CHAOS, AND WHAT TO DO ABOUT IT           
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         Georgia Tech PHYS 7123                  Fall semester 2003

         Slaven Peles
	 Lecture 18           9:35-10:55 Thu Oct 23 2003 in Howey S106 
	 Lecture 19           9:35-10:55 Tue Oct 28 2003 in Howey S106 

Periodic orbits: how to get them      --       Homework Assignment #13
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Reading:
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         chapter 14: Fixed points, and how to get them

Exercises: (any - all are optional)
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	* Exercise (1) - not in the webbook: 
This homework problem is optional, but as I said before it is so easy that it 
is a matter of honour to do it. The problem is to solve the Rosenbrock system 
of equations

f_1 (x, y) = a (1 - x)
f_2 (x, y) = b (y - x^2)

numerically, with a=1 and b=10. It is obvious that the solution is x=1, y=1. 
Your assignment is to reproduce this result numerically using multiroot 
solving routines from GSL [1], MINPACK [2], or your own code made from 
scratch. Test the convergence of your code for a bad initial guess. If you 
are using a numerical library try all multiroot solvers available in it. 
Please submit your code along with your explanations, comments and 
observations.
	[1] Gnu Scientific Library http://sources.redhat.com/gsl/
	[2] MINPACK (MINimization PACKage) http://www.netlib.org/minpack/
- Have fun - Slaven -

Exercises from the webbook:
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        * 14.1 +14.2 Cycles of the Ulam map

        * 25.3 Find cycles of the Henon map: compute prime
cycles with up to lenght 6 by your multi-shooting Newton routine for maps
(or whatever gets you through the night)

        * 14.7 Periodic orbits of the Rossler attractor:
implement the Newton-Raphson method to find several periodic orbits of the
Rossler flow with a=b=0.2 and c=5.7. Plot these orbits and compute their
period.


	Homework assignment #13 will be graded by Rytis Paskauskas

        due Thu Oct 30 


Project:
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complete the part planned for Oct 30, write up a current status (I
recommend you write results into a continuously updated LaTeX file, it is
a necessary skill if you plan to do research in physics or math). If you
failed to attain your goal for Oct 30, explain why.