In this homework, we will explore the Feigenbaum number, also known as the Feigenbaum constant

This universal constant applies to all one-dimension, one-parameter maps with a single quadratic maximum

This constant describes a limiting ratio of each bifurcation interval to the next between every period doubling, as described by this formula

In the following four problems, we will make a first estimate of the Feigenbaum constant; that is, we will calculate this equation

To do this, we will need to calculate b1, b2, and b3

Where b1, for example, is the r value where the dynamics bifurcate from a fixed point to a two-cycle

Similarly, b2 is the r value where the dynamics bifurcate from a two-cycle to a four-cycle, and so on

Weve seen several times now, both in homeworks and in quizzes, that the bifurcation from fixed point dynamics to a two-cycle occurs at r = 3

So b1 is equal to 3. This is the answer to 1a

To obtain b2, we need to use the bifurcation tool you generated in Homework 2.2 to zoom in to this region

If we do that, we see that the bifurcation from two-cycle dynamics to four-cycle dynamics occurs at 3.44948

This is the answer to 1b

Similarly, to obtain b3, we need to zoom in to this area

which is the area where it seems a four-cycle to an eight-cycle occurs

If we do this with our bifurcation tool, we see that the bifurcation from a four-cycle to an eight-cycle occurs at 3.54409

And this gives us the answer to 1c

We can now combine b1, b2, and b3 to create a first approximation of the Feigenbaum constant

We get a first approximation of 4.7514

This is the answer to question 1e