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By Edward Norton Lorenz. Journal of the Atmospheric Sciences 20(2), pages 130–141.

Edward Norton Lorenz showed how a deterministic low-dimensional model of convection can produce bounded, nonperiodic evolution with strong sensitivity to its initial state. His 1963 paper became a central example of chaos: the equations determine a trajectory, while finite initial precision can still limit long-time prediction.

The model retains three coupled variables from a simplified convection description rather than attempting to simulate the entire atmosphere. This distinction matters: the result demonstrates a mechanism of unpredictability without making every weather pattern random or every forecast impossible.^[ref]

The Lorenz system is the three-variable autonomous differential equationIt arose as a low-dimensional model of thermal convection. Edward Norton Lorenz used its numerical solutions in 1963 to study nonperiodic motion and sensitivity to the initial condition, with consequences for long-range weather forecasting.^[ref]

The equilibrium points follow directly from the equations. For $\sigma >0$ and $\beta >0$, the first gives $y=x$ and the third gives $z=x^2/\beta$. The second then becomes $x(\rho -1-x^2/\beta )=0$. Thus the origin always exists, while for $\rho >1$ there are two additional equilibriawith matching signs in the first two coordinates. The symmetry $(x,y,z)\mapsto (-x,-y,z)$ exchanges them.

The Equilibrium bifurcations of the Lorenz system determine how their stability changes. The separate instantaneous Lorenz error growth at a zero-y crossing calculation concerns a perturbation at one instant; it should not be confused with the long-term Lyapunov exponent.