Skip to the content
Chaos, Drawn

Chaos theory, drawnA small nudge,
a whole new world

Chaos is not disorder, and it is not hard. It is what a plain, exact rule does when a tiny change in where you start turns into a large change in where you end up. Weather does it. A dripping tap does it. A herd, a market, a creek in spring. This page draws every bit of it from the equations — and then hands you the two tools that let you stand in the middle of it and still get on with your life.

0.92 /show fast tiny errors grow in the weather model
4.669the number that shows up in every road to chaos
3.499average of 3,000 die rolls — the law of large numbers at work
6.92→0.61a shock, before and after one feedback loop

The whole idea in four sentences

A chaotic system follows a fixed rule with no dice in it, so in principle its future is set. But two starts too close to tell apart drift apart fast, so in practice you cannot see far ahead. That is the catch, and it is the whole of it. The rest of this site is what the catch looks like, why it happens, and what to do about it.

Play with it → turn the knobs yourself

Walk it in order

1 · The one idea2 · One rule into chaos3 · Weather & the butterfly4 · Tool one: large numbers5 · Tool two: build to bendPlay · turn the knobs6 · Do it yourself
2.62.833.23.43.63.8400.20.40.60.81growth rate rvalues it settles onThe logistic map, every r at oncechaos beginsperiod 2
One equation, every setting at once. Turn the knob left to right and a single steady value splits to two, to four, and then into a fog — that fog is chaos. · computed here · Nan · hongdam.net · CC BY 4.0
-20-100102001020304050xzThe Lorenz attractor · σ=10, ρ=28, β=8/3
The Lorenz weather model, traced out in the air. The line never crosses itself and never repeats, yet it stays on these two wings the whole way. · computed here · Nan · hongdam.net · CC BY 4.0

The part most people miss

Chaos gets sold as a reason to throw up your hands. It is the opposite. Once you know a system is chaotic, you stop wasting effort on predicting the one thing you cannot predict, and you spend it on the two things that work every time.

Count the many, not the one

You cannot say how one die lands. You can say, to a hair, what a thousand of them average. Chaos lives in the single case; order lives in the pile. That is the law of large numbers, and it is a tool you can pick up.

Build it to bend

You cannot stop the shock you did not see coming. You can build the thing that takes it: give it slack, give it spares, spread its bets, and wire in something that pulls it back to centre. That is a resilient system, and it is the second tool.