Set a system running and it rarely wanders forever. It falls — into stillness, into a rhythm, or into a shape so intricate it has no volume and never repeats. That destination is the attractor: not a place the system is pushed toward, but the only place left once everything transient has died away. In 1963 a meteorologist truncated three equations, ran them twice, and found weather that diverged from itself. This chapter is about what he saw.
A mass on a spring, with friction. Start it anywhere — pull it far, flick it hard — and it always ends in the same place: at rest, at the centre. That point is an attractor, and the whole plane is its basin.
What you are looking at is not space. It is phase space: horizontal is position, vertical is velocity. A single dot holds the entire state of the system, and the arrows show where that state must go next.
Turn the friction to zero and the attractor vanishes — the system orbits forever, remembering exactly how it started.
Van der Pol built this to describe a vacuum-tube oscillator: damping that is negative near the centre and positive far out. Small motions are pumped up; large ones are bled down. Nothing can rest.
So the system settles instead onto a closed loop — a limit cycle. Release a state from anywhere, inside or outside, and it spirals onto the same ring and then circles it forever. This is a heartbeat, a firefly, a hunted population.
Raise μ and watch the smooth circle sharpen into the jerky, snapping rhythm of a relaxation oscillator.
Now give the system two resting places — a bead in a double well, sliding into the left hollow or the right. Every starting state belongs to one basin of attraction or the other. The picture colours each point by where it will end up.
With heavy friction the border is a clean curve: near the left well, you fall left. Lower the friction and the bead overshoots, sloshes, and the boundary shreds into filaments — arbitrarily close to a point bound for the left well sits a point bound for the right.
The equations are still perfectly deterministic. Prediction is what breaks, not causality.
Drop the differential equations. Take the simplest map that can bite its own tail: next year's population is r times this year's, times the room that's left. x → r·x·(1−x).
Turn r up slowly. The population settles to one value. Then, at r = 3, that single value splits in two — boom, bust, boom, bust. Then four. Then eight. The splits come faster and faster, each gap about 4.669 times smaller than the last — Feigenbaum's constant, the same number for almost any such map.
By r ≈ 3.5699 the rungs have piled into a continuum: chaos. And yet look inside it — clean windows of order, a period-3 band near 3.83, out of nowhere.
Three equations, no randomness, no memory. Yet the trajectory never crosses itself, never repeats, and never leaves a bounded region — it winds forever on a surface of fractional dimension. Lorenz found it in a weather model he had cut down until it was almost nothing.
Drag to turn the shape. Change the constants and watch the wings appear, collapse, or fold into something else entirely. Then press Sensitivity: two states are released a millionth apart, and you can watch determinism stop being prediction.