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A 0.001-radian change—about 0.057 degrees—in one modeled double pendulum’s starting angle was enough for two browser-simulated trajectories to stop looking alike after seconds. In a 2026 DEV Community article, Lucian (LKB) reports that the paths remained visually aligned for about 5.6 seconds and fully decorrelated by 7.2 seconds. That is an outcome of one simulation and its chosen conditions, not a universal countdown for real pendulums.
What the 0.057-degree difference means
The “0.057 difference” is an angular offset in degrees, not radians. The author says both simulated pendulums began with the simulator’s default angles—173.12° and 178.85° from hanging—and one initial angle was nudged by 0.001 radians, which is approximately 0.057 degrees. The two systems then evolved under the same model.
For the setup described, the article reports about 5.6 seconds of visual alignment and full decorrelation at 7.2 seconds. “Full decorrelation” is the author’s description; the indexed article text does not specify a numerical threshold for deciding when the trajectories count as fully decorrelated.
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The example is a continuous double-pendulum model integrated numerically, not a measurement from physical pendulums. The article’s displayed equations specify equal masses and lengths, gravitational acceleration of 9.8, and a timestep of 1/240 second. The author says the simulator uses a fourth-order Runge–Kutta (RK4) solver to advance the two states.
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The original article is the primary source for these calculations. Its indexed text includes methods and code fragments, but the numerical results have not been independently reproduced here; the available evidence does not establish numerical convergence across different timesteps or experimental validation. Treat the times as the author’s reported output for the stated model and starting conditions.
Why a tiny offset grows—and what the Lyapunov exponent says
A double pendulum is nonlinear: its motion follows deterministic equations, but nearby starting states can diverge rapidly. That sensitivity makes long-term prediction difficult when initial conditions are known only approximately. It does not mean the simulation is generating randomness.
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Lucian reports an estimated largest Lyapunov exponent of approximately 1.095 s−1, corresponding to a Lyapunov time of about 0.91 seconds. The exponent describes an exponential separation rate for nearby trajectories in the modeled system; its reciprocal gives the characteristic Lyapunov time. It is not a promise that every pair of trajectories will separate at that exact rate at every moment. A finite-time visual outcome also depends on the starting state, model, numerical integration and the criterion used to label trajectories decorrelated.
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The author also reports that a larger starting nudge of 0.05 radians produced full divergence at 2.8 seconds, compared with 7.2 seconds for the 0.001-radian nudge. This illustrates how a larger offset can reach a chosen divergence threshold sooner in this run. It does not establish a general proportional relationship between offset size and divergence time.
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How the logistic map illustrates a different route to chaos
The article’s second demonstration uses the discrete logistic map, xn+1 = r xn(1 − xn). Here, the control variable is the parameter r, rather than an initial angle. As r increases, the reported behavior progresses from a stable value to cycles whose periods double, then toward chaos near r ≈ 3.5699.
| Behavior reported by Lucian | Approximate parameter value |
|---|---|
| Period-2 cycle | r ≈ 3.00 |
| Period-4 cycle | r ≈ 3.449 |
| Period-8 cycle | r ≈ 3.544 |
| Period-16 cycle | r ≈ 3.564 |
| Chaos reported near | r ≈ 3.5699 |
These are approximate iteration results reported in the article, not exact universal cutoffs for every numerical procedure. The author estimates successive period-doubling interval ratios of 4.75 and 4.65. Wolfram MathWorld describes the Feigenbaum constant, approximately 4.669, as the limiting ratio of parameter-space intervals in period doubling. A short list of rounded intervals can illustrate the approach to that limit, but it is not itself the limiting value.
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What the two demonstrations do—and do not—show
The pendulum and logistic-map examples make chaos visible in different ways. One evolves a continuous mechanical model and shows initially nearby trajectories separating; the other iterates a discrete equation and shows cycles doubling as a parameter changes. Both help explain sensitive, deterministic behavior, but the article’s particular browser results should not be treated as independent experimental evidence or as a general prediction about physical pendulums.
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