= Solution
At a <stationary bifurcation>, a real <eigenvalue> of the equilibrium linearization crosses zero; the critical motion has zero frequency. At a <Hopf bifurcation>, a conjugate pair crosses the imaginary axis at $\pm i\omega$ with $\omega>0$, permitting a nearby <limit cycle>.
The <Hopf normal form> has a small cycle when $R^2=\mu/a+O(\mu^2)>0$. Its radial multiplier exponent is $\partial_r\dot r|_R=-2\mu+O(\mu^2)$, while the origin's radial exponent is $\mu$. Thus: for $\mu<0<a$, there is an attracting spiral and no small cycle; for $0<\mu,a$, a repelling spiral is surrounded by an attracting cycle; for $\mu,a<0$, an attracting spiral is surrounded by a repelling cycle; for $a<0<\mu$, there is a repelling spiral and no small cycle. Outside the repelling cycle in the third case the local radial flow points outward; a cubic normal form makes no global prediction about where it goes.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-2-hopf.png]
{title=Four local Hopf phase portraits and the unstable subcritical radius branch}
The <subcritical bifurcation> has $a<0$ and the unstable radius branch $R=\sqrt{\mu/a}+O(|\mu|^{3/2})$ on $\mu<0$, meeting the origin at $\mu=0$. On either existing branch, the angular speed is $\omega+(c-b/a)\mu+O(\mu^2)$, so
$$
T=\frac{2\pi}{\omega+(c-b/a)\mu+O(\mu^2)},\qquad \boxed{\frac{dT}{d\mu}=-\frac{2\pi}{\omega^2}\left(c-\frac ba\right)+O(\mu).}
$$
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