Pendulum Lab V10 — Nonlinear Dynamics Research Platform

Main view — running

live

ΔE / E₀

λ (Benettin)

Phase portrait

θ₁–θ₂ angle projection (wrapped)

Angles over time — θ₁(t), θ₂(t)

Poincaré section (θ₁=0, θ̇₁>0) — wheel/pinch zoom

FFT spectrum (log)

Simulation

Speed 1.0×

Timing mode

⚛ System & Initial Conditions ›

System

Type

Seed

Initial Angles

θ₁ (rad) 2.000

θ₂ (rad) 2.500

θ₃ (rad) 1.000

Initial Velocities

ω₁ (rad/s) 0.0

ω₂ (rad/s) 0.0

ω₃ (rad/s) 0.0

⚖ Physical Parameters ›

Masses & Lengths

m₁ (kg)1.00

m₂ (kg)Theory1.00

m₃ (kg)1.00

ℓ₁ (m)1.20

ℓ₂ (m)1.00

ℓ₃ (m)0.80

Environment

g (m/s²) 9.81

γ damping 0.00

🎨 Visualization ›

Trail color

Trail length paint 1200

Phase axis

Glow FX GPU

Long exposure

Interpolated render

Continue while hidden

∫ Numerical Methods ›

Method

dt (s) 0.0030

tol (adaptive/implicit) 1.0e-6

Steps/frame 6

Quality mode

Use worker pool

Auto-quality scaling

👥 Ensemble ›

Ensemble members are perturbed copies of the main trajectory. They reveal chaos: nearby trajectories diverge exponentially.

N copies CPU0

ε (rad) 1.0e-4

♪ Audio Sonification ›

Enable audio

Volume0.08

⬇ Export & Record ›

Data Export

Replay and trajectory export retain the most recent 4,000 sampled frames. Earlier samples are dropped when full; CSV and run JSON exports report the retained and dropped counts.

📊 Live Diagnostics ›

time0.00 s

θ₁ / ω₁—

θ₂ / ω₂—

E₀ / E—

|ΔE/E₀| —

λ Benettin —

verdictanalyzing

memory—

recoveries0

⌨ Keyboard Shortcuts ›

Space pause R reset C trail
P Poincaré E ensemble ←/→ scrub
1-8 tabs ? help Ctrl+K palette

RK4 (cyan) · Leapfrog (orange) · RKF45 (green) · Yoshida4 (purple)

|ΔE / E₀| log₁₀

‖Δx‖ vs RK4 (log)

Benchmark — steps / ms

Full Lyapunov spectrum — QR-orthonormalization (Benettin's algorithm)

Lyapunov Spectrum

λComputation Parameters›

Computes {λ₁,…,λₙ} via QR-decomposition of perturbed orthonormal basis evolved alongside the trajectory. For conservative systems, Σλᵢ ≈ 0.

Renorm Δt0.50 s

Integrate T120 s

ε perturb1.0e-7

📊Spectrum Results›

λ₁—

λ₂—

λ₃—

λ₄—

Σ λ—

Kaplan–Yorke dim —

λ(θ₁, θ₂) chaos map · click to set initial conditions

Bifurcation diagram — θ₂ at Poincaré section vs g

Bifurcation Analysis

∿Bifurcation Parameters›

g min2.0

g max12.0

Steps400

T (sim)60 s

⤳Driven Orbit: Floquet & Pitchfork›

Drive A1.005

Damping γ0.50

◯Neimark–Sacker invariant circle›

NS invariant circle — delayed-logistic map, a: 2.05 → 2.01 (onset 2)

3D phase space (θ₁, θ₂, ω₂) — drag to rotate · scroll to zoom

Phase density (θ₁ vs ω₁) — GPU additive blending

initializing…

Validation Suite

Automated tests verifying determinism, energy conservation, integrator convergence, replay reproducibility, and long-run NaN resilience.

Long-run energy drift by integrator

Committed benchmark:energy curves; bounded oscillation explains why structure-preserving methods matter.

0–1 test translation variables (p_c, q_c) — bounded ⇒ regular · Brownian ⇒ chaos

0–1 Test for Chaos

◐Test Parameters›

Gottwald–Melbourne test on the bounded observable cos(θ₁). Independent of the Lyapunov/variational machinery, so K ≈ 1 (chaotic) / K ≈ 0 (regular) is a genuine cross-validation, not the same computation reported twice.

Samples3000

📊Result›

K—

verdict—

CLV exponents (top) and hyperbolicity-angle distribution (bottom) — Ginelli algorithm

Covariant Lyapunov Vectors

⇌About›

The true (non-orthogonal) Oseledets stable/unstable directions — not the orthogonal Gram–Schmidt frame. The hyperbolicity angle is the minimum angle between an expanding and a contracting CLV; angles approaching 0 signal homoclinic tangencies.

📊Result›

λ₁—

⟨angle⟩—

min angle—

Double-pendulum flip basin — colour = which rod flips first (fractal boundary)

Flip Basins (Fractal)

❋Basin Parameters›

Each initial (θ₁, θ₂), released from rest, is coloured by which rod flips over the top first. The Daza basin entropy (Sb, Sbb) and the Minkowski–Bouligand box-counting dimension quantify the boundary — a measured fractal, not "it looks fractal". Double pendulum only.

Resolution100

WebGPU acceleration (f32, CPU-cross-validated)

📊Result›

Sb—

Sbb—

box-dim—

fractal—

Recurrence plot of cos(θ₁) — long diagonals ⇒ regular · speckled texture ⇒ chaos

Recurrence Quantification

▨Embedding›

Time-delay embeds cos(θ₁) and quantifies the recurrence plot (Marwan et al.). Non-variational, so DET (structure vs noise) and DIV = 1/Lmax (a finite-size λ proxy) cross-check the Lyapunov verdict. ε is fixed at a 10% recurrence rate.

Dimension m2

Delay τ5

📊RQA Measures›

RR—

DET—

LAM—

Lmax—

DIV—

ENTR—

Finite-time Lyapunov exponent σ_T(θ₁, θ₂) — bright ridges are Lagrangian Coherent Structures

FTLE Field (LCS)

⩜Field Parameters›

σ_T = (1/T) ln σ_max(∂x(T)/∂x(0)) — the largest finite-time stretching rate of the flow map, from the singular value of the variational state-transition matrix. Ridges are Lagrangian Coherent Structures (transport barriers). Distinct from the Sweep, which time-averages λ₁. Double pendulum only.

Resolution70

Horizon T3.0 s

LCS ridge overlay

WebGPU acceleration (FD flow-map, f32)

📊Field Range›

σ_T min—

σ_T max—

horizon—

LCS ridge cells—

SDE ensemble: log₁₀ Var[θ₁](t) of the double pendulum under a stochastic torque

Inverse · Surrogate · Stochastic

∿Stochastic ensemble (SDE)›

Langevin ensemble of the double pendulum with additive angular-velocity noise (a stochastic torque), advanced with Euler–Maruyama and accumulated with Welford online moments.

⊶Inverse problem (parameter fit)›

Recover g from a synthetic (θ₁, θ₂) trajectory by nonlinear least squares (Levenberg–Marquardt) with the engine's own RHS in the fit loop, reporting the standard error.

∑PCE surrogate (Sobol)›

Polynomial-chaos surrogate on a total-degree Legendre basis (independent uniform inputs) gives an analytic Sobol variance decomposition of a test response.