DIRECTORY / SANDBOXES / DOUBLE-PENDULUM.CHAOS

DOUBLE PENDULUM CHAOS DYNAMICS

Extreme sensitivity to initial conditions simulated via 4th-order Runge-Kutta numerical integration.

🌀 LIVE PHASE SPACE (θ vs ω) θ₁ / ω₁ & θ₂ / ω₂

60 FPS

// PENDULUM MODE

// 4-REGIME PLUS LABORATORY RK4

// GRAVITY PRESETS

GRAVITY ACCELERATION (g) 9.8 m/s²

SIMULATION SPEED (TIME SCALE) 1.00x

DAMPING (FRICTION) 0.0002

TRAIL PERSISTENCE LONG

// TRAIL COLOR PALETTE

TOTAL MECHANICAL ENERGY: 0.00 J

DIVERGENCE RATE (λ): 0.00 ENERGY DRIFT: ±0.01%

💡 TIP: Grab and fling pendulum bobs with your mouse/touch. Explore 100-pendulum ribbons and live phase portraits in Plus Mode!

📚

Lagrange's equations of motion, Runge-Kutta 4th Order sub-stepping, and the butterfly effect.

ACADEMIC & ALGORITHMIC REFERENCE

// HISTORICAL ORIGINS

A quintessential model in nonlinear dynamics and deterministic chaos, demonstrating extreme sensitivity to initial conditions (Lyapunov exponent).

// GOVERNING EQUATIONS

\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta}_i}\right) - \frac{\partial L}{\partial \theta_i} = 0

// BROWSER IMPLEMENTATION

Solves coupled nonlinear second-order differential equations via 4th-Order Runge-Kutta (RK4) integration with 10 sub-steps per frame.

// GUIDED EXPERIMENTS TO TRY IN THIS SANDBOXNORMAL

1100-Swarm Lyapunov Divergence Ribbon

🎯 Action:Switch to 100-LYAPUNOV regime and observe the ethereal glowing ribbon.

✨ Observe:100 coupled pendulums differing by only 0.00002 radians stretch exponentially into an organic phase-space ribbon.

2Real-Time Phase Space Poincaré Portrait

🎯 Action:Observe the bottom-right phase canvas plotting (θ₁, ω₁) in cyan and (θ₂, ω₂) in pink.

✨ Observe:Complex toroidal attractors and stroboscopic trajectory loops reveal deterministic chaos geometry.

3Fractal Flip-Time Basin Explorer

🎯 Action:Open the FRACTAL BASIN regime and click any coordinate on the 80x80 initial condition map.

✨ Observe:The pendulum launches instantly into the corresponding phase trajectory with matching flip time.

// CURATED PAPERS, RFCS & RESOURCES

Standard derivation of generalized coordinates and kinetic/potential Lagrangian mechanics.