Looking at the logistic map a little at a time
This weekend I wrote two blog posts about the logistic map, working left to right on the diagram below.
The horizontal axis in the image is r, the parameter in the logistic map, running from 0 to 4.
Yesterday I wrote The simple part of the logistic map isn’t that simple, looking into the diagram for r going from 0 to 3. The diagram represents stable convergence points for iterations of the map rx(1 − x). What I’m calling the simple part is the flat part on the left end, corresponding to r between 0 and 1, and the arc in the middle, corresponding to r between 1 and 3.
Starting from almost any initial point, iterations converge to these curves in the diagram. The details of how the iterations converge are interesting. Convergence is fast and monotonic at first, then slows way down at r = 1, then speeds up, then starts to oscillate after r = 2, then slows way down again at r = 3.
When r > 3 we start to have bifurcations. This morning I published a post looking at just the first fork, going into some detail regarding what’s happening in the region where r is between 3 and 3.44.
Enjoy!