fput-chain-chaos#

Struck FPUT-beta chain, step 2 of "Listening for chaos" (after double-pendulum-chaos). Work in progress.

Current state: N masses (2 to 64, default 4) between two fixed walls, joined by springs with tension k d + k3 d^3. The chain sits at rest until Strike, which gives it energy E (log slider, 10 µJ to 100 J, default 2.51 J) as velocity, either on one mass j alone (default mass 1, a hammer blow at one end) or in the shape of one normal mode n. The left card animates the chain (top: to scale, rest spacing a; below: the same displacements drawn sideways as a profile, scaled to fit, with the scale given) with the time and the relative energy error. Speed: 0.1x to 5x real time (default 1x). The time and current energy show at the top left. The Ghost chip (as in the pendulum app) adds a second chain started from the same state with every mass moved by the same amount, 10^k / sqrt(N) m (total offset 10^k m, k = -12 to -1, default -10; moving only the struck mass would put the offset along the motion itself, which does not grow), run in lockstep without renormalization and drawn in translucent orange underneath; restarted on Strike, on "Restart ghost" (from the chain's current state), when switched on or when the offset changes; a readout gives the separation (root sum of squared displacement differences, m) and the time since the ghost started, and a "Separation" chart (log, last 60 s; offered only with the ghost on) plots it. A "Lyapunov" chart (last 60 s; the first 5 s after a strike left out of its y range) shows the largest Lyapunov exponent over the last 10 s and since the strike (src/lyapunov.ts, two-trajectory Benettin method as in the pendulum app: a hidden shadow chain 1e-8 away, displacements in units of d0 and momenta in units of sqrt(m k) d0, renormalized every step). Checked (N = 4, mass-1 strike, 300 s): without damping 0.1 J ~0.05 to 0.1 /s (weak chaos; N = 4 onset is near 3 k d0^2 = 75 mJ), 0.5 J ~0.5 /s, 2.51 J ~0.9 /s, 25 J ~1.7 /s; with tau 540 s, 0.1 J falls to ~0.01 /s and 2.51 J falls from ~1.0 to ~0.3 /s by 300 s. The slider is labelled "Strike energy". Parameters (m, k, k3, a, tau) fold out underneath with derived values (mode frequencies, d0, nonlinear energy scale, integrator step); Apply or changing N returns the chain to rest. The "Real time" card on the right has toggle chips for its charts. "Mode bars" shows the current harmonic energy of each normal mode relative to E on a log scale (12 decades), with the number of modes sharing the energy (inverse participation, 1 / sum (E_n / sum E)^2). The other chips are charts over time (from src/traces.ts and src/cursor.ts of double-pendulum-chaos, with per-group colors and a legend capped at 8 series): displacements, velocities, spring stretches, mode energies (log, last 60 s; values below 1e-12 of E left out), energies (E, T, V, W dissipated) and the energy balance error ((E + W - E0) / E0); others last 10 s; sampled every 10 ms; hover for values, click to pin. Mode bars, displacements and mode energies are on by default. Masses, springs and modes share one color per index (four fixed colors for up to 4, a blue-to-red sweep beyond).

"Sped up · listen" (open by default, below the cards; otherwise as in the pendulum app): the header says how much faster it runs ("the whole decay, 321× faster: 62 min of motion in 12 s"; without damping "321× faster than the chain above"). The page starts muted (the mute button is tinted red and reads "Muted"; muted notes are still computed and shown); the audio context is created at the first Strike; closing the panel stops the note. While it is open each Strike plays a note, the same strike rendered in a Web Worker (src/render-worker.ts) sped up so the lowest linear mode sounds at the Pitch (30 to 1000 Hz, default 100; the readout gives the lowest and highest mode pitches). A damped note is the whole decay, ending when its energy has fallen to 1e-6 of the strike's (-60 dB; predicted length (tau/2) ln 1e6 of physical time, 11.6 s of sound by default, measured 11.3 s; capped at twice that); undamped notes last 10 s. "Strike to hear the whole decay." shows until the first note; the status line gives the signal and note length ("computing…" while rendering). Replay plays the last strike again with the current settings (to compare signals). Signal (src/signals.ts): velocity (default), displacement or acceleration of mass j (default 1, the end mass, which is never at a node), force on the left wall (the first spring's tension), or sum of accelerations (the net wall force, in which even modes cancel). In one mass's motion mode n is weighted by sqrt(E_n) / omega_n, sqrt(E_n) or sqrt(E_n) omega_n. Each note is made zero-mean, scaled to a 0.9 peak (all signals equally loud; Volume sets the level) and faded out over its last 20 ms (src/sound.ts). Chips (as in the pendulum app): Spectrogram (on; pendulum's NoteSpectrogram: 2048-sample Hann, zero-pad 4096, hop 512; dashed lines at the linear mode pitches, drawn only while at least 6 px apart, since crowded together their dashes merged into a solid on-off bar at the bottom), Lyapunov (the note's largest Lyapunov exponent over the last 10 s of physical time, from a shadow run in the worker alongside the note; the first 5 s of physical time left out of the y range), Energy (log scale, so the decay is a straight line) and λ vs E (src/scatter.ts from the pendulum app: one point per 256 samples, accumulated over notes, log energy axis, colored by time into the note on a log scale, earlier notes faded, hover for values, Clear button). The first three span the whole note (strike at left, end at right) and fill in as the audio clock reaches each sample; one shared cursor (playhead while playing; hover for values, click to pin). Pause (or space) pauses the animation and the sound (audio context suspended). Render time in the browser pane (with the Lyapunov shadow): ~0.6 s for N = 4, ~2 s for N = 32 (default strike). First look (N = 4, 2.51 J): λ ~1.3 /s at the strike, falling to ~0 by ~4 s of sound as the spectrogram resolves into lines; on λ vs E the note's points sit near 0 below ~0.1 J and rise to ~1.4 /s at the top.

