double-pendulum-chaos#

Moved (2026-10-09). This app now lives in the chaos-apps repository, in src/double-pendulum/, with code shared with the other apps in src/common/; its history was carried over. Live site: https://chaos-apps--listening-for-chaos.vault1-sites.magland.org/double-pendulum/. This repository is kept for reference and is no longer updated; its site now redirects to the new one (redirect/index.html).

Interactive double pendulum, step 1 of "Listening for chaos". Work in progress.

Current state: the pendulum hangs at rest until Strike, which starts it from hanging rest with the energy on the slider (default 50 J), plus a random relative jitter of up to 1e-9 in each angle and momentum (so chaotic strikes differ from one another while regular ones sound alike); Pause (or the space bar) pauses everything: the animation and the sound (the audio clock is suspended, so the note and every display fed by it stop where they are); Resume or Strike continues. The Ghost toggle adds a second pendulum (translucent orange) started from the same state with the outer arm's angle offset by 10^-12 to 10^-1 rad (default 10^-10), restarted on Strike or "Restart ghost", with a readout of the angular separation and the time since the ghost started: it grows exponentially when the motion is chaotic and stays tiny when it is regular. Charts "Separation" (ghost separation, log scale, last 30 s; its chip shows only with the ghost on) and "Lyapunov" (largest Lyapunov exponent over a 10 s window and since the strike, last 30 s; the first 5 s after a strike, where the values are large, are left out of the y range, here and in the sound's Lyapunov chart). Chart legends show 4 significant digits, ranges 3. Parameters (masses, lengths, g, decay time, presets) fold out underneath. The plots card is labelled "Real time" and the "Sped up · listen" panel's header says how much faster its run is (e.g. the whole decay, 262x faster: 62 min of motion in 14 s), since the two run on different clocks; the Follow options read "real time" and "sped up". Toggle chips choose the plots: live traces of energies, energy balance error, angles, momenta and bob positions. Each chart shows its time span at the bottom right ("last 10 s" when scrolling; the note length in the sound panel). The Poincare section panel (closed by default) has its own energy slider (default 17 J) and uses the same masses and lengths, without damping: each time the inner arm swings down through vertical moving right, the outer arm's angle and momentum are plotted. 42 orbits start on a grid inside the allowed region (outlined) and leave 300 dots each, computed in a Web Worker and drawn as they arrive; regular orbits trace closed curves, chaotic ones scatter. Hover to pick out an orbit (with its Lyapunov exponent), click inside the outline to start a new orbit there (with "Click to strike" on, the default, this also strikes the pendulum above from that exact state, at the section's energy, without jitter). The animated pendulum's own crossings are drawn as dark rings (the latest larger), so you can see which orbit it is on; only those within 5% of the section's energy are drawn (others would fall outside the allowed region; the legend then says "pendulum at 48 J, not shown"), and they clear on each strike. Follow stays selected; setting the section's energy by hand pauses everything, clears the recent crossings (rings and note trail) and shows the section at that energy, and Follow takes over again on resume. Follow: "real time" makes the section's energy track the pendulum's as it decays (recomputed after a 3% change, at most every 2 s); "sped up" (which opens the "Sped up · listen" panel) shows the playing note's crossings (sent by the audio worker with the samples, drawn as the audio clock reaches them) as a fading trail of the last 150, with the axes fitted to the note's current energy (zooming as it decays) and nothing computed live. Without an atlas only the outline is drawn behind the trail; with one, the atlas section nearest the current energy, fitted to the current allowed region (each point scaled by the ratio of the regions' extents in th2 and p2: exact at low energy, where sections are scaled copies, an approximation over the <~15% gaps between atlas energies). Trail dots are fitted the same way from their own energies. "Precompute sections" (shown only with Follow "sped up", until the atlas is complete) builds the atlas (src/atlas.ts): sections at 24 energies log-spaced from 0.4% of to twice the both-inverted energy, 42 orbits x 150 crossings each, highest energy first, in its own worker (~80 s in the desktop app's browser pane), stored in IndexedDB per parameter set and deleted after 7 days. "Color by λ" colors orbits by Lyapunov exponent (log scale, blue regular to red chaotic). The "Sped up · listen" panel (above the section) starts closed; opening it switches the sound engine on (closing it switches it off) without playing anything; a line of text says to Strike (or click the section) until the first note. While it is open, every Strike plays a note: a separate, sped-up copy of the motion from the same strike; a damped note ends once it has decayed by 60 dB. Save WAV saves the current or last note as heard (as received from the audio worker, up to 120 s; 16-bit mono PCM, peak at 0.9; src/wav.ts; named e.g. double-pendulum-50.0J-x2-100Hz.wav). The mute button (speaker icon; when muted it is tinted red and reads "Muted") starts muted and unmutes the first time the panel is opened; muted notes are still computed and shown, just silent. The panel also has "Signal" (which coordinate is heard), volume, pitch (where the small-amplitude lower mode lands, 30 to 1000 Hz, default 100 Hz; the readout gives both modes' pitches), and optional whole-note displays that do not scroll (strike at the left edge, end of the note at the right): a spectrogram of the sound (frequency axis up to just above the highest frequency that comes within 40 dB of the peak at any time in the note, so the broadband start of a chaotic note stays in view), the largest Lyapunov exponent of the motion being heard (10 s window, physical time), its energy (log scale, so the exponential decay is a straight line), and a scatter of lambda against energy accumulated over notes (log energy axis, since a note spends most of its time at low energy; colored by time into the note, earlier notes faded, Clear button). Spectrogram on by default. Each card and panel (and Parameters) has a small "?" button that opens a short help dialog (src/help.ts; the texts are <template>s in index.html). Hovering over any chart (real time or sped up) shows a time cursor and the values there; a click pins it. Charts in the same card share the cursor. The time axis spans the predicted note length ((tau/2) ln 1e6 of physical time, divided by the speedup), is extended if the note runs longer, and is fitted to the note when it ends.

