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🌀 Triple Pendulum

Three-link pendulum on a Blender-modelled brass-and-aluminium rig with a protractor wall: exact Lagrangian mass-matrix dynamics (RK4), a twin trajectory, Benettin Lyapunov exponent, Poincare section, energy graphs and normal-mode checks.

Chaos & Dynamics3DAdvanced60 FPS📱 Mobile-adapted🟠 Blender⇄ 2D version

About this simulation

💡 Did you know?

Adding just one extra link to a double pendulum roughly doubles its Lyapunov exponent (from about 1-3 s⁻¹ to 3-8 s⁻¹ here), meaning a triple pendulum forgets its starting condition and becomes unpredictable in only a fraction of the time.

Frequently asked questions

What makes a triple pendulum chaotic?

Its three coupled nonlinear equations of motion have no closed-form solution and are extremely sensitive to initial conditions, so trajectories that start almost identically diverge exponentially fast, which is the defining signature of deterministic chaos.

What does the Lyapunov exponent λ mean here?

λ measures how fast nearby trajectories separate over time via |δ(t)| ≈ |δ₀|·e^(λt); for this triple pendulum λ is roughly 3-8 s⁻¹, meaning an initial difference of 10⁻⁴ rad grows by a factor of e in well under a second.

Why does the ΔE drift stat matter?

The true physical energy of an undriven pendulum should stay constant, so any visible ΔE drift reflects numerical integration error rather than real physics — a smaller timestep Δt would reduce that drift.

How is this different from a double pendulum?

Adding a third rigid link increases the degrees of freedom from 2 to 3 and roughly doubles the Lyapunov exponent, so the triple pendulum's third bob appears to move almost randomly almost immediately, making it an even sharper demonstration of the butterfly effect.

⚙ Under the hood

Canvas 2DChaosPendulumRK4LyapunovTriple Pendulum

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install