Double Pendulum and Chaos

Release two almost identical deterministic systems and watch a microscopic difference grow into radically different motion.

Near-identical comparisonRunning

System ASystem BTrails trace the second bobs

Simulation time0.00 s

Angular separation0.0000°

Total energy A—

Energy drift0.000%

MotionOverlapping

Angular divergence · log₁₀(degrees)

Mechanical energy · joules

What is a double pendulum?

It is one pendulum attached to another. Each arm influences the other, producing coupled, nonlinear motion.

Sensitivity to initial conditions

The equations contain no randomness. Yet a tiny initial difference can grow rapidly, making long-term prediction difficult while remaining deterministic.

Phase space and energy

A complete state needs both angles and both angular velocities—its point in phase space. The energy graph helps distinguish physical damping from numerical drift.

Model note: Rigid massless rods and point masses are integrated with fixed-step fourth-order Runge–Kutta. Angular separation combines the wrapped differences of both arm angles.