= Hopf bifurcation
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A Hopf bifurcation creates or destroys a small <periodic orbit> near an <equilibrium point> as a pair of <complex conjugate> <eigenvalues> crosses the imaginary axis. It explains how a resting state can give way to sustained <oscillations>: the <linearization of a dynamical system> supplies rotation and growth, while a nonlinear term can saturate that growth at a nonzero <amplitude>.
The simplest <supercritical Hopf bifurcation> is the exact <Hopf normal form>
$$
\dot z=(\mu+i\omega)z-|z|^2z,\qquad z=x+iy,\qquad \omega>0.
$$
Here $z$ is a <complex number>, $\mu$ is the <bifurcation parameter>, and $\omega$ is the <angular frequency>. In <polar coordinates> $z=re^{i\theta}$, the equations become $\dot r=\mu r-r^3$ and $\dot\theta=\omega$ for $r>0$. At $\mu<0$, each turn shrinks towards the <equilibrium point>. At $\mu>0$, small radii grow and large radii shrink towards $r=\sqrt\mu$, producing a <limit cycle>. For example, $\mu=0.04$ and $\omega=2$ give radius $0.2$, period $\pi$, and <frequency> $1/\pi$ cycles per unit <time>. The limiting <angular frequency> stays nonzero while the <amplitude> tends to zero: a tiny newborn <oscillation> can still repeat at a finite rate.
\Image[/hopf-crossing-phase-portraits.png]
{title=Computed phase portraits across a supercritical Hopf bifurcation}
{description=Exact <orbits> of the displayed <Hopf normal form> with $\omega=1$, at $\mu=-0.2,0,0.2$. Arrows follow increasing <time>; the two starting radii are $0.82$ and $0.13$. The green circle at $\mu=0.2$ has radius $\sqrt{0.2}$. At $\mu=0$, nonlinear decay still attracts <orbits>, despite purely imaginary linear <eigenvalues>. These are dimensionless model calculations, not measured data.}
For a precise generic statement, consider a <smooth function> $f$ defining an <autonomous differential equation> $\dot x=f(x,\alpha)$ on $\mathbb R^n$, with a smooth branch of <equilibrium points> $x_*(\alpha)$ and critical parameter $\alpha=0$. Require:
* the <Jacobian matrix> $D_xf(x_*(\alpha),\alpha)$ to have a simple pair of <eigenvalues> $\beta(\alpha)\pm i\omega(\alpha)$, with $\beta(0)=0$ and $\omega(0)>0$;
* every other <eigenvalue> to have nonzero <real part> at $\alpha=0$;
* a transverse crossing, $\beta'(0)\ne0$;
* a nonzero cubic radial coefficient $a=\operatorname{Re}\ell(0)\ne0$ in the reduced <Hopf normal form>.
Then a locally unique branch of small <periodic orbits>, up to <phase>, emerges on the side $\beta a<0$. On the two-dimensional <centre manifold>, the branch attracts for $a<0$ and repels for $a>0$. Attraction in the full <phase space> additionally requires all remaining <eigenvalues> to have negative <real part>; any transverse unstable direction makes the <periodic orbit> unstable there.https://www.scholarpedia.org/article/Andronov-Hopf_bifurcation[Yuri A. Kuznetsov's authored account of the generic Hopf theorem]{ref}
An <eigenvalue-crossing bifurcation test> alone cannot establish these nonlinear conclusions. For example, the purely linear equation $\dot z=(\mu+i\omega)z$ has the same crossing but no isolated <periodic orbit>: at $\mu=0$ every nonzero radius is constant, and at $\mu\ne0$ none is. If the cubic coefficient vanishes, higher-order terms must be examined; a <generalized Hopf bifurcation> is one possible degeneracy. The usefulness of the generic theorem is that a large system can have its small oscillatory onset understood through one <Hopf normal form>, subject to these hypotheses and local limits.
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