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.