= Hopf normal form
{c}
{title2=$\dot z=(\mu+i\omega)z+\ell z|z|^2+\cdots$}
The Hopf normal form is a local <normal form of a dynamical system> on the critical two-dimensional <centre manifold>. After shifting the <equilibrium point> and changing coordinates, its cubic truncation can be written
$$
\dot z=(\mu+i\omega(\mu))z+(a+ib)z|z|^2,\qquad
\dot r=\mu r+ar^3,\qquad
\dot\theta=\omega(\mu)+br^2,
$$
where $\mu$ is the critical <eigenvalue> <real part>, $a=\operatorname{Re}\ell(0)$, $b=\operatorname{Im}\ell(0)$, and $\omega(0)>0$. Smooth parameter dependence of the cubic coefficients and higher-order terms are omitted in this truncation. The real coefficient $a$ controls radial saturation; the imaginary coefficient $b$ changes the <angular frequency> with <amplitude>. This finite-order reduction, rather than literal equality to a cubic equation everywhere, is the meaning of the <Hopf normal form> for a general system.https://www.ma.imperial.ac.uk/~dturaev/bif_theory3.pdf[Dmitry Turaev's bifurcation-theory notes, section 3, parameter-dependent complex normal form]{ref}
In the cubic truncation, $r_*^2=-\mu/a$ whenever $\mu a<0$. Substituting into the angular equation gives $\Omega=\omega(\mu)-b\mu/a$. For the full smooth system, the leading <normal form> radius satisfies $r_*^2=-\mu/a+o(|\mu|)$ and the period tends to $2\pi/\omega(0)$. The exact circle and exactly constant <angular frequency> of the special model $a=-1$, $b=0$, $\omega(\mu)=\omega$ are conveniences of that model. In the original coordinates the small <periodic orbit> need not be a circle. Radial scaling can normalize a nonzero $a$ to $-1$ or $+1$, so $\sqrt{|\mu|}$ is a normalized leading <amplitude>, not a universal physical radius.
\Image[/hopf-criticality-radius-branches.png]
{title=Computed radius branches and radial drift for the two Hopf criticalities}
{description=The exact cubic <Hopf normal forms> $\dot r=\mu r-r^3$ and $\dot r=\mu r+r^3$, with nonzero angular motion understood. The upper <bifurcation diagrams> distinguish attracting solid branches from repelling dashed branches; $r=0$ is an <equilibrium point>, while a nonzero constant radius is a <periodic orbit>. Vertical arrows show radial motion at fixed $\mu$. Below, the sign of $\dot r$ shows why the nonzero roots attract or repel. Stability is within the reduced plane. These are original dimensionless calculations, not measured data; continuation away from zero is a property of the chosen cubic equations, not a global prediction of the local <Hopf bifurcation> theorem.}
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