Chaos and Sensitive Dependence: From Lorenz's Rounded Number to the Butterfly Effect

Put the same numbers into the same equations and you get the same answer. In the 1950s, meteorologists starting to compute the weather on machines were counting on exactly that. Measure the atmosphere more densely, build faster computers, and forecasts should reach as far ahead as you like. What undid that hope was, of all things, typing three fewer digits.

Winter 1961: 0.506 instead of 0.506127

Edward Lorenz, a meteorologist at MIT, was running a twelve-variable model of the atmosphere on a small Royal McBee LGP-30 computer. One day he wanted to look at part of an earlier run in more detail. Instead of starting from the beginning, he took the numbers from the middle of the old printout and typed them back in. The machine carried six decimal places internally, like 0.506127, but the printout showed only three, like 0.506. A difference of one part in a thousand seemed harmless.

He went down the hall for coffee. When he came back about an hour later, the new run had tracked the old one at first, then drifted, and within a few simulated months the two bore no resemblance to each other. As APS News later recounted the episode, the important point was that nothing had malfunctioned. A tiny rounding error had grown steadily as the calculation proceeded until it dominated the solution.

Lorenz boiled the phenomenon down to a much simpler system and published “Deterministic Nonperiodic Flow” in the Journal of the Atmospheric Sciences in 1963. It modelled convection in a fluid layer heated from below with just three variables, x, y and z, and it contains the shape now called the Lorenz attractor. The abstract’s conclusion still reads clearly: in systems whose solutions stay bounded, nonperiodic solutions are ordinarily unstable with respect to small modifications, so slightly differing initial states can evolve into considerably different states. Lorenz drew the consequence for weather directly. If the atmosphere behaves this way, forecasting the sufficiently distant future is impossible unless present conditions are known exactly.

Deterministic but not predictable

In the early nineteenth century Pierre-Simon Laplace imagined an intellect that knew every force and the position of every particle; for it, he wrote, nothing would be uncertain and the future would be as present as the past. Chaos does not dispute the first half of that picture. The Lorenz equations contain no randomness, and the double pendulum obeys plain Newtonian mechanics. Feed in identical starting values and you get identical trajectories every time. What fails is the assumption that anyone can know those starting values well enough.

It helps to separate two properties:

• Determinism: exactly the same initial state gives exactly the same future.
• Predictability: a slightly different initial state gives a slightly different future.

Most everyday calculations have both. Aim a cannon 0.1° off and the shell lands only a little off. A chaotic system has the first property and lacks the second. Since no real measurement is infinitely precise, a deterministic law does not buy you the distant future.

Lorenz was not the first to notice this. Henri Poincaré, studying the three-body problem around 1890, found that orbits could depend with extreme delicacy on their starting conditions, and in Science and Method (1908) he wrote that small differences in initial conditions may produce very great ones in the final phenomena, making prediction impossible. Lorenz’s contribution was to show, with a computer, that this was not an exotic quirk of celestial mechanics. It turned up in a few lines of equations and in a very practical problem, the weather.

How fast is fast? Lyapunov exponents

The Lyapunov exponent λ, named after the Russian mathematician Aleksandr Lyapunov, who studied the stability of motion at the end of the nineteenth century, puts a number on “slightly different becomes very different.” If the distance δ between two nearby trajectories grows on average like δ(t) ≈ δ₀·e^{λt}, a positive λ means the system is sensitive to initial conditions.

For Lorenz’s parameters (σ = 10, ρ = 28, β = 8/3) the largest exponent is about 0.906, so an error doubles every ln 2 ÷ 0.906 ≈ 0.77 units of model time. The important consequence is the time it takes an error to reach some tolerance Δ:

t ≈ (1/λ) · ln(Δ/δ₀)

The forecast horizon grows only with the logarithm of the initial precision. Make your initial error a thousand times smaller and you gain ln(1000)/λ, about 7.6 time units for the Lorenz system. Measured against the cost of building instruments a thousand times better, that is a poor return.

You can check this relationship in the double pendulum simulator. Start two pendulums with 1 m arms from a large angle and shrink the difference between them by factors of 1,000, from 10⁻³ rad down to 10⁻¹² rad; each step delays the visible split by only about three seconds. The effect has been measured on real hardware too: in 1992 Shinbrot and colleagues reported that uncertainties in a physical double pendulum grew with an exponent of λ = 7.5 ± 1.5 per second.

