Chaos 1

Many of us must have looked into chaos theory because it seems so mysteriously promising, but how are we supposed to relate it to the real world? After all it can seem somewhat remote, complex and mathematical.

What helped me with the what does it all mean aspect of chaos theory was putting the logistic map into MS Excel.

Xn+1 = rXn(1-Xn)

Very easy to do and just fiddling* around with it brought home to me how a simple equation can become amazingly complex when the next result depends on the previous value. As we see in natural systems of course.

* fiddling - a mathematical term.

I use a bubble graph because I think it highlights certain features such as bifurcation quite well. Here are a few Excel bubble graphs for different values of r with a seed value of X = 0.3.

Firstly a simple plot with r = 2.00

Next we increase r to r = 3.20. X increases then we see a bifurcation where X settles into an oscillation between two stable values.

When we increase r to r = 3.50, we see two stable values of X split into four. A rapidly progressive series of bifurcations being characteristic of the onset of chaotic behaviour.

Increasing r to r = 3.55 leads to more pronounced bifurcations.

Increase r to r = 3.60 and we see the onset of chaotic behaviour. Remember that this is a plot of a very simple equation in MS Excel. 

Increase r to r = 3.70 and the graph is more chaotic, although there are obvious patterns such as an upper and lower boundary imposed by the mathematical structure and values chosen. 

Round about r = 3.82 to 3.86 we see an island of regularities such as r = 3.851.

If we then plot two graphs with r= 0.390, X = 0.300000000 and X = 0.300000001 we see the graphs diverge after about 30 iterations. The red bubbles are initially hidden by the blue, but soon become visible as values of X diverge. This is a graphical illustration of the so-called butterfly effect - major changes evolving from minutely different starting values.