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.

## 2 comments:

Interesting, how well it applies to very large organisations which when they get into trouble the political answer is usually to make them bigger.

Demetrius - yes, it takes talent to manage a large organisation and keep chaos at bay.

Bigger makes it worse which is why the civil service needs to be smaller and why the NHS is probably not manageable.

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