CHAOS, AND WHAT TO DO ABOUT IT ----------------------------------------------------------- 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 -------------------------------- Reading: -------- chapter 14: Fixed points, and how to get them Exercises: (any - all are optional) ---------- * 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: --------- * 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: -------- 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.