Release two nearly identical pendulums. Watch certainty split into two universes — then measure exactly how fast prediction disappears.
An interactive laboratory for understanding and measuring nonlinear dynamics.
1,597 tests SciPy-validated reproducible by design
First, follow a single release. The reference is the state we will keep unchanged while we ask what one tiny difference can do.
The still image preserves the same reference path when motion or graphics preferences keep the live renderer off.
Reference trajectory · cyan · unchanged Reference trajectory · static viewThe violet trajectory copies the reference, then adds Δθ₁ = 8×10⁻⁴ rad to its first angle. Everything else stays identical.
The static traces preserve the reference and the one-variable perturbation without implying that animation is running.
At first the two paths overlap. Then the same deterministic rules carry them apart. Cyan always means reference; violet always means perturbed.
Cyan marks the reference and violet marks the perturbed path; their growing distance is visible without relying on motion alone.
The console below reports the wrapped angular difference |Δθ₁(t)| for a planar double pendulum. Enter its exact reference state and perturbation, then continue that exact planar experiment in the full Lab without retyping it.
The hero uses a constrained double-spherical model at 240 Hz. The hands-on console below integrates a planar double pendulum with RK4 at dt=0.001 and samples its trail at 150 Hz, matching the experiment continued in the full Lab.
Enter exact values or use the sliders for quick exploration. The reference and perturbed states are named, not left for colour alone to explain.
Referenceθ₁ 2.18 rad · θ₂ 2.64 rad · ω₁ 0 · ω₂ 0
Perturbedθ₁ 2.181 rad · θ₂ 2.64 rad · ω₁ 0 · ω₂ 0
Each entry opens the smallest useful workspace for that goal. You can reveal more controls later without restarting the experiment.
Release two double pendulums a hair apart and they trace the same arc — until they don't. Trajectories peel away exponentially, prediction collapses into noise. That rate of divergence is the largest Lyapunov exponent, and measuring it honestly — every parameter disclosed, every uncertainty owned — is the entire discipline.
Identical start · 1.0e-3 rad apart → exponential divergence
The Lab is organised around four jobs. Method names stay one level deeper until you need them.
Choose a pendulum system, set exact initial conditions, and watch its state evolve. Presets give beginners a safe start; direct controls remain available for deliberate experiments.
Move from “the paths look different” to quantities that describe divergence, recurrence, stability, and phase-space structure—with uncertainty where the method defines it.
Compare resolution, convergence, residuals, conservation, and independent references before treating a computed pattern as a scientific conclusion.
Carry exact setup data in a link, save manifests and research bundles, and keep each public result tied to source evidence and explicit caveats.
The advanced workspace includes fifteen primary integrators, Lyapunov spectra and vectors, Poincaré sections, recurrence analysis, FTLE, basin entropy, continuation, data-driven models, and CPU-oracle-gated GPU paths. Each method explains what it measures and where it can fail.
Each launch carries its intended workspace and mission hint into the lab. A five-step spotlight tour greets first-time visitors, every menu entry explains itself in one plain-language line — in English or Korean — and a command palette (Ctrl+K) reaches everything.
A focused simulator: the live pendulum, one-click presets from Butterfly to Whirling, and the safest physical controls. No jargon, no clutter — just motion.
Adds the analysis workspaces: energy and Lyapunov plots, chaos maps, bifurcation sweeps, 3D phase space, validation runs, and exports — with every method labeled by what it actually measures.
The full surface: chaos diagnostics, Trust Inspector provenance on every quoted number, reviewer kit, governance, research bundles, and the certified workbench with persisted studies.
The interactive laboratory is the primary product. These sections document the broader research library and the separate engineering portfolio connection for readers who need them.
The headless core ships as a typed library with a CLI for batch studies. Every frontier module carries its own falsifiable test contract — closed-form anchors, convergence orders, or cross-method agreement.
The analytic chaos threshold vs the measured period-doubling onset — an engine-sized gap map, certified against literature anchors.
Multipliers on corrected periodic orbits, Mathieu stability tongues, arclength continuation with branch switching, Neimark-Sacker tracking.
Data-driven operator views of the flow: dynamic mode decomposition and Hankel-alternative analysis on a shared thin-SVD core.
Sparse regression rediscovers the equations of motion; polynomial-chaos surrogates yield analytic Sobol sensitivity decompositions.
Restarted thick-restart Lanczos for the symmetric case and an Arnoldi–Schur solver for non-symmetric spectra — matrix-free, test-pinned.
Coupled pendulum chains reproduce the analytic dispersion relation — the same normal-mode physics that underpins solid-state phonons.
Finite-dimensional quantum Floquet quasi-energies beside their classical chaos counterparts — one engine, both regimes.
Stochastic resonance, Euler–Maruyama ensembles with Welford moments, and Kramers escape rates for thermal-noise physics.
Pendulum dynamics and semiconductor devices solve different equations, but trustworthy simulation demands the same engineering habits: expose solver assumptions, quantify discretization error, and validate against an independent reference.
