We are considering the following system $$(1)\hspace{25pt} \begin{split} \dot x = A x + F(x,y),~\\ \dot y = B y + G(x,y), \end{split}$$ where $x\in\mathbb{R}^n,y\in\mathbb{R}^m$, $A$ and $B$ are the blocks in the Jordan canonical form with eigenvalues $Re\lambda=0$ and $Re\lambda<0$, respectively. We also assume that $F$ and $G$ are $\mathcal{C}^r$ ($r\geq 1$) functions and $$F(0,0)=0,\,G(0,0)=0,\,\quad \grad_{x,y} F(0,0)=0,\,\grad_{x,y} G(0,0)=0.$$ Then we have **Theorem 1** (Center Manifold Theorem) There exists a $\mathcal{C}^r$-center manifold $$W_{loc}^c = \{(x,y)\,:\, y=h(x),\,\abs{x}<\delta,\,h(0)=0,\grad_{x} h(0)=0 \},$$ such that the dynamics of (1) restricted to the center manifold are given by $$\dot x=A x + G(x,h(x)).$$ The proof of Theorem **1** can be found in [Car81, Section 2.3]. **Remark 1** If we substitute $y=h(x),\dot y=\grad_x h(x) \dot x$ into (1), we then find $h$ satisfies $$Bh(x)+G(x,h(x))=\grad_x h(x)\big(Ax+F(x,h(x))\big).$$ Expressed differently we have $$N(h(x)): = \grad_x h(x)\big(Ax+F(x,h(x))\big) - Bh(x) - G(x,h(x)) =0.$$ We have the following approximation theorem for the center manifold. **Theorem 2** Let $\phi:\mathbb{R}^n\rightarrow\mathbb{R}^m$ be a $\mathcal{C}^r$ function with $\phi(0)=0$ and $\grad_x\phi(0)=0$. Assume that $N(\phi(x)) =O(\abs{x}^k)\quad (\abs{x}\rightarrow 0)$ for some $k>1$. Then $$\abs{h(x)-\phi(x)} = O(\abs{x}^k),\quad (\abs{x}\rightarrow 0).$$ **Remark 2** The proof of Theorem **2** is proved in [Car81] by considering the operator $N$ on centain space. Here we give a more direct proof. //Proof// === Reference W. O. Bray [[http://www.math.umaine.edu/~bray/Archive_res/Lecture3.PDF|Dynamical System Reduction: The Center Manifold]]. [Car81] J. Carr, Applications of Center Manifold Theory, Springer-Verlag, 1981.