If you have done some reading on chaos you will be very familiar with
the startling and very beautiful diagrams mathematicians and scientists
are able to come up with. Though the preceding function looked very
simple, a sampling of its output will result in just one of those diagrams.
In order to view the full measure of what the logistic map is capable of
I have modified the previous code (see code) to
run through a wide range of m values, namely
1.5 to 4.0. You will notice that these values include both the chaotic
and the non chaotic ranges of the function. What is interesting is
that now the function appears to keep to a definite range of possible values,
it is not merely sprayed all over the map, but appears to keep to several
distinct territories.
Another property of chaotic functions most people are familiar with is the property of self-similarity, which is the fact that the above shape is actually made up of tiny exact replicas of itself. For example look at the following graph;
The same graph as above right (only flipped, whoops)? It actually is not, a closer look at the range of x and m values reveals that it is only a small piece of the above, in fact merely about a third. Now for an even closer look;
Now in addition to changing the programs range of output, I needed to change the program so that it would run to a greater degree of precision at the "close-up" values. A lot of the points you see on the above graph are details too small to be shown on the previous. If you were to create a computer program which outputs something like a few million points however, you would merely need to zoom in and see each layer of self-similarity. Even at that height of precision though you would not see everything, the logistic map has infinite precision, down to the smallest scale imaginable, while human life must be satisfied with a mere few billion processor cycles. Just to get a good appreciation of this depth let's look at a scale which is really too small to be seen on the very first graph;
An interesting animal, the bifurcation diagram. Check this space later for further investigations, namely to the Lyapunov Exponents.
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