Lorenz system ($\dot{x}=\sigma (y-x),\dot{y}=x(\rho -z)-y,\dot{z}=xy-\beta z$)

The Lorenz system is a three-variable autonomous differential equation modeling an idealized thermal convection flow. In Edward N. Lorenz's "Deterministic Nonperiodic Flow" (1963), published in the Journal of the Atmospheric Sciences 20, pp. 130–141, numerical solutions of the model illustrated nonrepeating motion and instability to changes in the initial condition. Lorenz used this example to examine obstacles to very-long-range weather forecasting.^[ref]

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.

At $y=0$, the symmetric Jacobian sum is ${\mathrm{diag}}_{\mathrm{block}}([[-2\sigma ,\sigma +r-z],[\sigma +r-z,-2]],-2b)$. Its planar largest eigenvalue is positive exactly under the displayed condition. Thus both $z<\sigma +r-2\sqrt{\sigma }$ and $z>\sigma +r+2\sqrt{\sigma }$ permit instantaneous error growth. The vertical direction initially contracts, but skew couplings can rotate errors later; this local criterion is not a claim of persistent growth along a trajectory. For $\sigma =10,r=28,z=26,b=3$, the eigenvalues are $4,-26,-6$ and the growing direction is $(1,2,0)/\sqrt{5}$.