The proof uses the invariance equation and the contraction structure of the center manifold construction to show that approximate solutions of the invariance equation are close to the true solution. **Step 1: The invariance equation.** A [function](/page/Function) $\psi: \mathbb{R}^c \to \mathbb{R}^s$ with $\psi(0) = 0$, $D\psi(0) = 0$ defines a center manifold if and only if $\mathcal{M}(\psi)(x) = 0$ for all $x$ near $0$, where the **invariance operator** $\mathcal{M}$ is defined by \begin{align*} \mathcal{M}(\psi)(x) := D\psi(x)\bigl[Ax + f(x, \psi(x))\bigr] - B\psi(x) - g(x, \psi(x)). \end{align*} This equation expresses the chain rule condition: for the graph $y = \psi(x)$ to be invariant, the time [derivative](/page/Derivative) $\dot{y} = D\psi(x) \dot{x}$ must equal $By + g(x, y)$ when $y = \psi(x)$. **Step 2: The approximate solution.** By hypothesis, $\phi$ satisfies $\mathcal{M}(\phi)(x) = O(|x|^p)$ as $x \to 0$. Let $h$ be the true center manifold (from the [Center Manifold Existence theorem](/theorems/617)), so $\mathcal{M}(h)(x) = 0$. **Step 3: Estimate of $h - \phi$.** [claim:Closeness Of Approximate And True Manifolds] $\|h(x) - \phi(x)\| = O(|x|^p)$ as $x \to 0$. [/claim] [proof] Define $\rho(x) := h(x) - \phi(x)$. Subtracting: $\mathcal{M}(h)(x) - \mathcal{M}(\phi)(x) = -\mathcal{M}(\phi)(x) = O(|x|^p)$. Expanding the left side: \begin{align*} &D h(x)[Ax + f(x, h(x))] - Bh(x) - g(x, h(x)) \\ &\quad - D\phi(x)[Ax + f(x, \phi(x))] + B\phi(x) + g(x, \phi(x)) = O(|x|^p). \end{align*} Rearranging and using the [mean value theorem](/theorems/186) to estimate the nonlinear terms: the difference $f(x, h(x)) - f(x, \phi(x))$ is bounded by $\|Df\|_\infty |\rho(x)|$, and similarly for $g$. The terms involving $Dh - D\phi$ contribute at most $|D\rho(x)| \cdot O(|x|)$ (since $Ax + f(x, h(x)) = O(|x|)$ near the origin). After collecting, one obtains a differential inequality of the form \begin{align*} |B\rho(x) + \text{(lower order in $\rho$)}| \le C|x|^p + L(\delta)(|\rho(x)| + |D\rho(x)| \cdot |x|), \end{align*} where $L(\delta) \to 0$ as $\delta \to 0$. Since $B$ is invertible (no zero eigenvalues) and $L(\delta)$ is small, this can be inverted to give $|\rho(x)| \le C'|x|^p$ for $|x|$ small. More precisely, this follows from the contraction mapping structure: the operator $\mathcal{T}$ from the existence proof satisfies $\|h - \phi\|_\eta \le (1 - \kappa)^{-1} \|\mathcal{T}(\phi) - \phi\|_\eta$ (where $\kappa < 1$ is the contraction constant), and $\|\mathcal{T}(\phi) - \phi\|_\eta$ is controlled by $\|\mathcal{M}(\phi)\|_\eta = O(|x|^p)$. [/proof] This proves $\|h(x) - \phi(x)\| = O(|x|^p)$. In particular, $h$ and $\phi$ agree up to order $p - 1$ in their Taylor expansions at the origin. Since $p$ is arbitrary (as long as $\phi$ approximates the invariance equation to that order), all center manifolds share the same Taylor expansion to every finite order — even though the manifolds themselves may differ globally.