The weather shows the same structure. Zhang and colleagues (2019) ran high-resolution global models with almost no initial error and estimated the ultimate limit of midlatitude day-to-day predictability at about two weeks. Today’s practical skill is about 9 to 10 days; reducing initial errors by an order of magnitude could add a few more days, but improving the observations cannot push far past that limit.

Motion that stops can still be unpredictable: fractal boundaries

Chaos usually brings to mind something that moves erratically forever. But a system that is certain to come to rest can still leave you unable to say where.

The magnetic pendulum is the standard example. Hang an iron bob over three magnets and friction guarantees it will eventually stop over one of them. Colour each starting point by the magnet it ends on and you get broad patches of solid colour separated by bands where the three colours are finely interleaved. Zoom into a band and you find narrower bands inside it. These are fractal basin boundaries.

McDonald, Grebogi, Ott and Yorke (1985) worked out what such boundaries mean for prediction. If you know the starting point to within ε, the probability of predicting the wrong outcome falls like ε^α, and for a fractal boundary α is less than 1. Measuring ten times more precisely does not cut your error rate tenfold. This has nothing to do with running the calculation for a long time: whether the pendulum takes 30 seconds or three minutes to stop, if you do not know the start precisely enough, you do not know the answer.

Where “the butterfly effect” came from

Lorenz’s 1963 paper says nothing about butterflies. The phrase comes from the title of a talk he gave on 29 December 1972 at the 139th meeting of the American Association for the Advancement of Science in Washington, D.C.: “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” Lorenz later explained that when he failed to supply a title in time, the session convener, the meteorologist Philip Merilees, made one up. That the Lorenz attractor looks a little like a pair of wings is a coincidence.

The talk itself was more careful than the slogan it spawned. Lorenz said that, lest he appear frivolous for even posing the question, he would first offer two propositions. First, if a single flap of a butterfly’s wings could help generate a tornado, so could all the flaps before and after it, the flaps of millions of other butterflies, and the activity of countless other creatures, including people; and a flap could just as well help prevent a tornado. Second, such tiny disturbances do not increase or decrease the frequency of events like tornadoes; at most they may change the order in which they occur. He also admitted that whether the atmosphere was really unstable in this way could not yet be settled.

So reading the butterfly effect as “the butterfly caused the tornado” gets it backwards. Lorenz’s point was that countless small differences are mixed together, too many to single out a cause, and there is no way to know in advance which of them will decide the outcome.

Four common misunderstandings

Chaos is randomness. It is not. Chaos arises in systems whose rules are completely fixed. The particles in the Lorenz attractor do not wander anywhere they like; they collect on a thin two-winged structure. Individual paths are unpredictable, yet the overall shape is the same every time.

Small changes always have big effects. They do not. A double pendulum swinging through small angles keeps its errors small. Lower ρ in the Lorenz equations below about 24.74 (more precisely, below about 24.06) and every trajectory eventually settles onto a fixed point. Sensitive dependence is a property that appears under particular conditions.

If you can’t forecast the weather, you can’t project the climate. A particular day’s weather and the statistics of many years are different questions. In the Lorenz attractor the position of any one particle soon becomes unknowable, but statistics such as the fraction of time spent on each wing are stable. Climate projection works with statistics of that kind. Lorenz’s remark in 1972, that small disturbances change the order of events but not their frequency, makes the same distinction.

Better computers will fix it. Because the forecast horizon grows only with the logarithm of the initial error, huge gains in computing power and measurement precision buy relatively little time. That is why operational forecasting moved from a single run to ensemble forecasting: dozens of runs from slightly different starting points, with the spread of the results used to judge how much to trust the forecast. It is the same idea as running a hundred pendulums at once in the double pendulum simulator.

Try it yourself

• In the double pendulum, time the split as you change the initial offset from 10⁻³ to 10⁻¹² rad, then repeat at small angles (10°) and compare.
• In the Lorenz attractor, set ρ to 28, 24.5, 20 and 0.5 to watch the chaos switch on and off.
• In the magnetic pendulum, lower friction from 0.4 to 0.05 to see the borders spread, and tap two neighbouring points on a border to see where their paths part ways.