Strike, or click a point on the Poincaré section, to hear the whole decay.
A double pendulum, shown in real time. Strike starts it from hanging rest with the energy on the slider. At low energy the motion is regular; at higher energies most motions are chaotic, so a small change in the start leads to a completely different motion.
With damping (τ under Parameters) the energy slowly decays, and the current energy is shown at the top left. Without damping the slider changes the energy as it runs.
Ghost adds a second pendulum started a tiny angle away. In chaotic motion the two separate exponentially; in regular motion they stay together. The space bar pauses.
Masses, arm lengths, gravity, and the decay time τ of the amplitude (drag on the bobs; 0 for none). Apply restarts the pendulum at rest. They apply to everything on the page.
The presets change the outer arm's length, which sets the ratio of the two normal-mode frequencies (listed below the form). These are the pitches heard at low energy.
Charts of the pendulum on the left, on the same clock. Click a chip to show or hide a chart. Hover over a chart to read values; click to pin the cursor.
Error is the integrator's relative error in the energy balance (the energy plus what damping has removed); it should stay tiny. Lyapunov is the largest Lyapunov exponent: the rate (1/s) at which nearby motions separate, positive for chaos and near 0 for regular motion. Separation, shown with the ghost on, is the ghost's distance from the pendulum on a log scale.
A copy of the same strike, run fast enough that the swinging becomes sound: Pitch sets where the lower normal mode lands. The whole decay plays in a few seconds. While this panel is open, each Strike plays a note.
Save WAV saves the current or last note as heard (16-bit mono, normalized). Signal chooses which coordinate is heard. Chaotic motion sounds noisy; as the energy decays it settles into steady tones.
The charts span the whole note, strike at the left: spectrogram, Lyapunov exponent, energy, and λ against energy collected over notes.
A map of the motions at one energy, without damping. Each time the inner arm swings down through vertical moving right, the outer arm's angle θ2 and momentum p2 are plotted as a dot. Regular motions trace closed curves; chaotic ones scatter over a region. The outline bounds what is reachable at this energy.
The pendulum's own crossings are dark rings. Follow makes the section's energy track the pendulum (real time) or the playing note (sped up). Hover to pick out an orbit; click to start a new one there, which with Click to strike also strikes the pendulum from that point.