INTERACTIVE COMPANIONVOLUME 3, ISSUE 1FALL 2026

The Divergence Bench

A live model accompanying “Measuring Exactly How Unpredictable Three Gravitating Bodies Are”

Study: Computational study
Review: Peer-edited by the club review board
Article: 12 sectionsabout 26 min
Files: LaTeX source29 kB
Analysis code46 kB
Raw output19 kB

Nothing on this page is pre-rendered. The two panels below run the article's own gravity simulator — the same equations of motion, the same 4th-order Yoshida integrator, the same adaptive step tied to the local free-fall time, the same Benettin two-trajectory renormalisation — directly in this browser tab, in plain JavaScript, every time you press Run. Leave every control at its default and you are running the article's own configuration: you should land close to λ = 0.037 for the figure-eight and λ = 0.47 for the Pythagorean problem, the same two numbers §6 of the article reports. They will not match to the last digit — this page draws its own random perturbation direction from its own random number generator, not the analysis script's numpy stream — but the physics is identical, and a chaotic system's Lyapunov exponent is a property of the dynamics, not of any one perturbation direction, so repeated runs should cluster tightly around the article's value.

Model 1. Measure a Lyapunov exponent yourself

Pick a configuration. Figure-eight and Pythagorean use the exact initial conditions §5 and §6 use; random bound triple draws a fresh triangle the way §6's 40-trial ensemble did, scaled to a chosen fraction α of the local virial speed. Press Run to integrate a reference trajectory and a copy nudged by δ₀ in a random direction of the full twelve-dimensional phase space, exactly as Benettin's method (article ref. 11) prescribes.

Configuration

figure-eight (§3, exact periodic orbit)Pythagorean problem (Burrau 1913)random bound triple

perturbation size δ₀10⁻⁸The article's default is 10⁻⁸. A Lyapunov exponent should not depend on this choice at all — try 10⁻⁵ and 10⁻⁹ back to back and confirm the measured λ barely moves.step-size fraction η0.00500§10's sensitivity check moved the Monte Carlo ejection fraction by 50% between the extremes of this slider. Push it up on the Pythagorean case and watch the closest-encounter distance and the measured λ both drift.independent trials n10The article averaged 10 trials for the figure-eight and the Pythagorean case, 40 for the random ensemble. This page caps at 15 so a run finishes in a second or two in a browser tab.

random seed

trials completed: –mean λ: –median λ: –SEM: –Lyapunov time: –closest encounter: –

Press Run to measure a Lyapunov exponent. Nothing has been computed yet.

Watch the running-estimate panel rather than only the final number. Each faint line is one trial's ln(D/δ₀) running average; the bold line is their mean. Early on every trial swings wildly, driven by whichever direction the perturbation happened to point. That the swings settle down onto roughly the same value by the end of the run, for every independent trial, is the actual evidence that a real exponent is being measured — a single trial's number, on its own, would not tell you that.

Model 2. From an exponent to a horizon, and to who gets thrown out

The two calculators below take a Lyapunov exponent as an input rather than measuring one. The first turns it into the prediction horizon §7 derives; the second runs the article's own Monte Carlo ejection experiment live, body-by-body, with the same escape criterion §2 defines.

use Model 1's measured mean λ (falls back to the article's random-triple median, 1.326, until you run Model 1)initial uncertainty δ (fraction of system size L)10⁻⁶

λ used: 1.326horizon: – time unitshorizon / Lyapunov time: –

t_{\text{horizon}} = \frac{1}{\lambda}\ln\!\left(\frac{L}{\delta}\right)

α0.9These are the article's own six settings from §9's table. At α=0.9 the plot also draws the article's own 170-trial convergence data (§9's Figure 5) as a grey reference line, so you can watch your own live run settle onto the same published curve.trials to run50

random seed

trials run: 0 ejected: –fraction: –± SE: –mean E: –

Every trial here is the article's real escape test: a body counts as ejected once it clears 20 length units from the center of mass of the other two, still receding, with positive specific energy relative to that pair. Twenty time units is a hard cutoff, not a verdict on eventual fate — see §9 of the article for why a triple that survives the window is reported as “not yet ejected,” never as permanently stable.