Implicit integrators report convergence instead of silently accepting a bad Newton step — the same honesty required by nonlinear device solvers.
dt-halving, order checks, and energy-drift curves turn numerical resolution into evidence rather than a hidden default.
SciPy, symbolic identities, literature anchors, hashes, and replay manifests form the equivalent of a reproducible golden deck.
Every integrator is cross-checked against closed-form, energy, and reference-method criteria, then externally cross-validated against an independent SciPy DOP853 reference for both the double and triple pendulum.
| Evidence | Measured result | Scope | Status |
|---|---|---|---|
| Best energy profile | DOP853 8(5,3)1.612e-14 max relative drift | 14 methods profiledCompare each method against its documented order and structure-preservation behavior. | measured |
| SciPy DOP853 agreement | ~6e-14 | regular reference casesChaotic trajectories use time-amplified tolerances and are not claimed bitwise-identical. The flagship study centers one frequency, uses a restricted scan and warm-started attractor branch, may miss coexisting basins, and uses first-order Melnikov theory. The independent Python check uses coarser steps and finite-difference monodromy, supporting tolerance-level reproduction rather than bitwise equality. MATLAB validation is optional and unpinned; any release claim using it must pin release, toolbox, script, and checksum manifests. | validated |
| Period-doubling anchor | 1.066372 vs 1.066300 | computed vs literature | passed |
| Physical GPU matrix | 1 / 3 vendors | NVIDIA + AMD pendingExpired or undated driver evidence cannot retain a validated hardware level. Partial vendor coverage is a measured subset and cannot imply complete GPU certification. The matrix is intentionally incomplete until missing physical vendor runners upload evidence. Software adapters do not satisfy this contract. Physical GPU evidence currently covers Intel only; NVIDIA or AMD behavior and complete vendor certification must not be inferred. The N-chain GPU path is planar N at most eight and hybrid; nonlinear trajectories and Jacobian tapes remain CPU f64, so it is not a full-GPU trajectory implementation. The AssemblyScript f64 SIMD N-chain path remains an unpromoted candidate until parity and speedup evidence satisfy its promotion gate. SharedArrayBuffer side-plot claims require Cloudflare mirror COOP and COEP headers plus a positive crossOriginIsolated deployment probe. This evidence is expired and retained only as historical information. | informational |
| Public release chain | withheld | withheldFailed or unavailable endpoint evidence cannot support a publication claim. npm registry does not resolve this exact package version. No matching public Zenodo DOI record resolves for Pendulum Lab. SharedArrayBuffer side-plot claims require Cloudflare mirror COOP and COEP headers plus a positive crossOriginIsolated deployment probe. MATLAB validation is optional and unpinned; any release claim using it must pin release, toolbox, script, and checksum manifests. Windows headless WebKit audio instability is environment evidence; user-facing audio needs a fallback and an explicit browser matrix. | withheld |
Evidence current · 2 validated · 1 measured · 2 informational · 1 withheld · valid through 9 Sept 2026
Windows headless WebKit audio instability is environment evidence; user-facing audio needs a fallback and an explicit browser matrix. Plus Chromium, Firefox, WebKit and mobile end-to-end suites, with a machine-readable Stryker aggregate of 65.32% · low band · 29 shards (informational). Expired or undated mutation evidence cannot retain a validated public level. A mutation score below the quality target remains a measurement, not a validated quality claim. Below the 70% quality target; the 65% regression floor is enforced. This evidence is expired and retained only as historical information.
Regular orbits agree to ~6e-14 over 20 s; chaotic orbits to the e^{λ₁t}-amplified tolerance floor. Chaotic trajectories use time-amplified tolerances and are not claimed bitwise-identical. The flagship study centers one frequency, uses a restricted scan and warm-started attractor branch, may miss coexisting basins, and uses first-order Melnikov theory. The independent Python check uses coarser steps and finite-difference monodromy, supporting tolerance-level reproduction rather than bitwise equality. MATLAB validation is optional and unpinned; any release claim using it must pin release, toolbox, script, and checksum manifests.
Engine-measured against the published value of 1.0663 — a literature anchor, not a fit.
These highlights are synchronized from the simulation repository at the same evidence commit used by this page.
the entire docs/ tree moved to documents/ via git mv (history preserved).
the companion pendulum-landing page (EN + KO) and its docs were repointed to .../blob/master/documents/...
older CHANGELOG entries keep their original docs/...
No install. No account. Open it in your browser, choose a workspace, run it, and export diagnostics anyone on earth can reproduce.
Pick Beginner, Student, or Research, then a system from double pendulum to N-link, driven, and elastic — and one of fifteen primary integrators, each labeled with its order and character.
Dial in initial conditions and dt, then watch the trajectory, energy drift, and residuals update live. Summon chaotic, periodic, or resonant regimes from presets in a single click.
Run the Lyapunov spectrum, Poincaré sections, and bifurcation sweeps, then export a hash-verifiable research bundle — PNGs, CSVs, SHA-256 checksums, and a replay manifest.
The full simulator runs in your browser — fifteen primary integrators, every analysis tab, and hash-verifiable research exports. No install, no account. Just the unvarnished mathematics of chaos.
