Required Project II: exploration of the logistic map

1) make a plot like Figure 2.2 showing the set of x values the map produces after a large number of iterations. Write an account of how you made this map and turn in source and .class files you used. Explain how you avoided plotting early x values that depend on the starting value of x. Your plot doesn't have to include scales like Figure 2.2.

2) Analyze the bifurcation point in r where a stable fixed point gives way to a two-cycle. Find the value of r, called r₂, and the corresponding x^* analytically.

3) Find the two stable x values numerically for some value in the range of r for which two-cycles are stable, using the Newton Raphson method. Do not pick an r value that is the same as someone else's. You can start using the Newton Raphson applet presented in class. However, you don't want to type in the new trial values for x manually throughout the whole iteration, but you want this to be done in some kind of a loop construction. Formulate a condition for the loop (i.e. the Newton Raphson iteration) to stop.
Find the four stable x values numerically for some r value in the range of stable four-cycles, using the Newton-Raphson method.

4) stability near bifurcations. Find analytically the Floquet multiplier (The derivitive Λ defined in Chapter 3) of the 2-cycle of the logistic map at r₂, where the two cycle first appears. Use your work on 3) to write an applet that plots the Floquet multiplier as a function of r for the N-times iterated map between r_N, where the N (=2^k) cycle is born, and r_2N, where this cycle becomes unstable, for N = 2, 4, 8. Compare with your analytic result at r₂. Can you formulate a general rule for the stabilities of the 2^k and 2^k+1 cycles at the bifurcation point where the 2^k cycle becomes unstable and the 2^k+1 cycle is born? Such a rule was proposed in class.

5) There are several changes of behavior at specific r values for r₂ < r < 4. Find three of the most dramatic changes, other than those found in 4). Describe the change that occurs and report their r values to 2 significant figures. One of these should be the point at which the number N of values in the cycle goes to infinity.

6) Three-cycles. Write an applet that finds the elements of the cycles of length three in the logistic map for r=3.95 using the Newton-Raphson (or secant) method. How many period-3 cycles are there? What are the elements of each cycle? Are these cycles stable or unstable? Answer the same questions for r=4. As r decreases what happens to these three-cycles? At which value of r?

6) Using the theta-i representation for r=4 described in chapter 3, find two N-cycles, state their theta values and determine their stability.

7) Chaos. Let us imagine that we compare two different developments of population at r = 4 starting from almost the same values of x. One population starts with the value x_o, the other with the value y_o = x_o + ε, where ε is very small. Then the populations develop through many iterations. At the beginning the populations remain close to one another. Then, after some large number of steps they begin to be more and more different. In fact after J steps the values x_J and y_J which respectively developed from x_o and y_o have begun to differ by an amount of order unity. Question: How does x_i-y_i depend upon ε and i? You may take i to be much less than J, so that x_i-y_i remains small. (You may answer this question by writing a program, or you can take advantage of the variable change discussed at the end of Chapter 3 and find a solution analytically.) You have noticed that x_i has a sensitivity to the value of x_o which grows very rapidly as a function of i. This kind of sensitivity to initial conditions is a hallmark of chaotic systems.