October 30, 2007 --- Class 18 --- Finding Superstable Trajectories with GSL,
			Instability and Chaos, Lyapunov Exponents

	Finding Superstable Trajectories with GSL

	In the last class we used Mathematica to find the superstable
	trajectories of small period.  I also demonstrated a program I
	wrote using root finding routines from Numerical Recipes (NR).  The
	GNU Scientific Library also has root finding routines and there is
	no restriction on the use of their library.  To find out about
	what is available from GNU, go to:

	http://www.gnu.org/software

	or for the GNU Scientific Library, go here:

	http://www.gnu.org/software/gsl/

	Documentation for the one-dimensional root finding routines 
	are found here:

	http://www.gnu.org/software/gsl/manual/html_node/One-dimensional-Root_002dFinding.html

	In ~sg/gslroots you will find code for the example in the documentation
	and for my code to solve for the values of r that result in 
	superstable trajectories.  The code is more verbose that that using
	the NR routines in that for each solve details of the convergence
	are printed out.  When high enough accuracy is required, the results
	are seen to be that same as found with the NR based code.  As pointed
	out before, my values of r are more accurate that found in Table 6.2.
	This was verified in one case by using my value of r and the one in
	the book.  We found that after the expected period with my value
	x returned to 0.5, but with the value in the book, it was slightly off.

	Instability and Chaos

	A key issue in chaotic systems is sensitivity to initial
	conditions.  This is discussed in CSM in Problem 6.3 on page
	134.  We looked at r=0.91 as described in part (a) of that
	problem.  It is easy to write a program that will follow two
	trajectories starting at nearby values and see how they converge
	or diverge.  We found that for r=0.91, nearby starting points
	diverge.  The divergence is roughly exponential.  We reproduced
	Fig. 6.8 on page 147 using my code mapdiff.c which is in my home
	directory.
	
	Lyapunov Exponent

	Near the end of class we briefly discussed the Lyapunov exponent
	which describes the exponential growth or decay of the
	difference between two nearby trajectories.  Equations (6.15)
	and (6.16) are the basis of a computer code to calculate the
	Lyapunov  exponent.  We reproduced Fig. 6.9 on page 149.
	Bifurcation points correspond to the Lyapunov exponent going to
	zero.  The big dips correspond to the superstable attractors.
	The transtion to chaos occurs when the Lyapunov exponent first
	becomes positive.  Note some islands of stability above that
	transition point where the Lyapunov exponent becomes negative.