npm install
npm run dev

Model#

• SI units. Defaults m = 1 kg, k = 10 N/m, k3 = 4000 N/m^3, a = 0.5 m. The cubic tension equals the linear one at d0 = sqrt(k / k3) = 5 cm; how nonlinear the motion is depends only on E / (k d0^2) (25 mJ by default). The rest spacing only enters the drawing.
• Fixed ends. Normal modes of the linear chain: shapes sqrt(2 / (N + 1)) sin(pi n j / (N + 1)), angular frequencies omega_n = 2 sqrt(k / m) sin(pi n / (2 (N + 1))); 0.311, 0.592, 0.814, 0.957 Hz for the defaults. Mode energy P_n^2 / 2m + m omega_n^2 Q_n^2 / 2.
• Damping: linear drag on each mass proportional to its mass, force -gamma m v = -gamma p, so every linear mode's amplitude decays as exp(-t / tau), tau = 2 / gamma (default 540 s: ~10% amplitude per minute in the animation; 0 = off). Applied exactly in half steps around each Verlet step (p scaled by exp(-gamma h / 2)), with the kinetic energy removed added to W. Checked: modes 1 and 4 at 1e-8 J decay to 0.13539 and 0.13536 of their energy after tau (e^-2 = 0.13534); energy balance error 3e-5 over the default note (6000 s); -60 dB reached at 3635 s vs 3730 s predicted.
• Integrator: velocity Verlet (order 2, symplectic; H is separable, so an explicit method suffices). Step from the energy (stepFor): min(0.005 sqrt(m / k), 0.02 sqrt(m / k_eff)), k_eff = k + 3 k3 d^2 with d = (4 E / k3)^(1/4) the largest stretch a spring holding all of E could reach. Set at each strike in the animation (1.58 ms when weak, 0.81 ms at 2.5 J, 0.32 ms at 100 J); re-chosen every 256 samples in the sound, so steps lengthen as a note decays. Relative energy error 1e-5 to 5e-5 from 1 mJ to 100 J (N = 4 and 32, 300 s). Render time in the pane: 0.2 to 0.3 s (N = 4), 0.6 to 0.8 s (N = 32), 2.5 to 100 J. At 100 J stretches reach ~0.55 m, more than the drawing's rest spacing, so masses pass each other in the to-scale drawing (the model has no contact).
• Chaos for N = 4 (two-trajectory Lyapunov, 5000 / sqrt(k/m) s): none below E ~ 1 k d0^2; onset near 3 k d0^2; at the default 10 k d0^2, lambda ~0.1 sqrt(k/m) = 0.32 /s, energy shared by ~2.5 modes.
• Symmetry: the quartic force couples odd modes only to odd modes, so a strike in an odd mode leaves the even modes at exactly zero (they stay off the bottom of the chart). A strike on one mass, other than the middle one, excites all modes.

Code: src/chain.ts (physics, modes, strikes), src/main.ts (page), src/traces.ts and src/cursor.ts (charts), src/signals.ts, src/render-worker.ts, src/sound.ts, src/spectrogram.ts, src/scatter.ts (sound), src/lyapunov.ts (Lyapunov exponent, animation and sound).