Equilibrium bifurcations of the Lorenz system ($r_H=\sigma (\sigma +b+3)/(\sigma -b-1)$)

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 $\sigma \le b+1$ 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.