Activity 2-7: The Logistic Map and Chaos

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In maths, we are used to small changes producing small changes.

Suppose we are given the function x2.

When x = 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.

When x = 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...

The logistic function is one possible model.

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.

Pn = kPn-1(1 Pn-1), where k > 0, 0 < P0< 1.

Pn here is the population in year n, with k being a positive number that we can vary

to change the behaviour of the model.

Task: try out the spreadsheet below and see what different population behaviours

you can generate as k varies.

Population Spreadsheet

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

dies out.

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 3 < k < 3.45, the population seems to

oscillate between two values.

For 4 < k, the population

becomes negative,and diverges.

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.

For 3 < k < 3.45, we have oscillation between two values.

But if we examine with care the early part of this range for k, we see that

curious patterns do show themselves.

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.

It’s worth examining the phenomenon of the doubling-of-possible-values more carefully.

We call the values of k

wherethe populationsthat we oscillate between double

in number points of

bifurcation.

If we calculate successive ratios of the difference between bifurcation points, we get the figures in the right-hand column.

With the help of computers, we now have that d = 4.669 201 609 102 990 671 853 203 821 578...

A mathematician called Feigenbaum showed thatthis sequence converged, to a number now called

(the first) Feigenbaum’s constant, d.

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…

Mitchell Feigenbaum,

(1944-)

With thanks to:Jon Gray.

The Nuffield Foundation, for their FSMQ resources,including a very helpful spreadsheet.

MEI, for their excellent comprehension past paper on this topic.Wikipedia, for another article that assisted me greatly.

Carom is written by Jonny Griffiths, mail@jonny-griffiths.net