2,500 double pendulums

Chaos vs order

Every pendulum obeys identical physics and differs only in its two starting angles — lower arm across, upper arm down, both spanning −π to π. Watch determinism dissolve at the edges while islands of order hold.

Speed1×

Loop

Grid

Trails

lower initial angle −π → πupper initial angle: π top, −π bottom

t = 0.0 s

Inspector — tap any cell

Initial angles: –

First flip: –

Twin separation: –

The faint trace is the path of the lower tip.

About the physics

Each cell is a frictionless double pendulum with equal masses and arm lengths, integrated with fourth-order Runge–Kutta at 240 steps per simulated second. Nothing is prerecorded — every frame is computed live.

Hue colors each pendulum by the current angle of its lower arm, so coherent regions flow together. Chaos runs an invisible twin of every pendulum, offset by one hundred-millionth of a radian, and colors by how far the pair has drifted apart — the signature of sensitive dependence on initial conditions. Flips records when each lower arm first swings over the top; the boundary between early and never is a fractal.

The calm horizontal band through the middle is low energy: those pendulums swing gently and predictably forever. The corners start inverted, carrying enough energy to go chaotic almost instantly.