilibrium in $(x,y,z,\mu) = (0,0,0,0)$. From the center manifold theorem I know there exists a neighbourhood $U_{n_0} \subset \mathbb R^{n_0}$ of the origin and a $n_0$-dimensional center manifold $c: U_{n_0}\to \mathbb{R}^{n_-}\times \mathbb{R}^{n_+}$. The standard trick is to extend the system to
$\dot{x} = Ax + f_A(x,y,z,\mu) \\ \dot{\mu} = 0 \\ \dot{y} = By + f_B(x,y,z,\mu) \\ \dot{z} = Cz + f_C(x,y,z,\mu) $
and the center manifold theorem implies there exists neighbourhood $\mathbb{R}^{n_0} \times \mathbb{R}^{p}$ of the origin and a $n_0+p$-dimensional center manifold $\tilde{c}: U_{n_0} \times U_{p}\times \mathbb{R}^{p} \to \mathbb{R}^{n_-}\times \mathbb{R}^{n_+}$.
My question is, how are $c$ and $\tilde{c}$ related? Is there a way to express $c$ in terms of $\tilde{c}$ or vice versa?