The Logistic Map and Chaos www.carom-maths.co.uk Activity 2-9: The Logistic Map and Chaos

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...

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. 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 4 < k, the population becomes negative, 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.

But if we examine with care the early part 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.

that we oscillate between double It’s worth examining the phenomenon of the doubling-of-possible-values more carefully. We call the values of k where the populations that we oscillate between double in number points of bifurcation.

A mathematician called Feigenbaum showed that 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 converges, to a number now called (the first) Feigenbaum’s constant, d. With the help of computers, we now have that d = 4.669 201 609 102 990 671 853 203 821 578...

Mitchell Feigenbaum, (1944-) 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…

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