A while back @alexanderalexis wrote about the beauty of order. So that inspired me to write about the beauty of chaos. In this post I will give an introduction to deterministic chaos.
We will be looking at the logistic map. You can think of this map as describing time evolution of a one dimensional point. Here time moves at discrete steps so time 0, time 1, time 2, time 3 etc. We will be looking at how the position of the point changes of these time steps, this is called its evolution. The logistic has a parameter which we can tweak to generate different evolutions of our point over time. Instead of looking at the whole evolution will be just looking at what the evolution looks like after a long time. We then end up with the image below:
On the x-axis is the parameter, mu, and on the y-axis the evolution. I starts with a single point then as we increase the parameter it doubles we get 2 points for a single parameter, then it doubles again, we get 4 points, then doubles again, 8, then 16, then 32, etc..but then from the red line the points go crazy! We can zoom in a bit and then see what is going on:
The parameter corresponding to the red line is called the onset of chaos. Not all states beyond this parameter are chaotic. You can see there are parameters where it seems to be filling a line-segment and then suddenly it reduces to a couple of points. It seems to be fairly unpredictable what state corresponds to which parameter.
What is chaos?
So how to identify chaos? Informally, points which start very close exhibit different behaviour as time passes. In the image below we follow the chaotic evolution of two points which start very close to each other. We observe that for about the first 10 time steps they look very similar by then over time evolution 1 and evolution 2 behave completely different.
The evolution of a single point is deterministic. Meaning that the past determines all future positions of the point. Chaos here means that an inexactness in determining the starting position will yield a completely different evolution. Quoting Lorenz, the father of deterministic chaos,
Chaos: When the present determines the future, but the approximate present does not approximately determine the future. source
What is the driving mechanics of chaos?
We look at the mu=1 case. We graph the state at time t against the next state at time t+1. If we study what happens to the curve projected on the x-axis versus the y-axis we see that the curve gets stretched and folded. The stretching makes sure that points which are close to each other will move away from each other and the folding makes sure that the points don't run off to infinity.
The logistic map is a toy model. There are many more complicated systems that exhibit chaos. A famous example is the butterfly attractor in the Lorenz system. The Lorenz system is derived from a weather-type model. So probably it comes as no surprise that in weather modelling chaos plays an important role. To me it is remarkable that chaos underlying complicated weather models can be captured by something as simple as the equations for the logistic map. But this is one of the marvels of math: a couple lines of equations can contain a whole theory.