Hopf normal form ($\dot{z}=(\mu +i\omega )z+\ell z|z{|}^2+\cdots$)

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 writtenwhere $\mu$ is the critical eigenvalue real part, $a=\mathrm{Re}\ell (0)$, $b=\mathrm{Im}\ell (0)$, 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.^[ref]

In the cubic truncation, $r_{\ast }^2=-\mu /a$ 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 $r_{\ast }^2=-\mu /a+o(|\mu |)$ 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.