= Supercritical Hopf bifurcation
{title2=$\dot r=\mu r-r^3$}
In a supercritical <Hopf bifurcation>, a small <periodic orbit> attracts within the critical <centre manifold> on the side where the <equilibrium point> repels within that manifold. Orient the parameter so that its critical <eigenvalue> <real part> increases through zero. For the exact <Hopf normal form> $\dot z=(\mu+i\omega)z-|z|^2z$, with $\omega>0$, the nonzero constant radius solves
$$
0=\mu r-r^3,\qquad r_*=\sqrt\mu\quad(\mu>0).
$$
For $0<r<r_*$ the radius grows; for $r>r_*$ it shrinks. A radial disturbance $\delta r$ therefore has leading equation $\dot{\delta r}=-2\mu\,\delta r$, since $\partial_r(\mu r-r^3)|_{r_*}=-2\mu<0$. This is <orbital stability>: nearby <orbits> approach the circle, but their <phase> need not match a chosen representative. The exact <angular frequency> is $\omega$, so the period is $2\pi/\omega$ and the <frequency> is $\omega/(2\pi)$.
The <Jacobian matrix> at zero is $\begin{pmatrix}\mu&-\omega\\\omega&\mu\end{pmatrix}$, with <eigenvalues> $\mu\pm i\omega$. Zero attracts for $\mu<0$ and repels for $\mu>0$. Exactly at $\mu=0$, $\dot r=-r^3$ gives
$$
r(t)=\frac{r_0}{\sqrt{1+2r_0^2t}},\qquad t\geq0.
$$
Thus zero still has <asymptotic stability> in this model, with algebraic decay, and there is no nonzero <limit cycle> at the threshold itself. As $\mu\downarrow0$ on the cycle branch, both its radius $\sqrt\mu$ and its radial attraction rate $2\mu$ vanish. Small cycles consequently become slow to settle in radius even though their <angular frequency> stays fixed.
// Target: dynamical-systems.bigb
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