Chaos in computational simulations

I originally studied chaos in the context of pure mathematical thought experiments like the Cantor set. I had no clue that the extremely abstract material from my dynamics class would repeatedly inform the way I write code.

I first experienced applied chaos theory while writing an app that maps light past rotating black holes.

As I’ve seen more computational simulations in HPC applications, I’ve realized that understanding the app’s runtime dynamics is critical to performance, and should determine the app’s design.

What is convergence?

In the previous post on stability, we saw a few familiar properties that indicate stability in the system. However, what we really mean when we are looking for stability is a system that is robust to small rounding errors. In other words, we want convergence.

By exploring fixed points, we will be able to properly define what this means, and build an intuition for what convergence looks like.

Easy indications of a stable system

Some systems can be easily identified as stable, and not chaotic. In these special cases we can easily find hints that indicate that we can sleep easily at night knowing that our system is well behaved.

For our purposes, stable simply means that rounding errors do not change the output drastically.

We’re going to look at a few types of stable dynamics that are very good indications that what we expect is actually happening.

Chaos 101: a small change in input is a BIG change in output

Many systems are chaotic.

Some are only interesting to physics classes, like the double pendulum. Some, like the three-body problem, are concrete physics problems that feel a bit more important. However, the weather is also a highly chaotic system, and the fact that weather prediction is so complex has an immediate effect on all of our daily lives.