In the course of the series of posts I’ve been writing on fractals, several people have either emailed or commented, saying something along the lines of “Yeah, that fractal stuff is cool – but what is it good for? Does it do anything other than make pretty pictures?”
That’s a very good question. So today, I’m going to show you an example of a real fractal that
has meaningful applications as a model of real phenomena. It’s called the logistic map.
The logistic map is a way of describing the expectations about the size of a population which
is primarily bounded by limited resources. In a population like this, there are two factors which determine the population size and the rate of population growth. The first is the rate of reproduction; the second is the rate of starvation due to lack of resources. There’s a bit of complexity that I’ll handwave past: the rates of reproduction and starvation are coded together into a parameter r; calculating A value for r is a complicated thing, which is dependent on a huge number of factors about the environment and the species being analyzed.
So suppose we’ve got an r value for some species in some environment. We can easily determine what the maximum population is, based on how much resources (food) are present. With that, we can describe the population at any time t by a value between PT between 0 and 1 which represents the saturation of the species – the percentage of the maximum population that exists at time T.
Given an r-value, the population saturation at time T is described by a iterated function system:
PT+1 = rPT(1-PT)
For such a simple relation, this describes a very complex behavior, dependent on the value of r:
- If r<1, the population will die out – the limit of PT as T increases is 0.
- If r>=1 and r<3, the population will converge on a single value, r-1/r.
- If r≥3 and r<sqrt(6), the population will be bimodal – it will oscillate between two values, dependent on the specific value of r.
- If r≥sqrt(6) and r< 3.54, the population will be quad-modal: it will oscillate between
4 different values of r.
- As r gets larger in the range slightly greater than 3.54, it will oscillate between 8 values, then 16, then 32. T r-value range for each power of 2 is smaller that the range for the preceeding value.
- When r reaches roughly 3.57, all semblance of order disappears, and the system becomes
highly chaotic. (This is in the formal sense of chaotic: deterministic, but incredibly sensitive
to initial conditions; with a fixed r, tiny changes in the initial value of P0 create huge changes in the behavior.) Within this chaotic
region, there are places where there are stable oscillations, but they appear in what seem to be
- When r>4, the equation diverges – which means that it’s not applicable for this kind of
So, what does this have to do with fractals?
You can plot a graph of the logistical function, where one axis is P-values, and one
is R-values. This graph is called a bifurcation map, which shows the values in
the oscillation sets for the R-value. If you plot this, it looks like the following example (copied from the Wikipedia entry on bifurcation diagrams):
It’s a fractal. Plow down at progressively smaller regions, and it always maintains
near-perfect self-similarity. And even better – when you cross the line into the chaotic region,
fractals can be used to describe the chaotic behavior, which is pretty much intractable by
other methods. This is typical of many of the real applications of fractals: many natural dynamical
systems that diverge into chaotic behavior can be described using fractals.