The Lorenz System and the Butterfly Effect

Can the flap of a butterfly's wings in Brazil set off a tornado in Texas? The question sounds like a riddle, but it captures one of the most surprising ideas in modern science: a system governed by simple, exact rules can still be fundamentally unpredictable.

A shortcut that changed everything

In 1961, the meteorologist Edward Lorenz at MIT was running a simple computer model of the weather. Wanting to look at one simulation again, he restarted it from the middle, typing in the numbers from an earlier printout. The printout showed three decimal places, 0.506, while the computer had stored six, 0.506127.

The difference was tiny, about one part in ten thousand. Lorenz expected the new run to follow the old one closely. Instead, after a short while, the two runs had nothing in common. The rounding error had not stayed small. It had grown until it took over completely.

Three equations

In 1963, Lorenz distilled the essence of this behavior into a model of a fluid heated from below, described by just three variables , and :

Roughly speaking, measures how fast the fluid rolls over, while and describe how the temperature varies across it. The constants are usually chosen as , and . There is no randomness anywhere: given the starting values, the future is completely determined.

The butterfly takes shape

Solve these equations on a computer and plot the result, and you get the famous Lorenz attractor. The point loops around one wing, then unpredictably switches to the other, again and again, never exactly repeating itself and never escaping. Fittingly, the shape looks like a butterfly.

t = 0.0

Figure 1. A single trajectory of the Lorenz system in three-dimensional space, starting from . Drag the slider to move through time.

Sensitive dependence on initial conditions

Now start two trajectories from almost the same point, differing in by just . For a while they are indistinguishable. Then, suddenly, they part ways and become completely unrelated.

Figure 2. for two starting points that differ by . The solid and dashed curves overlap at first, then diverge.

On average, a small separation grows exponentially:

where for these parameters is called the Lyapunov exponent. A positive Lyapunov exponent is the mathematical signature of chaos.

Why forecasts have a horizon

Suppose a forecast is useless once the error reaches some size . Solving for the forecast horizon gives

The initial error appears only inside a logarithm. Measuring the starting state a thousand times more precisely buys only extra time units. Better instruments help, but they can never push the horizon to infinity. This is why weather forecasts are reliable for days, not months.

Determinism does not mean predictability. The future can be fully determined by the present and still be impossible to calculate in practice.

The butterfly

In 1972, Lorenz gave a talk with the title “Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?” The phrase stuck, and the “butterfly effect” became a popular name for this sensitivity. Lorenz's honest answer was more subtle than the slogan: the point is not that butterflies cause tornadoes, but that tiny, unmeasurable differences make the long-term weather impossible to pin down.