Let $U \subseteq \mathbb{R}^2$ be open with $0 \in U$ and let $f \in C^k(U \times \mathbb{R}, \mathbb{R}^2)$ with $k \ge 5$. Consider the parameter-dependent map: \begin{align*} x \mapsto f(x, \alpha), \quad x \in \mathbb{R}^2, \quad \alpha \in \mathbb{R}, \end{align*} with $f(0, \alpha) = 0$ for all $\alpha$ in a neighbourhood of $0$. Let $A(\alpha) := D_x f(0, \alpha)$ and let $\lambda(\alpha), \overline{\lambda(\alpha)}$ denote the complex conjugate eigenvalue pair of $A(\alpha)$, written in polar form as $\lambda(\alpha) = r(\alpha) e^{i\theta(\alpha)}$ with $r(\alpha) > 0$ and $\theta(\alpha) \in (0, \pi)$. Suppose the following four conditions hold at $\alpha = 0$: **(NS1) Criticality.** The eigenvalues lie on the unit circle: \begin{align*} r(0) = 1, \quad \theta(0) = \theta_0 \in (0, \pi). \end{align*} **(NS2) Transversality.** The modulus of the eigenvalue crosses the unit circle at non-zero speed: \begin{align*} r'(0) := \frac{d}{d\alpha} r(\alpha)\bigg|_{\alpha = 0} \neq 0. \end{align*} **(NS3) Non-resonance.** No strong resonance occurs: \begin{align*} e^{ik\theta_0} \neq 1 \quad \text{for } k \in \{1, 2, 3, 4\}. \end{align*} Equivalently, $\theta_0 \notin \{0, \pi/2, 2\pi/3, \pi\}$ (excluding the $1{:}1$, $1{:}4$, $1{:}3$, and $1{:}2$ resonances). **(NS4) Non-degeneracy.** Let $g_{jl}$ denote the Taylor coefficients of the map in complex normal form coordinates $z = x_1 + ix_2$ at $\alpha = 0$: \begin{align*} z \mapsto e^{i\theta_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*} The First Lyapunov Coefficient: \begin{align*} L_1 := \mathrm{Re}\!\left(\frac{e^{-i\theta_0} g_{21}}{2}\right) - \mathrm{Re}\!\left(\frac{(1 - 2e^{i\theta_0})\,e^{-2i\theta_0}}{2(1 - e^{i\theta_0})}\,g_{20}\,g_{11}\right) - \frac{1}{2}|g_{11}|^2 - \frac{1}{4}|g_{02}|^2 \end{align*} is non-zero: $L_1 \neq 0$. Then the following conclusions hold: 1. **Existence.** There exists a neighbourhood of the origin in which a unique closed invariant curve $\Gamma_\alpha$ bifurcates from the fixed point. The curve encloses the origin and has diameter $O(\sqrt{|\alpha|})$ as $\alpha \to 0$. 2. **Supercritical case ($L_1 < 0$).** The invariant curve is attracting (stable). If $r'(0) > 0$, the curve exists for $\alpha > 0$, where the fixed point has become unstable ($r(\alpha) > 1$). Nearby orbits are attracted to $\Gamma_\alpha$ and exhibit quasi-periodic or phase-locked motion on it. 3. **Subcritical case ($L_1 > 0$).** The invariant curve is repelling (unstable). If $r'(0) > 0$, the curve exists for $\alpha < 0$, where the fixed point is still stable ($r(\alpha) < 1$) but has a finite basin of attraction bounded by $\Gamma_\alpha$.