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For $r>0$, polar coordinates give $\dot{r}=r({\mu }_1-2x)$ and $\dot{\theta }=1$; the remaining amplitude equation is $\dot{x}={\mu }_2+x^2+r^2$. Interpret $r=0$ in the original complex coordinate, where the phase is undefined.

If ${\mu }_2<0$, put $a=\sqrt{-{\mu }_2}$. The full equilibrium points are $E_{\pm }:(z,x)=(0,\pm a)$. Their eigenvalues areThus $E_{-}$ is stable for ${\mu }_1<-2a$ and has a two-dimensional unstable manifold for ${\mu }_1>-2a$. The point $E_{+}$ has a one-dimensional unstable manifold for ${\mu }_1<2a$ and is repelling for ${\mu }_1>2a$. On ${\mu }_2=0$ they merge in a saddle-node bifurcation; the simultaneous zero and imaginary eigenvalues at $({\mu }_1,{\mu }_2)=(0,0)$ give a fold-Hopf bifurcation.

A steady positive amplitude represents a full periodic orbit, not a full equilibrium. Solving the amplitude equations givesIt exists in the wedge ${\mu }_2<-{\mu }_1^2/4$. The transverse amplitude Jacobian matrix has characteristic polynomial $s^2-{\mu }_{1}s+4r_{\ast }^2$. Its two transverse Floquet exponents have negative real part for ${\mu }_1<0$ and positive real part for ${\mu }_1>0$; the phase direction supplies the neutral Floquet multiplier one.

The wedge boundaries are Hopf bifurcations at ${\mu }_1=-2a$ from $E_{-}$ and ${\mu }_1=2a$ from $E_{+}$. The left branch is supercritical, producing a stable periodic orbit as $E_{-}$ loses stability; the right branch is subcritical in the oscillatory subspace, with an unstable periodic orbit on the side where the transverse complex pair of $E_{+}$ is stable. The real unstable direction of $E_{+}$ persists throughout. There are no full equilibrium points when ${\mu }_2>0$, and $\dot{x}>0$ then excludes a local periodic orbit. The right panel of the bifurcation diagram in part 1(b) shows the primary curves and the perturbed global curve calculated below.