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The larger bifurcation value is $\mu =0$. For $\mu <0$ the origin is stable and an unstable periodic orbit of radiussurrounds it. That orbit collapses into the origin as $\mu \uparrow 0$, after which the origin is unstable. This is a subcritical Hopf bifurcation.

After a positive rescaling of radius, its local normal form isor equivalentlyThe full quintic radial Hopf equation also has the stable outer branchIn the $(\mu ,r)$ bifurcation diagram, the branch $r=0$ is stable for $\mu <0$ and unstable for $\mu \ge 0$. The unstable $r_{-}$ branch runs from $(0,0)$ backwards towhere it folds into the stable $r_{+}$ branch; the latter continues to larger $\mu$.

In a subcritical Hopf bifurcation, the small periodic orbit repels within the critical centre manifold and lies on the side where the equilibrium point attracts within that manifold. With the critical eigenvalue real part increasing through zero, the positive cubic sign gives the exact radial Hopf normal formFor $\mu <0$, the nonzero radius is $r_{\ast }=\sqrt{-\mu }$. Radii below $r_{\ast }$ shrink to zero, whereas radii above it grow. The radial derivative at the cycle is $\mu +3r_{\ast }^2=-2\mu >0$, proving that the periodic orbit repels. For example, $\mu =-0.04$ gives an unstable circle of radius $0.2$: a radius of $0.1$ decays while one of $0.3$ grows. The unstable circle shrinks into zero as $\mu \uparrow 0$; at $\mu =0$, $\dot{r}=r^3$ already drives every small positive radius outward.

For $\mu >0$ there is no nonzero constant radius in this cubic model, and zero repels. In a general dynamical system, this says only that the nearby orbits leave the local neighbourhood. It does not guarantee a distant attracting limit cycle, a jump to a particular amplitude, or hysteresis. Additional nonlinear or global information is needed; the quintic radial Hopf equation illustrates one possible stabilizing extension. Reversing the parameter orientation reverses which numerical side contains the branch, so the defining distinction between supercritical Hopf bifurcation and subcritical Hopf bifurcation is their reduced stability and their relation to the equilibrium point, rather than the bare sign of an arbitrarily named parameter.