Chaos Theory and the Butterfly Effect: Why Weather Defies Forecasts

Chaos theory studies systems that follow exact rules yet still behave in ways we cannot predict far ahead. The best-known idea from it is the butterfly effect: a tiny change now can grow into a huge difference later. At the heart of the story is a shape called the Lorenz attractor. This post links to our explainer Why Does Time Only Go Forward?, because chaos is one of the reasons the future is so hard to know even when the laws are known.

What is chaos in physics?

In everyday speech, chaos means a mess. In physics it means something precise. The Stanford Encyclopedia of Philosophy describes the core property as “sensitive dependence”: the smallest change in a system’s starting state can lead to very large differences in its later behaviour (Bishop, SEP “Chaos”^[1]). A chaotic system is deterministic. There is no dice-rolling in the equations. The trouble is that we can never know the starting state perfectly, and in a chaotic system the error does not stay small.

Edward Lorenz and the equations behind the butterfly effect

Edward Lorenz was an MIT meteorologist. In the early 1960s he realised that small differences in a model of the atmosphere could trigger large and unexpected results, and he was the first to recognise what is now called chaotic behaviour in weather modelling (MIT News, 2008^[2]). In 1963 he published “Deterministic Nonperiodic Flow” in the Journal of the Atmospheric Sciences (Lorenz, 1963^[3]). The paper boiled a model of convection (warm fluid rising, cool fluid sinking) down to three equations:

Ẋ = −σX + σY, Ẏ = −XZ + rX − Y, Ż = XY − bZ

The dot means “rate of change with time”. X measures the strength of the convective motion, while Y and Z measure the temperature pattern. All three are scaled, dimensionless numbers, as is time in the model. The constants are σ (the Prandtl number, a property of the fluid), r (how strongly the fluid is heated, relative to the value at which convection starts) and b (set by the shape of the convection cells). Lorenz ran the numbers with σ = 10, b = 8/3 and r = 28 on a Royal McBee LGP-30 electronic computer (Lorenz, 1963, scanned reprint^[4]).

What Lorenz found

For those values the solutions never settle down and never exactly repeat. They loop around two lobes, switching from one to the other at irregular moments. This is the butterfly-shaped figure, later named a strange attractor. With the paper’s constants, the steady convection solutions become unstable once r passes 470/19 ≈ 24.74, so r = 28 lies in the unstable region. Volumes in the model’s state space also shrink at a constant rate σ + b + 1 ≈ 13.67 per unit of model time, so every starting point is pulled onto the thin attractor (Lorenz, 1963^[4]).

How fast do small errors grow?

The standard way to measure sensitive dependence is the Lyapunov exponent λ. If two starts differ by a tiny distance |δ(0)|, then for a while their separation grows roughly as (Bishop, SEP “Chaos”^[1]):

|δ(t)| ≈ |δ(0)| e^(λt)

Here |δ(t)| is the separation at time t, and λ has units of inverse time (for example per day). If λ is positive, errors grow exponentially. Our worked example, with an illustrative λ of 1 per day: a starting error of one part in a million grows to the size of the whole signal in ln(10⁶) ≈ 13.8 days. Make the measurement ten times better and you gain only ln(10) ≈ 2.3 days. That is why better instruments help forecasts so slowly. The same article stresses that the growth cannot go on forever: in a bounded system the separation stops growing once it reaches the size of the attractor.

MIT’s obituary of Lorenz states the practical result for meteorology: it may be fundamentally impossible to predict weather beyond two or three weeks with a reasonable degree of accuracy (MIT News, 2008^[2]).

Where does the butterfly come from?

The name grew out of a talk Lorenz gave in 1972 called “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” (MIT News, 2008^[2]). The butterfly is a picture, not a claim that insects steer storms. The point is that in a chaotic system no change is too small to matter eventually. Lorenz received the Crafoord Prize in 1983, shared with Henry Stommel, and the Kyoto Prize in 1991.

Chaos is not randomness, and weather is not climate

The Nobel committee’s 2021 popular background explains that the equations governing air movement are entirely deterministic, yet weather is chaotic, “where small deviations in initial values result in huge differences at a later stage” (NobelPrize.org, Physics 2021 popular background^[5]). That year’s prize honoured Syukuro Manabe and Klaus Hasselmann for climate modelling, and Giorgio Parisi for hidden patterns in disordered systems, all for “groundbreaking contributions to our understanding of complex physical systems” (NobelPrize.org, Physics 2021^[6]). Climate is the long-term statistics of weather, and statistics can be reliable even when the day-to-day path is not. The same idea, that averages over huge numbers of possibilities behave well, is at the heart of entropy and Boltzmann’s formula.

Chaos you can see on a desk

You do not need an atmosphere for chaos. A double pendulum, one pendulum hanging from the end of another, is enough. Release it twice from almost the same large angle, high enough for the arms to flip, and within seconds the two swings look nothing alike. A Poincaré section is one of the tools used to study such motion, and our chaos and waves designs live in the Waves, Chaos & Matter collection. Two more faces of chaos: the Arnold’s Cat – Cat Map Chaos Physics T-Shirt, a map that scrambles a picture by stretching and folding, and the Duffing’s Strange Attractor – Plate 082 Physics Poster, a driven oscillator with its own strange attractor.

Quick FAQ

Is chaos theory just about weather?

No. Weather was where Lorenz found it, but sensitive dependence shows up in pendulums, fluids, orbits and many other systems.

Does chaos mean physics cannot predict anything?

No. Chaotic systems are predictable for a while, and their long-run statistics can be predictable even when single paths are not.

Can a butterfly really cause a tornado?

That is a metaphor from the title of Lorenz’s 1972 talk. It means tiny differences can grow, not that any one butterfly can be blamed for any one storm.

References

1. R. Bishop, “Chaos,” Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/chaos/
2. MIT News (16 April 2008), “Edward Lorenz, father of chaos theory and butterfly effect, dies at 90”. https://news.mit.edu/2008/obit-lorenz-0416
3. E. N. Lorenz (1963), “Deterministic Nonperiodic Flow,” Journal of the Atmospheric Sciences 20, 130–141 (DOI 10.1175/1520-0469(1963)020&lt;0130:DNF&gt;2.0.CO;2; metadata verified via Crossref). https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
4. E. N. Lorenz (1963), “Deterministic Nonperiodic Flow” — scanned reprint hosted by the University of Vermont (equations 25–27 and σ = 10, b = 8/3, r = 28 read from the scan). https://cdanfort.w3.uvm.edu/research/lorenz-1963.pdf
5. NobelPrize.org, Popular science background: “They found hidden patterns in the climate and in other complex phenomena” (2021). https://www.nobelprize.org/prizes/physics/2021/popular-information/
6. NobelPrize.org, “The Nobel Prize in Physics 2021” (S. Manabe, K. Hasselmann & G. Parisi). https://www.nobelprize.org/prizes/physics/2021/summary/

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.