Two oscillators with a quartic coupling#

with $\omega_2 = 1$ (equal frequencies, the system of Pullen and Edmonds 1981) or $\omega_2 = 1.618$ (an inharmonic pair, closer to the plate). Since rescaling the variables by $\sqrt{E}$ leaves only $\beta E$, the energy $E$ is the single parameter, the analogue of the plate's strike strength.

model.py integrates many orbits at once with a fourth-order symplectic scheme (Yoshida's composition of leapfrog steps; energy conserved to 1e-6 or better over 1000 time units), optionally with damping $p \to p\, e^{-\gamma \Delta t}$, and integrates the tangent dynamics for the largest Lyapunov exponent (Benettin's renormalization). An orbit counts as chaotic when its finite-time exponent over 2000 to 3000 time units exceeds 0.02; regular orbits give values near $\log(t)/t \approx 0.005$.

Results (2026-10-05)#

Poincare sections (results/sections.png; crossings of $y = 0$ with $p_y > 0$, in the $(x, p_x)$ plane). At low energy every orbit lies on a closed curve (a torus): the motion is quasi-periodic. As the energy grows, chaotic layers appear around the separatrices between families of tori, widen, and merge, until only small islands of regular motion remain.

Onset of chaos (results/lyapunov.png; 200 orbits per energy, 25 energies). Fraction of orbits that are chaotic:

E	0.42	0.56	0.75	1.0	1.3	1.8	2.4	3.2	5.6	10
$\omega_2 = 1$	0	0.03	0.14	0.48	0.69	0.87	0.91	0.94	1.00	1.00
$\omega_2 = 1.618$	0	0	0	0	0	0.17	0.60	0.81	0.96	0.98

The transition is gradual, not a sharp threshold, and the inharmonic pair needs about twice the energy before it starts: with equal frequencies the 1:1 resonance couples the two oscillators strongly from the start, while detuned oscillators exchange energy only through higher-order resonances. In the chaotic regime the Lyapunov exponent keeps growing with energy (median about 0.3 at E = 5 to 1.4 at E = 100 for $\omega_2 = 1.618$): stronger excitation, faster separation, as in the plate.

A damped strike and its frozen outcome (results/damped.png; $\omega_2 = 1.618$, $\gamma = 0.005$; damping the momenta by $e^{-\gamma \Delta t}$ makes the energy decay as $e^{-\gamma t}$). Both oscillators start at rest with momentum $\sqrt{2 E_0}$ in a fixed direction; an ensemble of 16 strikes differs in that direction by $10^{-10}$ times the member index. At $E_0 = 0.5$ and $E_0 = 3$ the strike lands on a regular orbit: the local Lyapunov exponent decays to zero, and all 16 strikes end with the same split of the energy between the two oscillators (0.557 and 0.398 in the x oscillator). At $E_0 = 20$ the motion is chaotic for the first 300 to 600 time units (local exponent 0.4 to 0.8), the 16 strikes separate completely, and as the energy falls below about 1 each is captured by a regular orbit and keeps the split it happened to have. At the end the share of energy in the x oscillator ranges from 0.05 to 0.95 across strikes that differed by $10^{-10}$ (standard deviation 0.31).

This is the plate's frozen fingerprint in its simplest form: a transient chaotic phase while the energy is high, capture onto a torus once damping has taken the energy below the onset of chaos, and a final state that is a random sample from a distribution, set by perturbations far below anything measurable. Note that $E_0 = 3$ lies where 81% of section orbits are chaotic, yet this particular strike is regular: whether a strike is chaotic depends on where in phase space it starts, not only on its energy.

Possible next steps#

• The distribution of the frozen split as a function of $E_0$ and of the damping rate (slower damping gives the chaotic phase longer to mix).
• Damping that grows with frequency (as in the plate), which makes the higher oscillator lose energy faster and biases the outcome.
• The many-mode version (FPUT chain) for the cascade.