= Subcritical Hopf bifurcation
{wiki}
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 form>
$$
\dot r=\mu r+r^3,\qquad \dot\theta=\omega>0.
$$
For $\mu<0$, the nonzero radius is $r_*=\sqrt{-\mu}$. Radii below $r_*$ shrink to zero, whereas radii above it grow. The radial <derivative> at the cycle is $\mu+3r_*^2=-2\mu>0$, 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 $\mu\uparrow0$; 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.
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