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In this homework, we will explore the Feigenbaum number, also known as the Feigenbaum constant
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This universal constant applies to all one-dimension, one-parameter maps with a single quadratic maximum
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This constant describes a limiting ratio of each bifurcation interval to the next between every period doubling, as described by this formula
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In the following four problems, we will make a first estimate of the Feigenbaum constant; that is, we will calculate this equation
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To do this, we will need to calculate b1, b2, and b3
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Where b1, for example, is the r value where the dynamics bifurcate from a fixed point to a two-cycle
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Similarly, b2 is the r value where the dynamics bifurcate from a two-cycle to a four-cycle, and so on
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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
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So b1 is equal to 3. This is the answer to 1a
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To obtain b2, we need to use the bifurcation tool you generated in Homework 2.2 to zoom in to this region
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If we do that, we see that the bifurcation from two-cycle dynamics to four-cycle dynamics occurs at 3.44948
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This is the answer to 1b
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Similarly, to obtain b3, we need to zoom in to this area
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which is the area where it seems a four-cycle to an eight-cycle occurs
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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
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And this gives us the answer to 1c
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We can now combine b1, b2, and b3 to create a first approximation of the Feigenbaum constant
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We get a first approximation of 4.7514
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This is the answer to question 1e