Two identical double pendulums are released from angles that differ by a tiny ε. Their equations of motion are fully deterministic — same rules, same masses, same gravity — yet the paths separate exponentially fast because the system is chaotic.
|δ(t)| ≈ |δ(0)| · e^(λt) (λ = Lyapunov exponent, λ>0 ⇒ chaos)
θ̈₁, θ̈₂ from the coupled double-pendulum Lagrangian (no closed form — integrated numerically)
- Initial divergence ε — how different the two starting angles are; smaller ε delays but never prevents the eventual split.
- Gravity — stronger gravity swings the pendulums faster, accelerating how quickly small differences compound.
- Damping — friction that bleeds energy out; enough damping eventually pulls both pendulums to the same resting state, ending the chaos.
- Reset — re-rolls a fresh, tiny random ε and restarts both pendulums from rest.
Real-world relevance: this exact sensitivity-to-initial-conditions is why weather forecasts are useless past about two weeks — the atmosphere is governed by equations just as chaotic as this pendulum.