By which I of course mean mathematical chaos--sensitive dependence on initial conditions. With the right groundwork, it can be easier than you think! We often teach middle school students about linear and exponential functions using the concept of recursive sequences, sequences in which the next term is generated from the previous one: In a linear sequence, you add a constant to the previous number; in an exponential sequence, you multiply the previous number by a constant instead. This is a great, intuitive framing for illustrating the difference between these two basic models--and curiously, a great way to introduce chaos too!

The logistic map is a simple population model from mathematical biology that pares down the famous logistic equation into a single recursive definition:

$$x_n=r x_0 (1-x_{n-1})$$

\\(x_n\\) is, roughly speaking, the population as a fraction of carrying capacity, and \\( r \\) is an abstract growth-rate parameter between 0 and 4. This is a more complicated definition than what they’ve seen before, but it really only introduces one twist on the exponential sequence: Multiplying by \\( 1-x_{n-1} \\) . We can think of this as a force of decay--the trouble a species encounters as it nears carrying capacity and runs out of resources.

During the 1970s, the logistic map attracted a great deal of attention as a system that became chaotic through doubling bifurcation--limiting behavior that split into ever-more complex loops with increasing frequency until the behavior at any given time was, practically speaking, unpredictable. And it did so surprisingly photogenically as \\( r \\) increased--but not quite quickly enough that students could be expected to discover this behavior themselves working by hand. So, I built this simple Google Sheets document for my class so they could experiment with changing the value of \\( r \\) (cell A1 in the sheet). After allowing them some time to experiment, I asked them for their observations--what did they notice happened at various values of \\( r \\) ? The behavior was clear enough that they could identify many of the well-studied characteristics of the map--the limiting behavior, the period-doubling, the islands of stability.

Side note: I seeded my students with values to try--ones that would produce good-looking values. If I did this again, I don’t think I’d do this! They had little enough trouble experimenting that I think more of their interest could have been captured with a simple “try any numbers between 0 and 4, go!”

I then showed them these two diagrams, asking them to try and make sense of them:

(Bifurcation diagram by Geoff Boeing.)