The Chaos of Double Pendulums

What is chaos? Chaos is seen a crowded room, a shattered glass and a uncontrollable dog. At its deepest level, chaos is disorder and unpredictability.

In mathematics and physics, chaos means something very specific:

• sensitive to starting conditions
• nonrepeating

These two ideas mean that chaotic problems are impossible to solve exactly and difficult to solve approximately. But they're very important problems to work on because chaos theory is the reason that things like the weather, the stock market and fluid flow is so hard to predict. If we can get better at predicting hurricanes or stock market crashes, we can do a better job preparing for them.

The atmosphere and the stock market are large systems with many moving parts, but we can find chaos in very simple systems. Pendulums are the essense of regularity. They swing so consistently that they were used by Galileo to make the earliest reliable clock. In the diagram directly below, we add a second pendulum to the end of the first. This adds a slight disruption to the oscillation, but it still swings regularly.

If we increase the length of oscillation, the system takes a few swings, but it soons loses all regularity and becomes untrackable. Below is a demonstration of how sensitive and unpredictable double pendulums are. This system starts out with 50 pendulums almost exactly ontop of each other. Their bobs are just 0.03 degrees appart at the furthest (If you zoom into the image, you should be able to see the edges of the ones in the back). After just a few seconds, these pendulums split and take a totally unique path.

There is a certain beauty to the chaos. The few moments of instability before the bobs diverge is mesmerizing. In the demo below, the paths of the pendulums are plotted behind them.

Below I added a demonstration where you can play with some of the parameters.