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 x 2. When x = 1, x 2 = 1, and when x is 1.1, x 2 is A small change in x gives a (relatively) small change in x 2. When x = 1.01, x 2 = : A smaller change in x gives a smaller change in x 2. 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. P n = kP n-1 (1  P n-1 ), where k > 0, 0 < P 0 < 1. P n 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. carom/carom-files/carom-2-7.xlsx

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 where the populations that 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  A mathematician called Feigenbaum showed that this sequence converges, to a number now called (the first) Feigenbaum’s constant, .

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, American, (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,