Double Pendulum - How close does it get from the initial conditions?
23Each pixel in this diagram represents the time evolution of a double pendulum with initial angles and angular velocities X0 = (theta1, theta2, 0, 0), from -pi to +pi in either direction. (The time matches the video's. The simulation runs for 40 seconds.)
The brightness of the pixels indicate how close that pendulum gets from its initial condition in phase space X0 (that is, both initial position AND speed are taken into account), with white being "it goes back exactly where it started". As the pendulum moves farther away in phase space, the pixel gets darker. The color mapping is a decaying exponential exp(-0.5*|X(t) - X0|) using Euclidean norm.
In other words, for a given time t, white regions indicate points where the pendulum is completing a closed orbit. As the double pendulum is chaotic, things get messy really quickly.
Music is a harp arrangement of the left-hand part in Scott Joplin's "The Entertainer".