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.