fput-chain#

(See also ../gong-wire, which adds bending stiffness and a dome to this chain to get a gong-like cascade to high frequencies.)

A struck Fermi-Pasta-Ulam-Tsingou chain: the many-mode companion of the two-oscillator toy in ../chaos-toy-examples, and a simple stand-in for the struck plate of cymbal-elastodynamics/flat-plate-2.

The model#

$N$ unit masses on a line between two fixed walls, joined by springs with force law $f(\Delta) = \Delta + \beta \Delta^3$ ($\beta = 1$), and optionally held by on-site springs of stiffness $\mu^2$. The normal modes are $\sin(jk\pi/(N+1))$ with $\omega_k^2 = \mu^2 + 4\sin^2(k\pi/(2(N+1)))$: for $\mu = 0$ the low modes are nearly harmonic (like a string), and $\mu > 0$ detunes them (like a plate). The stiffness of a stretched spring is $1 + 3\beta\Delta^2$, so stretching couples the modes, as stretching the midplane does in a plate. Damping is friction $c_0$ on every mass plus dashpots $c_1$ between neighbours; mode $k$ decays (in amplitude) at $(c_0 + c_1(\omega_k^2 - \mu^2))/2$, the chain's version of the plate's $\sigma_0 + \sigma_1 k^2$ damping. The strike gives a few masses near one end a smooth velocity profile with kinetic energy $E$.

model.py integrates ensembles of strikes with a fourth-order symplectic scheme (Yoshida) and the exact propagator of the damping, integrates the tangent dynamics for the largest Lyapunov exponent, and computes mode energies and the effective fraction of modes $e^S/N$ ($S$ the entropy of the mode shares; 1 for equal shares, about 0.7 for random equipartition, $1/N$ for a single mode).

Results (2026-10-05; $N = 32$, strike on mass 8, width 2 masses)#

Equipartition and chaos vs strike energy, undamped (results/scan.png, 4000 time units, 6 strikes per energy):

Below $E \approx 0.5$ the harmonic chain keeps the strike's energy in the modes it started in, the classic FPUT observation. From $E \approx 1$ to 2 it spreads to equipartition, and the Lyapunov exponent grows roughly in proportion to $E$. The detuned chain ($\mu = 1$) holds the energy in its initial modes up to $E \approx 30$, some twenty times more: detuning removes the near-resonances through which the modes exchange energy (results/cascade.png shows the mode energies over time). The same effect, much weaker, separated the equal and detuned pairs of the two-oscillator toy, and the plate's spectrum is strongly inharmonic.

Damped strikes and the frozen spectrum (results/damped_mu0.png, results/damped_mu1.png; lowest mode decaying at 4e-4, highest at 3e-3 per time unit; 16 strikes whose strengths differ by $10^{-10}$ times the member index):

• $\mu = 0$, $E_0 = 0.5$: regular; the energy stays in the lowest four modes and all 16 strikes end identical.
• $\mu = 0$, $E_0 = 5$: a real cascade, energy spread over about 70% of the modes for the first 1000 time units, and a positive local Lyapunov exponent (about 0.015) for a few hundred. Yet all 16 strikes end identical. The chaotic phase amplifies differences by about $e^{10}$, which takes $10^{-10}$ to $10^{-6}$, still invisible.
• $\mu = 0$, $E_0 = 30$: local exponent up to 0.08 for about 600 time units, amplification far beyond $e^{23} \approx 10^{10}$, and the 16 strikes end with different spectra: the share of energy in each of the main modes differs by about 0.57 decades (about 6 dB, standard deviation across strikes).
• $\mu = 1$: no strike up to $E_0 = 30$ ends differently from its neighbours.

So an energy cascade is not the same as an audible fingerprint. Whether a tiny difference between strikes grows to a visible one depends on the amplification accumulated over the chaotic phase, $\exp(\int \lambda\, dt)$, against the size of the difference; a strike can share its energy among all modes and still be perfectly repeatable. The damping enters directly: the web app shows that with the highest mode damped twice as fast (ring time 0.15 s instead of 0.3 s at 110 Hz), the strong strike at $E_0 = 30$ no longer separates at all.

Web application#

web/index.html (single page, no build step; published as a private artifact at https://claude.ai/artifact/C6oaNiG2zCRX9VR7Wn4Wfx) simulates 8 strikes in the browser and plays strikes A and B. Controls: strike energy, number of masses, on-site springs, strike position and width, the A/B difference, ring times of the lowest and highest modes, the pitch of the lowest mode, and length. Panels: the chain with displacements drawn sideways and springs drawn by stiffness (with slow motion), the energy of strike A per mode over time, the effective fraction of modes for all 8 strikes, their final spectra, the local Lyapunov exponent with the energy, and a spectrogram of any of the 8 strikes (a selector above it, or the arrow keys, flips between strikes on a shared color scale, highlights the chosen strike in the two plots above, and plays it). Presets: Gentle ($E = 0.5$), Cascade only ($E = 5$), Strong ($E = 30$), Strong, detuned ($\mu = 1$, $E = 30$). Locally: cd web && python -m http.server.

Time in the app is scaled so the lowest mode sounds at the chosen pitch, so a ring time in seconds means different damping in the chain's own time units for different $\mu$: with $\mu = 1$ the lowest mode is at $\omega \approx 1$ instead of 0.095, and one second of sound is about ten times fewer time units.

Running#

Experiments run on the office workstation (~/work/fput-chain, Python 3.9 with numpy and scipy), plots locally:

WORKERS=32 python experiments.py all     # results/*.npz
python plots.py                          # results/*.png