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