# Thanks

And thanks to the guest who liked my Frozen saga of Florida Woman!

Writing about writing isn’t that interesting, so I’ll talk about what else I’m doing.

I’m running iterative solutions of the logistics map through an FFT. Let’s unpack that (talk to the duck programming).

The logistics map is the equation Xn+1 = a * Xn * (1-Xn). The constant a is selected at the beginning, and describes the behavior of the function. I’ve been using 3.6. The first input is Xn, where n is 1. So X1 is the first input. In programming, it’s often X0, but this difference is not meaningful.

Suppose I use 0.5 as X1. X2, or X1+1, is 3.6 * 0.5 * (1-0.5) or .9. X3 is 3.6 *0.9*(1-0.9) or 0.324.
N X
1 0.5
2 0.9
3 0.324

Each number results in a deterministic iteration that gives the next number. One can see that for n=1, X=0.5 is the same as n=3456, X=0.5.

With me? The n doesn’t matter. Each X must give the same next X if the coefficient a remains the same.

But it is very difficult, if not impossible, to conclude what the nth X will be, even if you have an earlier X, without going through each step. So if X1 =0.5, I can’t figure out what X5324 is without calculating all 5324 steps in between. Mathematicians call this chaos, and physicists call it non-linear behavior.

But that only happens during chaotic behavior, and whether or not the series will give chaotic behavior is determined by a. So for some values of a, the numbers will bounce between two common numbers. (Sometimes more than two. Any bouncing is called harmonic behavior) Then you don’t need to know what Xn-1 is to figure out what Xn is, you just need to know the n. This is exactly different from chaotic behavior.

What’s really interesting is that these two different behaviors, chaotic and harmonic, depend exclusively on the parameter a, and there seems to be a hard and sharp distinction between them. No one knows why. Wikipedia has a good article on it here.

I’m using C++ to do a bunch of these mappings (calculating lists of Xn’s) and putting the lists through little filters. It’s just a project of curiosity, and it buffs up my C++, which had atrophied. The C suite is probably the highest scoring programing language across three metrics: commonly used, powerful, and free.