Lorenz Attractor Explorer

In March 1963 the Journal of the Atmospheric Sciences printed twelve pages titled “Deterministic Nonperiodic Flow.” They hold three equations, two tables from a Royal McBee LGP-30 computer, and an argument about forecasting that still stands.

Edward Lorenz, a meteorologist at MIT, was not trying to draw a butterfly. He wanted the smallest set of equations that still behaved like a real fluid, one that never settles down and never repeats. He found it in a model of convection. Heat a layer of fluid from below and it turns over in rolls, warm fluid rising and cool fluid sinking. Barry Saltzman had reduced that problem to a set of ordinary differential equations and solved them by machine, and in some of his runs all but three of the variables faded to zero while the remaining three went on fluctuating without any pattern. Lorenz kept those three and threw the rest away.

Three variables and three numbers

The system is short enough to say aloud. The rate of change of x is sigma times (y - x). The rate of change of y is x times (rho - z), less y. The rate of change of z is x times y, less beta times z. In Lorenz’s words x is proportional to the intensity of the convective motion, y to the temperature difference between the rising and the sinking currents, and z to the distortion of the vertical temperature profile from a straight line.

The three constants carry the physics. Sigma is the Prandtl number of the fluid. Rho is the Rayleigh number divided by its critical value, which makes it a measure of how hard the layer is heated compared with the heating at which it first begins to turn over. Beta is a geometric factor fixed by the proportions of the rolls. Lorenz followed Saltzman in taking sigma as 10 and beta as 8/3, and then picked 28 for rho. Those values, 10, 28, 8/3, have been copied into nearly every picture of the attractor since.

He was blunt about what the model could not do. The equations may give realistic results when the heating is only slightly above the critical value, he wrote, but they cannot be expected to resemble strong convection, because so much was cut away to reach them. Warwick Tucker, decades later, called them a crude model of atmospheric dynamics. Nobody should read weather off this system. Its value lies elsewhere.

From rest to chaos

Turn the heating up slowly and the system passes through distinct regimes. With rho below 1 the only steady state is the origin, where nothing moves, and it is stable: every start decays to rest and the heat simply conducts. At rho equal to 1 that state loses its stability and two new ones appear, at x and y both equal to plus or minus the square root of beta times (rho - 1), with z equal to rho - 1. They are steady rolls, turning one way or the other.

Those two states do not stay stable either. Lorenz worked out the characteristic equation at each and found that a pair of complex roots crosses into growth when rho reaches sigma (sigma + beta + 3) / (sigma - beta - 1). The formula has a condition attached: sigma must exceed beta + 1, or steady convection stays stable at every rho. For 10 and 8/3 the critical value is 470/19, which is 24.74. Lorenz’s 28 sits just above it. At 28 the steady rolls lie at about (8.49, 8.49, 27) and its mirror image, both unstable, and the origin is unstable too. An orbit has nowhere to come to rest.

It cannot escape either. The divergence of the flow is the same at every point, minus (sigma + beta + 1), which is -41/3 for the classic values, so any blob of starting points loses volume steadily. Lorenz put a number on it: one circuit around a steady state takes roughly 0.7 time units, and in that time a volume shrinks to 0.00007 of what it was. Orbits are bounded, they contract onto a set with no volume at all, and they still never repeat. Lorenz pictured the result as an infinite complex of surfaces, each extremely close to one of two sheets that appear to merge.

Chaos is not the whole story at larger rho. In some windows the motion locks into a repeating loop; at 99.96, for instance, the orbit closes into a knot. There the measured exponent drops to zero.

Measuring the butterfly

The paper’s abstract states the key property plainly: slightly differing initial states can evolve into considerably different states. The Lyapunov exponent turns that sentence into a number. It is the mean rate at which two nearby orbits separate, counted as an exponent, so that a positive value means the gap multiplies by a fixed factor in each unit of time. A positive largest exponent is the usual test for chaos.

For the classic values the physicist J. C. Sprott computed it with a fourth-order Runge-Kutta method at a step of 0.001, over more than a billion steps. His three exponents are 0.906, 0 and -14.572. One of them must be zero for any flow, and the three must add up to the divergence, which they nearly do.

