Edward Lorenz, 1963 — three equations, the first clean specimen of chaos
Cited by: Topic 8 Chaos
01The Question It Poses
The consensus in the early 1960s ran roughly like this: a dissipative system — that is, anything real, anything that grinds energy away — eventually settles down. Either it comes to rest at some state (a fixed point) or it falls into an endlessly repeating cycle (a periodic orbit). There was no third fate.
Lorenz's own project was more mundane. He wanted to know whether the statistical methods then fashionable for long-range forecasting actually worked. For that he needed an example small enough to run on a desktop machine yet still carrying the nonlinear essence of the atmosphere. He took a set of equations from Barry Saltzman describing thermal convection — heat below, cold above, and the fluid starts turning over by itself — and threw away nearly all of it, keeping three.
The question then became: can a system this deterministic, with only three variables and with dissipation, manage to neither stop nor repeat?
02The Rules
dx/dt = σ (y − x) dy/dt = x (ρ − z) − y dz/dt = x y − β z
x ≈ the intensity of the convective overturning; its sign says which way the roll turns.
y ≈ the temperature difference between the rising current and the sinking one.
z ≈ how far the vertical temperature profile departs from a uniform top-to-bottom gradient.
σ is the ratio of viscosity to thermal diffusion (the Prandtl number), ρ measures how hard the bottom is being heated (a relative Rayleigh number), and β is a geometric constant set by the shape of the container.
Lorenz's numbers: σ = 10, ρ = 28, β = 8/3. The whole reputation of this system rests on that one triple — change it and the story changes completely (see the last section).
The nonlinearity hides in exactly two places: the products xz and xy. Nothing random appears anywhere; give it the same initial condition and it returns the same trajectory forever.
The equations are what a layer of heated fluid looks like once compressed into three numbers. They keep the nonlinearity and discard almost everything else.
03What You See When It Runs
The trajectory circles one centre a few times, abruptly jumps to the other side, circles there, and jumps back. Whether it jumps follows no rule: the loops before a jump look no different from the loops before no jump.
Draw that trajectory in the space whose coordinates are x, y and z (phase space → reference page) and you get the famous butterfly.
This is not one curve filling in a surface: zoom in and it is a fractal set of infinitely many stacked sheets, of dimension about 2.06.
Three things worth keeping:
One: it contracts and expands at the same time. The system dissipates, so a small blob of volume in phase space is squeezed flatter and flatter until its volume is zero; yet trajectories are sensitive to initial conditions, so points within the blob are pulled apart. Squeeze, stretch, fold back, repeated without end — that is what produces the fractal structure, and where the "strange" in strange attractor comes from.
Two: errors double about every 0.77 time units. At these parameters the largest Lyapunov exponent is λ ≈ 0.906, and a positive value is what chaos means.
Three: the sequence of lobe switches does not repeat. Write 0 for "on the left lobe" and 1 for "on the right," and the resulting binary string has no period.
Every crossing of the centre line is the convection reversing. When it will happen cannot be seen in advance.
04What It Explains
Strictly, it explains no particular phenomenon — it overturned a belief. Before 1963, "deterministic rules plus dissipation ⇒ eventual quiescence" counted as common sense; Lorenz produced three lines of equations as a counterexample. Nonperiodicity needs no external source of randomness; a system can keep failing to repeat all on its own.
The two concepts it left behind have travelled further than the equations: the strange attractor (erratic motion has a shape) and the predictability horizon (how long you can forecast is a computable quantity, not a matter of attitude). Ensemble forecasting works the way it does because of this lineage.
It also does not live only on paper. Willem Malkus at MIT built a chaotic waterwheel: a wheel hung with leaky buckets, water poured in from above. A bucket fills and drags its side down, water drains away, and the wheel turns one way, then the other, switching without pattern. Its equations of motion are the Lorenz equations. The single-mode laser (Haken, 1975) turns out to be isomorphic to them as well.
What It Cannot Explain
It is not a weather model. Three variables, with moisture, terrain, radiation and the Earth's rotation all thrown away — nearly everything, in other words. It can show why weather is hard to forecast; it cannot compute any actual episode of convection. Lorenz himself was clearer on this than most popular accounts.
Away from those parameters it is not chaotic. For ρ < 1 the system sits quietly at a fixed point (no convection at all). For 1 < ρ < 24.74 it settles into one of two steady convecting states — entirely predictable. Chaos appears only in a range above that, and 28 is a number Lorenz chose. "The Lorenz system is chaotic" is a sentence with "at these parameters" left out, and wherever it is used to make a point, that clause has to be put back.
It cannot predict any individual switch. What it offers is statistics: the fraction of time spent on each lobe, the distribution of residence times. Which loop will jump is a question the model explicitly declines.
Equations being isomorphic is not systems being isomorphic. The waterwheel and the laser really do obey the same equations, but only because both were abstracted onto the same skeleton. Seeing that some real system "looks a bit like a butterfly" and calling it a Lorenz system is treating analogy as derivation — to claim that, you must map variables and parameters one by one and produce a falsifiable prediction.
Even the butterfly effect itself is unsettled for the real atmosphere. In the model, arbitrarily small perturbations grow; but the real atmosphere has finite-scale effects, and whether a genuinely tiny disturbance can propagate all the way up to weather scales remains debated (what Palmer and colleagues call the "real butterfly effect" is not the classic picture). The model supplies a mechanism, not a cheque already cashed.