Lorenz Attractor (2D)

The three equations

Edward Lorenz (1963) reduced thermal convection in a fluid layer heated from below to three coupled ordinary differential equations:

x is proportional to the convection speed, y to the temperature difference between rising and sinking fluid, z to the deviation of the vertical temperature profile from linear. σ is the Prandtl number, ρ the Rayleigh number relative to its critical value and β a geometry factor. For the classic values (10, 28, 8/3) the solution never settles and never repeats: it winds around two unstable fixed points C_± = (±√(β(ρ−1)), ±√(β(ρ−1)), ρ−1) on the "butterfly" attractor.

Sensitive dependence

Two copies started 10^-6 apart separate, on average, like e^λt with the largest Lyapunov exponent λ ≈ 0.906. The page measures λ with the Benettin method: a shadow trajectory 10^-8 away is renormalised every 0.5 time units and the mean log growth is accumulated. A prediction horizon of about ln(1/δ₀)/λ follows: each factor of 10 gained in the initial accuracy buys only ln(10)/λ ≈ 2.5 time units.

The volume of any cloud of states contracts as e^{−(σ+1+β)t}, so the attractor has zero volume (its dimension is about 2.06).

The Lorenz map

Plotting each local maximum of z against the next one gives an almost one-dimensional curve with slope greater than 1 in magnitude: stretching and folding, the mechanism of chaos.

Honest simplifications

• This is a 3-mode truncation of the Navier-Stokes-Boussinesq equations, not a weather model; for ρ > 1 the real convection has many more modes.
• The numerical integrator is fourth-order Runge-Kutta with a fixed step; individual trajectories become wrong after the predictability horizon (the statistics stay right). This is the same sensitivity the page demonstrates.
• The finite-time Lyapunov estimate fluctuates by a few per cent for a few hundred time units.
• The regime test integrates one orbit from (1,1,1): between ρ = 24.06 and 24.74 chaos and steady states can coexist, so the check accepts both there.

• Run Classic chaos and read the measured Lyapunov exponent as it approaches 0.906.
• Reduce the initial offset to 1e-10: the two copies stay together longer, but only a few time units longer.
• Run Steady convection: the orbit spirals into a fixed point C₊ or C_- and the Lyapunov exponent turns negative.
• Run Just below chaos (rho = 22): long chaotic transients then settling (the point of no return is not sharp).
• Switch the view to x-y or y-z to see the same attractor from another side; open the Lorenz map graph to see the stretching and folding.