Let $n \ge 2$, let $\Omega \subseteq \mathbb{R}^n$ be open with $0 \in \Omega$, and let $F \in C^k(\Omega \times \mathbb{R}, \mathbb{R}^n)$ with $k \ge 3$. Consider the parameter-dependent system: \begin{align*} \frac{d}{dt}X = F(X, \mu), \quad X \in \mathbb{R}^n, \quad \mu \in \mathbb{R}, \end{align*} and assume $F(0,\mu) = 0$ for all $\mu$ in a neighbourhood of $0$. Let $A(\mu) := D_X F(0,\mu)$ denote the Jacobian at the origin. Suppose the system satisfies the Hopf Spectral Hypothesis at $\mu = 0$: the spectrum $\sigma(A(0))$ contains a simple pair of purely imaginary eigenvalues $\lambda_{1,2}(0) = \pm i\omega_0$ with $\omega_0 > 0$, and every other eigenvalue $\lambda_j(0)$ satisfies $\mathrm{Re}(\lambda_j(0)) \neq 0$. Then there exists an open interval $I \ni 0$ and a $C^{k-1}$ family of two-dimensional locally invariant manifolds $W^c_{\mathrm{loc}}(\mu) \subseteq \mathbb{R}^n$, defined for $\mu \in I$, with the following properties: 1. Each $W^c_{\mathrm{loc}}(\mu)$ passes through the origin and is tangent there to the center eigenspace $E^c := \ker(A(0)^2 + \omega_0^2 I)$. 2. Every sufficiently small periodic orbit of the full $n$-dimensional system near $(X,\mu) = (0,0)$ lies entirely on $W^c_{\mathrm{loc}}(\mu)$. 3. In a complex coordinate $z \in \mathbb{C}$ on $W^c_{\mathrm{loc}}(\mu)$ obtained by complexifying the restricted planar system, the dynamics are governed by the complex scalar equation: \begin{align*} \dot{z} = (\alpha(\mu) + i\omega(\mu))z + \sum_{j+k \ge 2} g_{jk}(\mu)\,z^j\bar{z}^k, \end{align*} where $\alpha(\mu) := \mathrm{Re}(\lambda_1(\mu))$, $\omega(\mu) := \mathrm{Im}(\lambda_1(\mu))$, $\alpha(0) = 0$, $\omega(0) = \omega_0$, and $g_{jk}(\mu) \in \mathbb{C}$ are the Taylor coefficients of the nonlinearity restricted to the center manifold.