October 18, 2007---Class 16--- Chaos and Logistic Map (continued)

Activities:
	
	Chaos and Logistic Map (continued)

	In the last class we saw this curious and beautiful behavior
	of period doubling.  We would like to understand how this comes about.

	We are looking at the mapping

		x_{n+1} = f(x_n)

	and when r <= 0.75 we find that the long term behavior is a
	constant.  We call this a fixed point.  Let

		x_{n} = x_f + \epsilon_n

	where x_f is the fixed point and \epsilon_n is how far we are from
	the fixed point at the n th generation. 

	By Taylor expanding around x_f, we found that if |f'(x_f)| < 1, we
	have a stable fixed point, otherwise the fixed point is unstable.
	It was easy to find the fixed points from the equation

	 	x_f = 4 r x_f (1-x_f)

	There are two solutions, x_f = 0 and x_f = 1- 1/(4r)

	We easily calculated the derivatives at the fixed points.  For r <
	0.25, x_f=0 is a stable fixed point, and the other value is not
	between 0 and 1.  For 0.25 < r < 0.75, x_f = 1- 1/(4r) is a stable
	fixed point and x_f = 0 is unstable.  When r > 0.75 neither is a
	stable fixed point.  However, when we have a trajectory of period
	2, then x_(n+2) = f(f(x_)) can have a stable fixed point.  We
	explored this using Mathematica.