Live site: https://double-pendulum-chaos--listening-for-chaos.vault1-sites.magland.org/ (built and deployed by .github/workflows/pages.yml on every push to main).

npm install
npm run dev

Model#

• Point masses on massless rods, absolute angles from the downward vertical, canonical momenta.
• SI units. m1, m2, l1, l2, g are set in the page (Apply restarts at the same energy relative to the both-inverted energy). Default 1 kg, 1 kg, 1 m, 1 m, 9.81 m/s^2, tau 540 s (normal modes 0.382 and 0.921 Hz, ratio 1 + sqrt(2)). Presets keep m1 = m2 = 1 kg, l1 = 1 m and vary l2; with equal masses the mode ratio is at least 1 + sqrt(2) (at l2 = l1) and is the same for l2/l1 and l1/l2. Presets: l2 = 1 (2.414), 1.5667 (5:2), 2 (golden ratio squared, 2.618), 2.1637 (8:3), 3.2476 (3:1), 6.886 (4:1). At a fixed lower-mode pitch only m2/m1, l2/l1 and the relative energy affect the sound.
• Energy is zero at hanging rest.
• Damping: linear drag on each bob, force -gamma m_i v_i (proportional to its mass; equal for equal bobs), so the generalized forces are -gamma p and every mode's amplitude decays with time constant tau = 2/gamma. Default tau = 540 s: about 10% amplitude per minute in the animation, about 2 s in the sound at x262. The state carries the dissipated energy W, so E + W is checked for conservation. With damping the energy slider (labelled "Strike energy") sets the strike energy and the current energy is shown at the top left of the animation; with tau = 0 it is labelled "Energy" and sets the energy live. The section's controls are in two rows (energy and Follow; Color by λ and Click to strike), and "computing…" shows in the plot's top-right legend.
• Live energy change: both momenta are scaled by one factor at fixed angles (a push along the current motion). A target below the current potential energy can't be reached at once; until it can, kinetic energy is removed at a rate of 6 per second.
• Lyapunov exponent: two-trajectory (Benettin) method in src/lyapunov.ts. A hidden shadow pendulum is kept 1e-8 away (angles in rad, momenta in units of sqrt((m1 + m2) l1^2 E_top)), moved back after every step, and the log growth averaged. The shadow keeps its own energy (forcing it onto the pendulum's energy by rescaling momenta inflates the separation near turning points). Checked (no damping, 300 s): 2 J and 10 J fall toward 0 (0.01/s), 30 J ~1.0/s, 50 J ~1.4/s; unchanged for shadow distances 1e-6 and 1e-10. With damping (tau 540 s) a 50 J strike's 10 s value falls from ~1.4 to ~0.1/s by 300 s.
• Poincare section (src/poincare-core.ts): on the section th1 = 0, a point (th2, p2) is reachable at energy E exactly when p2^2 <= 2 m2 l2^2 (E - V(th2)), V(th2) = m2 g l2 (1 - cos th2); p1 is the root with the inner arm moving right. Crossings are located by linear interpolation within a step. The step is 4x the animation's: over 300 crossings the energy error stays below ~2e-5, the regular/chaotic classification is unchanged, and regular orbits move by ~1e-4. Each orbit's exponent is the shadow method's mean over its run (regular orbits ~0.01/s or less).
• Integrator: 2-stage Gauss-Legendre (order 4, symplectic), fixed step h chosen per parameter set by a short calibration run so the relative energy error stays below 1e-6 up to the top of the energy slider (twice the both-inverted energy); 3.13 ms for the defaults, fixed-point stage solve (started from the previous step's stage derivatives extrapolated in time; tolerance 1e-12, below which the energy error does not change). Angles are wrapped to (-pi, pi] after each step. The physics costs well under 1 ms of CPU per second; drawing dominates, so traces draw at most a min/max pair per pixel column.

Audio: a Web Worker (src/audio-worker.ts) integrates the audio copy with substeps of at most 3.2 ms of simulated time per sample, high-passes at 20 Hz, and streams 1024-sample chunks through a MessageChannel to an AudioWorklet (src/audio-worklet.ts) that plays them from a ring buffer, keeping about 0.25 s queued. The page shows how many times faster than real time the worker computes, and any dropouts. The worker also sends each chunk to the page, which feeds it to the audio spectrogram as the audio clock reaches it (2048-sample window, new column every 512 samples). The worker runs the same Lyapunov shadow on the note and sends one exponent and one energy value per 256 samples with each chunk; the page plots them in step with the spectrogram. The whole-note spectrogram keeps every column (pairs are averaged past 4096 columns) and draws only new pixel columns unless the axis or color scale changes.

Code: src/pendulum.ts (physics), src/sim.ts (running simulation with energy target), src/audio.ts (audio setup), src/signals.ts (audible quantities), src/lyapunov.ts (Lyapunov exponent), src/scatter.ts (lambda vs energy), src/cursor.ts (shared hover/pin cursor), src/poincare-core.ts, src/poincare-worker.ts, src/poincare.ts (Poincare section), src/atlas.ts (cached sections for following the sound), src/audio-shared.ts (constants shared with the worker), src/traces.ts (strip charts), src/spectrogram.ts (whole-note spectrogram), src/main.ts (page).