= Solution
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> are
$$
E_+:\quad 2a,\ \mu_1-2a\pm i,\qquad
E_-:\quad -2a,\ \mu_1+2a\pm i.
$$
Thus $E_-$ 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 gives
$$
\boxed{x_*=\mu_1/2,\qquad r_*^2=-\mu_2-\mu_1^2/4,\qquad \text{period}=2\pi.}
$$
It exists in the wedge $\mu_2<-\mu_1^2/4$. The transverse amplitude <Jacobian matrix> has characteristic polynomial $s^2-\mu_1s+4r_*^2$. 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.
Back to article page
