Determinism and predictability
A system is deterministic when its rules are exact: if you know the start exactly, the future is fixed. A swinging pendulum, a falling stone and a planet orbit are like this.
For many such systems we can also predict the future far ahead. A small error at the start stays small. That is why we know eclipse times years before.
Determinism and unpredictability
Some deterministic systems are chaotic. The rules are exact, yet a very small error at the start doubles again and again. After a while the error is as big as the whole motion, and the prediction is useless.
This is the butterfly effect: a tiny flap of wings could, in a model, change the weather weeks later. Chaos is not randomness. If you restart from exactly the same start you get exactly the same motion. But we can never measure the start exactly.
A chaotic system has three signs: exact rules, strong sensitivity to the start, and motion that does not repeat.
Phase space and attractors
Phase space is a graph where one point shows the complete state of a system. For a pendulum, the state is its angle and its speed. As time passes the point moves and draws a path.
- A simple pendulum without friction draws a closed loop.
- With friction the path spirals into one point: the pendulum stops. That point is an attractor (the place the path is pulled towards).
- A chaotic system draws a tangled path that never closes and never crosses itself. Its attractor is called a strange attractor.
Fractal geometry
A fractal is a shape that looks similar at every zoom. Strange attractors are fractals, and so are coastlines, ferns and clouds.
The Koch curve is made like this: take a line, cut it into 3 equal parts, and replace the middle part with two sides of a small triangle. Repeat on every new line. At level n there are 4ⁿ lines, each 1/3 as long as before, so the total length grows by 4/3 each time and has no limit. See also Fractals and the Sierpinski triangle.
Why chaos matters
- Weather: forecasts fail after about 10 to 14 days.
- Heart rhythm and population ups and downs can be chaotic.
- Secure codes and random-looking numbers use chaotic maths.
Chaos teaches us a limit of science: knowing the rules is not always enough to know the future.
Try it
Hang a small weight from a pendulum string, and tie a second small weight to the first. Release it twice from almost the same position. Watch how the two runs differ. In the 3D, set the start difference to 0.0001 and press Restart. Does the gap still grow?
Key formulas and definitions
- Predictable system: small start error stays small
- Chaotic system: error grows roughly like e^(λt) (λ > 0). Example: doubling every second means error ×2ᵗ
- Koch curve: lines at level n = 4ⁿ; total length = (4/3)ⁿ × the first length
- Phase space of a pendulum: x-axis = angle, y-axis = speed
Worked examples
1. A gap between two double pendulums starts at 0.001 and doubles every second. What is it after 10 s?
0.001 × 2¹⁰ = 0.001 × 1024 = 1.024. The gap is now bigger than a whole rod.
2. How many lines does the Koch curve have at level 3?
4³ = 64 lines.
3. The first Koch line is 9 cm long. How long is the curve at level 2?
Length = 9 × (4/3)² = 9 × 16/9 = 16 cm.
4. A pendulum loses energy to friction. What does its phase-space path do?
It spirals in towards the centre point (angle 0, speed 0). That point is the attractor.
5. Say whether this is chaos: a dice roll. Give a reason.
A dice roll is called random in daily life, but physically it is deterministic and very sensitive to tiny changes in the throw. We cannot measure it well enough, so we treat it as random.
6. A weather model has 1% error today and the error doubles every 2 days. After how many days is the error about 16%?
1% → 2% → 4% → 8% → 16% needs 4 doublings = 8 days.
Common mistakes
- Thinking chaos means random. Chaos follows exact rules.
- Thinking the butterfly effect means a butterfly really causes a storm. It means the system is very sensitive.
- Believing a better computer fixes it. We can never measure the start perfectly, so long forecasts always fail.
- Thinking every fractal is chaotic. Fractals are shapes; chaos is behaviour. Strange attractors are fractal shapes.