In the process of learning about non-linear dynamics I created this image:

The bottom axis is the parameter r in the logistic map:

```
X = r*x(1-x)
```

The vertical axis shows the x-values produced by running the above equation many times. The equation starts with x = 0.2 (see code for all numbers and stuff). The output X from one iteration of the map is fed back into the map as x. These X values are being plotted. For each parameter r the map is being iterated 800 times and the first 400 iterates are being hidden. This removes the changing part of the x-values, and shows where the values tend to settle (or not) over a long period of time.

So for low r the equation results in an x-value that is fixed. As r increases the long-term behavior jumps to a osciallatory behavior with a low and a high value (period-2). Then it doubles, and again and again. But the doublings get closer and closer, so it is hard to see on the plot. For higher r, the x-values never settle down into a periodic pattern. BUT if you zoom into that non-periodic region, it has periodic sections nestled next to non-periodic sections.

Not only that, there is a section with period-3. Thats the sort of empty band near r=3.8. This period 3 section seems odd. Here it is zoomed in.

Notice that there are similar bands nested in the fuzzy parts (like at r=3.8). This keeps happening as you zoom in! So non-periodic sections have little chunks of order in them.

For some r you will see the progression of x-values fall into sections of period-3, and then back into chaos, and even little sections of higher-periods leading to chaos. I am pretty sure that happens because rounding errors make it so that “super zoomed in” is not possible. My understanding is that the iterated logistic map is *sensitive* to those errors. Even if I type in a r-value with a ton of decimals I can’t exaclty pin down those tiny sections of order amid the “chaos”. So instead the behavior of the system has pockets of order *and* chaos. Holy cow!

You can see this happen using x_0 = .2 and r = 3.82 in this webapp of the logistic map

Not sure how yet how these observation can be of practical use in modeling stuff outside, but that remains to be seen!