www.carom-maths.co.uk. Activity 2-7: The Logistic Map and Chaos. In maths, we are used to small changes producing small changes . . Suppose we are given the function x 2 . . When x = 1, x 2 = 1 , and when x is 1.1, x 2 is 1.21.
The Logistic Map and Chaos
Suppose we are given the function x2.
Whenx = 1, x2 = 1, and when x is 1.1, x2 is 1.21.
A small change in x gives a (relatively) small change in x2.
Whenx = 1.01, x2 = 1.0201 :
A smaller change in x gives a smaller change in x2.
With well-behaved functions, so far so good.
But there are mathematical processes where a small change to the input produces a massive change in the output.
Prepare to meet the logistic function...
Suppose you have a population of mice, let’s say.
As a mathematician, you would like to have a way of modelling how the population varies over the years,
taking into account food, predators, prey and so on.
The logistic function is one possible model.
Pn = kPn-1(1 Pn-1), where k > 0, 0 < P0< 1.
Pnhere is the population in year n,
with k being a positive number that we can vary
to change the behaviour of the model.
see what different populationbehaviours
you can generate as k varies.
Our first conclusion might be that in the main the starting population
does NOT seem to affect the eventual behaviour of the recurrence relation.
For 0 < k < 1, the population
For 1 < k < 2, the population seems to settle to a stable value.
For 2 < k < 3, the population seems to oscillate before settling to a stable value.
For 4 < k, the population becomes negative,
Which leaves the region 3.45 < k < 4.
The behaviour here at first glance does not seems to fit a pattern –
it can only be described as chaotic.
You can see that here a small change in the starting population can lead to a vast difference in the later population predicted by the model.
of this range for k, we see that
curious patterns do show themselves.
For 3 < k < 3.45,
we have oscillation between two values.
For 3.45 < k < 3.54, (figures here are approximate)
we have oscillation between four values.
As k increases beyond 3.54, this becomes 8 values,
then 16 values, then 32 and so on.
For 3.57 < k, we get genuine chaos, but even here there are intervals where patterns take over.
We call the
values of k
that we oscillate between double
points of bifurcation.
If we calculate successive ratios of the difference between bifurcation points, we get the figures in the right-hand column.
A mathematician called Feigenbaum showed that
this sequence converged, to a number now called
(the first) Feigenbaum’sconstant, d.
With the help of computers, we now have that
d = 4.669 201 609 102 990 671 853 203 821 578...
The remarkable thing is that Feigenbaum’s constant appears not only with the logistic map, but with a huge range of related processes. It is a universal constant of chaos, if that is not a contradiction in terms…