November 4, 2004 --- Class 20 --- Continuation of Logistic Map and Chaos

Activities:
	
	Period Doubling

	A graphical technique is useful to find fixed points.  If we plot
		y=x   and  y = f(x,r)
	then any point of intersection will be a fixed point, i.e., 
		x = f(x,r)
	If the abolute value of the slope at the fixed point is less than 1,
	it will be a stable fixed point.  We know that when r > 0.75, we no
	longer have a fixed point.  However, if we consider

		f^(2) = f( f(x,r), r)
	
	then this will have a fixed point if f has an attractor of period two.
	Under the previous notation f( f(x_n,r), r) is what we called x_(n+2)
	and for an attractor of period two, that would be x_n.

	Program iterate.c in ~sg will print f^(2) or a higher iterate of f
	as a function of x for a given r.  Usage:

	iterate r iterations dx

	where r is the value of r that you desire. iterations is how many
	times you want the functional mapping iterated, and dx is the
	spacing between the x values that are printed.  You may pipe the
	output to axis to make the graph.  It is also easy to do this
	with Mathematica.

	The program iterate.c is an example of a recursive program.
	The function f at the bottom of the code calls itself.  This
	can be done in C.  An example of an iterative solution is the
	towers of Hanoi problem.  This was described.  Thanks to Bruce Tepke
	for giving the reference of George Gamow's book "One, Two, Three, ...
	Infinity" for a discussion of this problem.

	Superstable fixed points

	We know that fixed points of higher iterates correspond to attractors
	of the corresponding period.  It is not that easy to find the
	r values where we have bifurcations because they correspond to
	instabilities.  However, it turns out that in each range of r, one
	of the branches of the bifurcation diagram will cross y = 0.5.

	That means that x = 0.5 is a fixed point of

		f^(n) (x,r) = x .
	
	We want to find out which values of r have 0.5 as a fixed point.
	We we are trying to find the root of this equation:

		f^(n) (0.5, r) = 0.5

	The program findroot.c uses routines from Numerical Recipes to
	accomplish this.  The routine is called zbrent and may be found in
	Chapter 9 on Root Finding.  This may be used on the homework on
	problem 6.6.  We discussed how to create a polynomial interpolating
	function to go through n data points.  Quadratic interpolation is
	used in the Brent algorithm, but is used to interpolate x and function
	of y.  The desired x value is the one for which y=0.  This results in
	a complicated, but linear function, for x the next guess at the root.
	The zbrent routine is useful when you can bracket the root and only
	want to evaluate the function whose root you want, not its derivative.