A chaotic system is one in which infinitesimal differences in the starting conditions lead to drastically different results as the system evolves.

Summarized by mathematician Edward Lorenz, “Chaos [is] when the present determines the future, but the approximate present does not approximately determine the future.”

There’s an important distinction to make between a chaotic system and a random system. Given the starting conditions, a chaotic system is entirely deterministic. A random system, on the other hand, is entirely non-deterministic, even when the starting conditions are known. That is, with enough information, the evolution of a chaotic system is entirely predictable, but in a random system there’s no amount of information that would be enough to predict the system’s evolution.

The simulations above show two slightly different initial conditions for a double pendulum — an example of a chaotic system. In the left animation both pendulums begin horizontally, and in the right animation the red pendulum begins horizontally and the blue is rotated by 0.1 radians (≈ 5.73°) above the positive x-axis. In both simulations, all of the pendulums begin from rest.

For more information on how to solve for the motion of a double pendulum, check out my video here.

Mathematica code posted here.