The double pendulum: deterministic, and still unpredictable

Each pendulum is two point masses on light rigid rods, the second hanging from the first, swinging without friction — or three or four, if you change Arms. All of them start at rest, at the angles in the controls panel, except that each upper arm starts slightly further round than the one before. The front pendulum is the one you can drag; the colours run through the rest.

The motion is calculated from the exact equations of motion with a fourth-order Runge–Kutta integrator, 400 steps per simulated second. Over twenty seconds the double pendulum's total energy changes by less than one part in a million, and even a quadruple pendulum falling from upright stays within one part in 10,000, so the divergence you see is the physics, not rounding error building up.

The chart at the bottom left plots the gap between the first two pendulums on a logarithmic scale. From the Raised high start it climbs in a straight line — the gap is multiplying by a fixed factor every second — until it is as large as it can get. A straight line on a log scale is exponential growth, and its slope is the Growth readout: about 1 per second here, meaning the gap grows roughly e ≈ 2.7 times each second. That rate is an estimate of the largest Lyapunov exponent of this motion.

It is why a billionth of a radian takes about eighteen seconds to show, and a millionth about eleven: shrinking the starting gap a thousandfold buys only seven seconds. To predict the motion twice as far ahead, you would need to know the starting position not twice as precisely but tens of millions of times as precisely. This is what Edward Lorenz ran into with weather models in the 1960s, and it is why forecasts have a horizon however good the models become.

At any instant a double pendulum is fully described by four numbers: the angle of each arm and how fast each is turning. Together they are a point in its phase space, and the motion is that point moving. Turn on Phase space in the controls to see a slice of it for one arm: every pendulum is a dot, at that arm's angle across and its angular velocity up.

Released together, all the dots sit on one spot and trace the same path. When the pendulums start to part, the spot stretches into a thread; the thread keeps stretching and folds back on itself, and soon it is a cloud spread over the whole panel. Stretching and folding, over and over, like a baker kneading dough, is the mechanism of chaos: without friction a patch of starting states can never grow in total volume, but it can be drawn out ever thinner in some directions while squeezed in others. From a Gentle swing the dots stay as one and keep looping round the middle. When an arm goes right over the top its angle passes ±180° and the dot reappears at the other edge.

Choose Gentle swing and the pendulums stay together indefinitely: with little energy the motion is quasi-periodic, and small differences stay small. Choose Normal mode and the motion is exactly periodic in the small-angle limit: with equal masses and rods, starting the lower arm √2 times further out than the upper makes both swing at the single frequency ω² = (2 − √2) g / l. Most starts with enough energy to throw the lower arm over the top are chaotic.

A triple or quadruple pendulum obeys the same kind of equations, only more tangled: every arm's acceleration depends on all the others through the chain's mass matrix, so the simulation solves a small system of linear equations at every step. More arms means more ways for energy to slosh between them, and usually faster divergence. Balanced Nearly upright, a billionth of a radian shows after about twenty seconds with two arms, six with three and three with four.

A chain of n arms has n normal modes. Normal mode starts every arm in proportion to the slowest one, found from the small-swing equations, so a triple or quadruple pendulum swings back and forth at a single frequency just as the double does.

• Set the gap between neighbours to 10⁻¹² and then to 10⁻⁶. The Growth readout barely changes: the rate belongs to the motion, not to the starting gap. (Start them much further apart and the gap fills the chart before there is enough of a straight line to fit.)
• Drag the lower bob of the front pendulum straight up from a hanging start, and compare how long the pendulums stay together with the upper arm low and high.
• Use one pendulum with trails on and slow motion to see the lower arm flip over the top.
• Turn on Phase space and watch from the release of Raised high: the single dot draws out into a thread at about the moment the pendulums visibly part.
• Choose Nearly upright, then switch Arms from Double to Quadruple and watch how much sooner the fan opens.

• E. N. Lorenz (1963), Deterministic Nonperiodic Flow, J. Atmos. Sci.
• T. Shinbrot, C. Grebogi, J. Wisdom and J. A. Yorke (1992), Chaos in a double pendulum, American Journal of Physics — a real double pendulum compared against simulation.