Two waterwheels start almost exactly the same. How long until they do something completely different?

What you'll discover

• A chaotic system follows exact rules, yet its long-term behaviour is still unpredictable.
• A tiny difference in the starting conditions grows faster and faster until the two runs look unrelated.
• Chaotic motion is not random: it keeps returning to a recognisable pattern without ever exactly repeating.

The science in plain words

Chaos theory studies systems that obey precise rules but behave in ways that are impossible to predict far into the future. The key feature is sensitive dependence on initial conditions: two starting points that differ by an almost unmeasurable amount end up on completely different paths.

A chaotic waterwheel shows this beautifully. Water pours into leaky buckets around the rim of a wheel. Depending on how the water is spread, the wheel may spin one way, slow, reverse and spin the other way, in an irregular pattern that never settles. The equations describing it are the same ones the meteorologist Edward Lorenz found in the 1960s while studying a simplified model of air currents. He noticed that rounding a number very slightly in his computer gave a completely different forecast weeks later.

That is the butterfly effect, the idea that a butterfly flapping its wings could in principle change whether a storm forms far away. In practice, the differences grow so quickly that weather forecasts lose their accuracy after about a week or two, however good the computers. Chaotic systems are not random, though. Their paths trace out a structured shape called a strange attractor. Chaos also appears in double pendulums, dripping taps and planetary orbits over very long times.

Where it fits

• India, NCERT: Goes beyond the NCERT syllabus for classes 6 to 10.
• United States: Grades 9 to 12

Common questions

What is the butterfly effect?

The butterfly effect is the idea that a tiny change at the start of a chaotic system can grow into a huge difference later. The name comes from the image of a butterfly's wingbeat eventually influencing whether a distant storm forms. It explains why long-range weather forecasting is so hard.

Is chaos the same as randomness?

No. A chaotic system follows exact rules, so running it from precisely the same start gives exactly the same result. It only seems random because any tiny difference in the start grows quickly, and we can never measure the starting conditions perfectly.

Why can't weather be forecast weeks ahead?

The atmosphere is a chaotic system. Small errors in measuring today's temperature, pressure and wind grow rapidly as the forecast runs forward. After about a week or two, those errors swamp the forecast, so detailed day-by-day predictions become unreliable.

Related labs

• Double Pendulum Twins: Release two double pendulums almost exactly the same. How long do they stay twins?
• Weather Patterns: Why does wind swirl round a storm instead of blowing straight in?
• Fractals: One simple rule, repeated, grows a whole tree. How many twigs does it make, and how much space does it fill?

See all maths labs or every Taz science lab.

Try it yourself

The Chaos Theory lab is one of the interactive 3D science labs inside Taz. Change one thing, predict what will happen, then run it and watch. Labs open after you sign in.

Try this lab on Taz