= Solution
The larger bifurcation value is $\mu=0$. For $\mu<0$ the origin is stable and an unstable periodic orbit of radius
$$
r_-=\sqrt{\frac{\lambda-\sqrt{\lambda^2+4\mu}}2}
\sim\sqrt{-\frac{\mu}{\lambda}}
$$
surrounds it. That orbit collapses into the origin as $\mu\uparrow0$, after which the origin is unstable. This is a <subcritical Hopf bifurcation>.
After a positive rescaling of radius, its local normal form is
$$
\boxed{\dot R=\mu R+R^3,\qquad \dot\theta=1},
$$
or equivalently
$$
\dot z=(\mu+i)z+z|z|^2.
$$
The full <quintic radial Hopf equation> also has the stable outer branch
$$
r_+(\mu)
=\sqrt{\frac{\lambda+\sqrt{\lambda^2+4\mu}}2}.
$$
In the $(\mu,r)$ bifurcation diagram, the branch $r=0$ is stable for $\mu<0$ and unstable for $\mu\geq0$. The unstable $r_-$ branch runs from $(0,0)$ backwards to
$$
\left(-\frac{\lambda^2}{4},\sqrt{\frac\lambda2}\right),
$$
where it folds into the stable $r_+$ branch; the latter continues to larger $\mu$.
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