What does 0.906 buy? A gap grows by a factor of e in about 1.1 time units and by a factor of ten in about 2.5 time units. Turn that around and it becomes the central fact of forecasting a chaotic system: measuring the start ten times more accurately extends the forecast by only 2.5 time units, between three and four loops of the orbit. A hundred times better buys twice that. Effort grows by multiplication while the reward grows by addition.

Here is the experiment as the tool on this page runs it by default. Start one orbit at (1, 1, 1) and a second at x larger by 1e-8, with a step of 0.005. The distance between them passes 1 at t = 27.73. Steady growth at 0.906 would have covered those eight decades in 20.3 time units, so this pair is slow. The log plot shows why: for its first stretch the orbit spirals gently outward from one steady state, where a gap grows only at 0.094, the real part of that point’s eigenvalues. The exponent is an average over the whole attractor, and any single stretch of orbit can run well above or below it.

That is why the two ways of estimating it differ so much in quality. The quick way fits a straight line to the logarithm of the distance while the orbits are still close. A worked example published with the Chebfun software does exactly this over 25 time units, gets 0.934, and remarks that averaging over many trajectories would do better. On the default run here the fitted slope is 0.634.

The careful way is the procedure Sprott describes: advance both orbits one step, record the logarithm of the ratio by which the gap changed, pull the second orbit back to its original distance of 1e-8 along the same line, and average over a long run. He warns that a flow typically needs millions of steps to do better than two significant digits. A million steps here, from eight different starts, gave between 0.902 and 0.907.

The orbit is wrong and the shape is right

The integrator errs too, and the equations cannot tell a rounding error from a second starting point. The classical Runge-Kutta formula, the best known member of its family, keeps each error tiny: the error of a single step shrinks with the fifth power of the step. Chaos multiplies it anyway.

The check is cheap. Integrate the same start twice, once with step h and once with h / 2, and watch the two copies. With h = 0.005 from (1, 1, 1) they pass a distance of 1 at t = 25.27. Beyond that time the curve on the screen is not the solution that starts at (1, 1, 1); it is some other path. A smaller step delays the moment and never removes it.

So why believe the picture at all? For about thirty-five years the honest answer was that nobody had proved it. Lorenz wrote that his conclusions rested on a finite stretch of a numerically determined solution and could not be regarded as mathematically proven. Steve Smale included the question in his list of challenging problems for mathematics. Tucker settled it in 1999 with a proof built from normal form theory and rigorous numerical computations: the Lorenz equations do support a strange attractor, and it persists under small changes to the coefficients. The individual path is unknowable. The object it wanders over is real, and its statistics, the exponent among them, come out the same from one start as from another.

Lorenz found a second way to see that order. He recorded each maximum of z and plotted it against the maximum before it. The points fall on a thin curve with a sharp peak, near a height of 38.5 in his run, and the curve is steeper than 1 everywhere. One maximum therefore fixes the next, and a small difference between two maxima is larger at the following pass, which is the butterfly effect again in one dimension.

What it says about weather

Lorenz drew his conclusion with care. If the atmosphere behaves like his system, he wrote, then “prediction of the sufficiently distant future is impossible by any method, unless the present conditions are known exactly.” He then declined to say how distant. “Our results do not give the answer for the atmosphere; conceivably it could be a few days or a few centuries.”

The butterfly came later, from the title of a paper he presented in 1972: “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” MIT’s account of his work gives the modern reading, that weather may be fundamentally impossible to predict beyond two or three weeks with reasonable accuracy. A 2019 study with high-resolution global models put today’s skillful lead time for midlatitude weather at about 10 days, and found that cutting the uncertainty in the starting state by a factor of ten would add up to 5 days. That is the same bargain as the 2.5 time units.

Forecasters have built that knowledge into their method. The European Centre for Medium-Range Weather Forecasts has issued ensemble forecasts since 1992, and now runs 51 at a time: one from the best available starting state and 50 from slightly perturbed ones. The spread between them shows how far ahead the forecast can be relied on.

The limits of this page should be as plain as its results. Three variables are not an atmosphere, and the time in these equations is dimensionless, so 0.906 cannot be converted into days of weather. The estimate here belongs to these equations, at these values, over a finite run, computed in arithmetic that is itself part of the experiment. What carries over is the form of the answer: a rate at which ignorance grows, and a shape that survives it.