da03127Two-oscillator chaos toy: model, experiments, figures, web appJeremy Magland 1# Two oscillators with a quartic coupling
3$$H = \tfrac12 (p_x^2 + p_y^2) + \tfrac12 (\omega_1^2 x^2 + \omega_2^2 y^2) + \beta x^2 y^2, \qquad \omega_1 = 1, \ \beta = 1,$$
5with $\omega_2 = 1$ (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.
7`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 $\log(t)/t \approx 0.005$.
9## Results (2026-10-05)
11**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.
13**Onset of chaos** (`results/lyapunov.png`; 200 orbits per energy, 25 energies). Fraction of orbits that are chaotic:
15| E | 0.42 | 0.56 | 0.75 | 1.0 | 1.3 | 1.8 | 2.4 | 3.2 | 5.6 | 10 |
16| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
17| $\omega_2 = 1$ | 0 | 0.03 | 0.14 | 0.48 | 0.69 | 0.87 | 0.91 | 0.94 | 1.00 | 1.00 |
18| $\omega_2 = 1.618$ | 0 | 0 | 0 | 0 | 0 | 0.17 | 0.60 | 0.81 | 0.96 | 0.98 |
20The 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.
22**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).
24This 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.
26## Possible next steps
28- 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).
29- Damping that grows with frequency (as in the plate), which makes the higher oscillator lose energy faster and biases the outcome.
30- The many-mode version (FPUT chain) for the cascade.