**Proof plan.** The analysis proceeds entirely in polar coordinates, where the radial and angular equations decouple. The radial equation is a one-dimensional bifurcation problem — a pitchfork for $r \geq 0$ — and determines existence, amplitude, and stability of the [limit](/page/Limit) cycle. The angular equation provides the frequency. **Step 1 (Polar coordinate reduction).** Substituting $z = re^{i\theta}$ into $\dot{z} = (\mu + i\omega)z + (\sigma + i\gamma)z|z|^2$ and separating real and imaginary parts of $\dot{z}/z = \dot{r}/r + i\dot{\theta}$: \begin{align*} \frac{\dot{r}}{r} + i\dot{\theta} &= (\mu + i\omega) + (\sigma + i\gamma)r^2. \end{align*} Equating real and imaginary parts gives the decoupled system \begin{align*} \dot{r} &= \mu r + \sigma r^3, \\ \dot{\theta} &= \omega + \gamma r^2. \end{align*} **Step 2 (Fixed points of the radial equation).** The radial equation $\dot{r} = r(\mu + \sigma r^2)$ has $r = 0$ as an equilibrium for all $\mu$, with linearisation $\partial_r[\mu r + \sigma r^3]|_{r=0} = \mu$. Hence $r = 0$ is stable for $\mu < 0$ and unstable for $\mu > 0$. A nontrivial equilibrium $r_0 > 0$ requires $\mu + \sigma r_0^2 = 0$, giving $r_0 = \sqrt{-\mu/\sigma}$. This exists when $\mu/\sigma < 0$. **Step 3 (Supercritical case: $\sigma < 0$).** Since $\sigma < 0$, the nontrivial radius $r_0 = \sqrt{\mu/|\sigma|}$ exists for $\mu > 0$. The linearisation of the radial equation at $r_0$ is \begin{align*} \frac{\partial}{\partial r}[\mu r + \sigma r^3]\bigg|_{r = r_0} = \mu + 3\sigma r_0^2 = \mu + 3\sigma \cdot \frac{-\mu}{\sigma} = -2\mu. \end{align*} For $\mu > 0$, this is $-2\mu < 0$, confirming that the limit cycle is asymptotically orbitally stable. The frequency on the limit cycle is $\dot{\theta} = \omega + \gamma r_0^2 = \omega - \gamma\mu/\sigma$. For $\mu < 0$, no nontrivial equilibrium exists and $r = 0$ is globally asymptotically stable (since $\dot{r} = r(\mu + \sigma r^2) < 0$ for all $r > 0$ when $\mu < 0$ and $\sigma < 0$). **Step 4 (Subcritical case: $\sigma > 0$).** Now $r_0 = \sqrt{-\mu/\sigma}$ exists for $\mu < 0$. The linearisation at $r_0$ gives $-2\mu > 0$ for $\mu < 0$, so the limit cycle is unstable. It coexists with the stable equilibrium $r = 0$ and bounds its basin of attraction: initial conditions with $r(0) < r_0$ converge to the origin, while those with $r(0) > r_0$ diverge (in the normal form; higher-order terms may prevent blow-up in the full system). **Step 5 (Amplitude scaling).** In both cases, the limit cycle amplitude scales as $r_0 = \sqrt{|\mu|/|\sigma|}$, which is $O(\sqrt{|\mu|})$ as $\mu \to 0$. This square-root scaling is the signature of the [Hopf bifurcation](/page/Hopf%20Bifurcation) and corresponds to the fact that the radial equation undergoes a pitchfork bifurcation at $\mu = 0$ (restricted to $r \geq 0$, this becomes a transcritical bifurcation).