A double pendulum is two rigid arms connected end to end, swinging freely under gravity. Despite its simplicity, it is one of the classic examples of deterministic chaos. The system obeys exact equations, yet tiny differences in starting conditions lead to completely different trajectories.
Two pendulums are launched simultaneously from nearly identical positions. Pendulum A starts at angles (2.5, 2.500) radians. Pendulum B starts at (2.5, 2.501). The difference is 0.001 radians, less than a tenth of a degree. For the first few seconds they move together. Then they diverge, dramatically and irreversibly.
This is sensitive dependence on initial conditions, popularly called the butterfly effect. The divergence grows exponentially, characterized by the system's Lyapunov exponent. No amount of measurement precision can predict the long-term behavior. The physics is exact, the predictions are not.
The simulation uses RK4 integration (fourth-order Runge-Kutta) with multiple substeps per frame for numerical stability. The equations of motion come from Lagrangian mechanics, treating the system as two coupled nonlinear differential equations.
Wikipedia: Double pendulum