Math 131

Spring 2000

This is due Friday, April 14 at the beginning of class. It is worth 10 points. Papers turned in Monday lose 3 points; papers will not be accepted after Monday. Write neatly or use a word-processor. Your grade will depend on the correctness of your work and how well you explain it. Don't just answer questions! Write a report that makes sense. You don't need to list every calculation, but describe what happens. You may discuss your work with me but not with others in the class.

The scientific field of chaos theory was introduced in assignment #8. You will be introduced to more of the basics of chaos theory in this assignment. Recall that you worked with the logistic equation x_n = x_n-1 ( c- x_n-1 ) with x₀ = 0.5 for various values of the parameter c.

You found in the previous assignment that with c=2.5 the second iterate was 1.5, and all iterates from there on equaled exactly 1.5. We call 1.5 a fixed point. Explain why a fixed point of the relation x_n = f(x_n-1) must be a solution of the equation f(x)=x. Solve this equation for the function f(x)=x(2.5- x). Note that there are two solutions. Find the two fixed points for the general equation f(x)=x(c- x).

Fixed points are classified as being attracting and repelling. To illustrate the difference, start with the function f(x)=x(2.5- x). One fixed point is x=1.5. Starting from an initial value near 1.5, like x₀ = 1.4, compute several iterates and observe how the iterates are attracted to 1.5. The other fixed point is x=0. Starting from an initial value near 0, like x₀ = 0.1, compute several iterates and observe how the iterates are repelled from 0.

Repeat this for c=3.2. That is, find the two fixed points (use your general formula from above), pick initial values near each one and observe whether the iterates are attracted to or repelled from the fixed point. You should find that the iterates are attracted to the 2-cycle {1.6417, 2.558}. We call this an attracting 2-cycle. Recall from assignment #8 that iterates starting with x₀ = 0.5 converged to a fixed point for c=2 and c=2.5, but converged to a 2-cycle for c=3.2. Explain why it seems fair to say that for a "typical" starting value like x₀ = 0.5, the iterates will converge to an attracting fixed point or cycle. Give an example of one atypical starting value for c=3.2 (that is, find an x₀ such that the iterates do not converge to the 2-cycle).

Comparing your results with c=2.5 and c=3.2, you have seen that the larger fixed point changes from attracting to repelling. It turns out that there is an exact value b such that if c<b the fixed point is attracting and if c>b the fixed point is repelling. The value b is called a bifurcation point. By trial-and-error, estimate the bifurcation point for the logistic equation x_n = x_n-1 ( c- x_n-1 ). In your report, list two values of c for which the fixed point is attracting and two values of c for which the fixed point is repelling. Also, comment on how you determined whether the fixed point was attracting or repelling (there are several methods, which require different numbers of calculations).

FYI: researchers in a wide variety of fields are trying to determine if events that were previously considered catastrophic failures might be better understood as bifurcations. For example, a common form of heart attack called fibrillation has puzzled cardiologists because of its seemingly random occurrence. Viewed as a bifurcation, there doesn't have to be a dramatic event causing fibrillation; instead, some parameter in the heart (such as heart rate) might cross over a bifurcation point separating healthy heart dynamics from unhealthy dynamics. Similarly, the Michael Crichton book The Lost World discusses the theory that the dinosaur extinction could have been caused by a seemingly innocent bifurcation point.