Bifurcation Diagrams

Well, most likely you already heard something about fractals called 'Bifurcation diagrams' or 'Verhulst diagrams'. If not, well, now you've heard something...

Perhaps you've even seen such a fractal. It does not look very aesthetic, so most people do not experiment with it. It makes much more fun creating beautiful images not caring about mathematical sense. If you agree to those people then there is no need to read on.

But if you don't agree with them, read on and try to understand the sense behind these diagrams.

Bifurcation diagram or Verhulst diagram?

Sometimes you will read about bifurcation diagrams, sometimes about Verhulst diagrams, do these two names mean just the same or are there differences? Well, as far as I know 'Bifurcation diagram' seems to be a general statement referring to a dynamic system which has bifurcations, whereas 'Verhulst diagram' most of the time simply references the model examined by Mr. Verhulst, namely y=a*x*(1-x) ...but I guess he examined many models.

The Model of Verhulst

Lets examine the model of Verhulst, an example of a bifurcation diagram. It describes how a population of animals (e.g. insects, hares, or something like that) grows from one generation to the next. This model is one of the most simple ones, and additionally it is very interesting and important (for science) so that very much has been written about it in almost every kind of magazines.

The model looks just like follows:

(1)    x_n+1 = a * x_n * ( 1 - x_n )

with 0 <= x_n <= 1 and 0 <= a <= 4

The biological meaning of x_n is the relative amount of animals in the n-th generation, where 1 specifies the maximal amount of animals possible and 0 no animals at all. The index n specifies the time which goes by in small steps. The parameter 'a' is a measure for the population growth. If a < 1 then the population will die out.

Now how can we work with this model? Lets start with an example: Lets set x=0.1, a=1.5, n=0, then lets apply the model to these values iteratively:

population (xn)

0.1

1

a*x0*(1-x0) = 1.5*0.1*(1-0.1) = 0.1350

2

a*x1*(1-x1) = 1.5*0.1350*(1-0.1350) = 0.1752

3

0.2167

4

0