We started in class today watching a Numberphile video about the Feigenbaum Plot and an interesting number that can be found in it that appears to have some surprising universality. The video does a great job of recapping what we’ve done over the past few days, so watch it if you’ve missed anything. You were also given an article to read about the plot and this constant.
The video also makes a point that the pattern we see in the Feigenbaum Plot is not unique to the Logistic function we’ve been iterating. In fact, any function that creates a bound area with the x-axis can exhibit such behavior. With the rest of the period, play around with Paul Fischer-York’s Bifurcation Diagram. Use the dropdown in the upper-right corner to examine diagrams for other functions. Some other functionality:
- Use the Darkness slider to make the image darker and easier to see.
- Left-click and drag a rectangle around any portion of the diagram to zoom in on that portion.
- Right-click anywhere in the diagram to run a series of iterations at that value of a along the x-axis. The software will identify the magnitude of the cycle you’re looking at (though sometimes it makes a mistake, so look at the list as well!)
Use this map to explore the Sharkovskii Ordering mentioned in the article above. Why does that ordering make sense given this picture?