The proof shows that the stable variables $y$ decay exponentially along any trajectory near the origin, so the long-time behaviour is governed entirely by the center variables $x$ restricted to the center manifold. **Step 1: Decomposition near the center manifold.** Write any point near the origin as $(x, y) = (x, h(x) + \eta)$, where $h(x)$ is the center manifold [function](/page/Function) from the [Center Manifold Existence theorem](/theorems/617) and $\eta = y - h(x)$ is the deviation from the manifold. On the center manifold, $\eta = 0$. We derive an equation for $\eta$. **Step 2: Equation for the deviation $\eta$.** Since $\dot{y} = By + g(x, y)$ and $\dot{h}(x) = Dh(x)(Ax + f(x, y))$, the deviation satisfies \begin{align*} \dot{\eta} = \dot{y} - \dot{h}(x) = B(h(x) + \eta) + g(x, h(x) + \eta) - Dh(x)(Ax + f(x, h(x) + \eta)). \end{align*} Using the invariance condition (which says $Bh(x) + g(x, h(x)) = Dh(x)(Ax + f(x, h(x)))$ on the manifold) and Taylor-expanding around $\eta = 0$: \begin{align*} \dot{\eta} = B\eta + R(x, \eta), \end{align*} where $R(x, \eta)$ collects the nonlinear remainder terms satisfying $R(x, 0) = 0$ and $D_\eta R(x, 0) = 0$. The crucial structure is that the linear part of the $\eta$-equation is $B\eta$, with $B$ having all eigenvalues in the left half-plane. **Step 3: Exponential decay of $\eta$.** [claim:Exponential Stability Of The Deviation] There exist constants $M > 0$, $\alpha > 0$, and $\delta > 0$ such that if $|(x(0), y(0))| < \delta$, then \begin{align*} |\eta(t)| \le M e^{-\alpha t} |\eta(0)| \quad \text{for all } t \ge 0 \text{ as long as } |(x(t), y(t))| < \delta. \end{align*} [/claim] [proof] Since $B$ has spectral radius $\max_j \operatorname{Re} \lambda_j(B) < 0$, there exists $\alpha > 0$ such that $\|e^{Bt}\| \le M e^{-\alpha t}$ for $t \ge 0$. The remainder $R(x, \eta)$ satisfies $|R(x, \eta)| \le L(\delta)(|x| + |\eta|) |\eta|$ where $L(\delta) \to 0$ as $\delta \to 0$ (since $R$ vanishes to second order). The variation-of-constants formula gives $\eta(t) = e^{Bt}\eta(0) + \int_0^t e^{B(t-s)} R(x(s), \eta(s)) \, ds$. For $\delta$ small enough, the Gronwall-type estimate absorbs the nonlinear perturbation into the exponential decay, yielding $|\eta(t)| \le 2M e^{-\alpha t/2} |\eta(0)|$. [/proof] **Step 4: Reduction of stability.** Since $\eta(t) \to 0$ exponentially, the trajectory $(x(t), y(t))$ approaches the center manifold exponentially fast. The long-time behaviour of $x(t)$ is therefore governed by the reduced equation $\dot{u} = Au + f(u, h(u))$ up to exponentially decaying errors. If the origin is asymptotically stable for the reduced equation, then $x(t) \to 0$, which combined with $\eta(t) \to 0$ gives $(x(t), y(t)) \to 0$. If the origin is unstable for the reduced equation, then trajectories leave a neighbourhood of the origin along the center manifold, so the full system is unstable. Stability (without the "asymptotic" qualifier) transfers similarly, since the exponential decay of $\eta$ means that any Lyapunov function for the reduced system extends to the full system near the manifold.