Let $F: U \subseteq \mathbb{R}^n \to \mathbb{R}^n$ be a $C^k$ vector field ($k \ge 3$) with an equilibrium at the origin: $F(0) = 0$. Let $A := Jf_0 \in \mathbb{R}^{n \times n}$ denote the Jacobian at the origin. Suppose: 1. $A$ has a simple pair of purely imaginary eigenvalues $\lambda_{1,2} = \pm i\omega_0$ with $\omega_0 > 0$. 2. Every other eigenvalue $\lambda_j$ of $A$ satisfies $\operatorname{Re}(\lambda_j) < 0$. Let $W^c_{\mathrm{loc}}(0)$ denote the local center manifold (guaranteed by the [Center Manifold Existence theorem](/theorems/617)), and let \begin{align*} \dot{z} = i\omega_0 z + \tfrac{1}{2}g_{20}\,z^2 + g_{11}\,z\bar{z} + \tfrac{1}{2}g_{02}\,\bar{z}^2 + \tfrac{1}{2}g_{21}\,z^2\bar{z} + \cdots \end{align*} be the complexified system restricted to $W^c_{\mathrm{loc}}(0)$, with Taylor coefficients $g_{jk} \in \mathbb{C}$. Define the **First Lyapunov Coefficient**: \begin{align*} L_1 := \frac{1}{2\omega_0}\,\operatorname{Re}\!\left(ig_{20}g_{11} + \omega_0\,g_{21}\right). \end{align*} Then: 1. If $L_1 < 0$, the origin is **asymptotically stable** for the full system on $\mathbb{R}^n$. 2. If $L_1 > 0$, the origin is **unstable** for the full system on $\mathbb{R}^n$. 3. If $L_1 = 0$, let $L_k$ denote the $k$-th Lyapunov coefficient, corresponding to the coefficient of the resonant monomial $z^{k+1}\bar{z}^k$ in the normal form. The stability is determined by the first nonzero $L_k$: $L_k < 0$ gives asymptotic stability, $L_k > 0$ gives instability. If all Lyapunov coefficients vanish, the origin is a nonlinear center (all nearby orbits on $W^c_{\mathrm{loc}}(0)$ are periodic).