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