Quintic radial Hopf equation
= Quintic radial Hopf equation
For
$$
\dot r=r(\mu+\lambda r^2-r^4),\qquad \lambda>0,
$$
nonzero periodic orbits have
$$
r_\pm^2=\frac{\lambda\pm\sqrt{\lambda^2+4\mu}}2.
$$
They are born together at $\mu=-\lambda^2/4$; the inner unstable orbit then vanishes in a subcritical Hopf bifurcation at $\mu=0$, while the outer orbit is stable.