= Equilibrium bifurcations of the Lorenz system
{c}
{title2=$r_H=\sigma(\sigma+b+3)/(\sigma-b-1)$}
For positive $\sigma,b,r$, the origin of the <Lorenz system> loses stability at $r=1$ and two equilibria $(\pm\sqrt{b(r-1)},\pm\sqrt{b(r-1)},r-1)$ emerge stably. Center-manifold projection gives $\dot q=\sigma(r-1)q/(\sigma+1)-\sigma q^3/[b(\sigma+1)]+\cdots$, proving a supercritical pitchfork. The nonzero-branch characteristic polynomial is $\lambda^3+(\sigma+b+1)\lambda^2+b(\sigma+r)\lambda+2\sigma b(r-1)$. If $\sigma\leq b+1$ these equilibria remain stable for all $r>1$. Otherwise their transverse Hopf crossing occurs at the displayed <parameter>, with frequency squared $b(\sigma+r_H)$. Its cubic Hopf coefficient is positive, so this Hopf is subcritical, producing an unstable small <periodic orbit> on the stable-equilibrium side. A zero <eigenvalue> detects the pitchfork, while the Hopf condition uses the Routh-Hurwitz equality; neither alone determines nonlinear criticality without the corresponding normal-form calculation.
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