MATH 489-501: CHAOS and Dynamical Systems HOME PAGE

Class meets: Tues: 2:20-3:35pm in Blocker 156.

Things of interest for class Math 489-501 "An intro to Chaotic Dynamical systems" will be posted here.

• Professor: Denise Kirschner
• the class Syllabus
• questionnaire Aug. 31, 1995
• Logistic Eqn(Maple file) Sept. 5
• Numerical Homework Sept. 12, 1995
• DEsolving(Maple file) Sept. 12
• Flows on a line(Maple file) Sept. 19
• Bifurcation Diagrams (Maple file) Sept. 25
• Phase Planes (Maple file) Sept. 25
• Nonlinear Phase Planes (Maple) Oct 3
• Limit Cycles (Maple) OCt. 10
• Lorenz Attractor (Maple) Nov. 13
• Lyapunov for iterative systems (MAPLE)
• LENNS (a tar file). This is a file containing subdirectories and file to run a program called LENNS which computes the lyapunov exponent from times series data. You will have to save it to your directory as lenns.tar then execute the command "tar xvf lenns.tar". then type make lenns.u You can then cd to the manual subdirectory and print out the manual.ps file. If you want to use this on your PC at home, then grab the lennsarc.exe file and ftp it to your home directory and type lensarc followed by install install. You must have F77 on your machine to run it on the PC. IMPORTANT NOTE! The two fortran files of LENNS lenns.for and objfun.for are updated. AFter you install LENNS, copy these files and replace those two with these.
• AUTO ( a tar-ed file). FIRST you must print out the postscript manual to help explain the procedure. I have already unedcoded it. This is a general program for bifurcation detections and diagrams of dynamical systems.
• Correlation Dimension Procedure . It requires two header files: nr.h and nrutil.h and two c files: nrutil.c and sort.c . This is a Borland C program. It computes the data (an (x,y) ordered pair) which you must plot out and fit a line through and find the slope of. REMEMBER, the hard part is finding the C(e) ! that is what this program does and returns (lnC(e),lne). Now, there is two more things: "Embedding dimension" and "LAG". These are things which need to be "figured out" i.e. The embedding dimension is roughly equal to two times to upper cartesean dimension plus 1. (i.e. for Lorenz, which is in between 2 and 3 would have an embedding dimension of 3x2+1=7). LAG?? Pick embedding (as above) and run through a bunch of lags and see which gives the best fit to a straight line....then do it for an embedding of one larger or smaller. Then compare your results.
• IMPORTANT! I ran havstadt.c on the X(t) variable of the Lorenz system with parameters (sigma = 10, b = 8/3, and r = 28) and integration time step of 0.009. (number of points = 4500) (it is in the file xte1.out . used a emdedding of 7 and a lag of 6. Plotting the first column as x with the second column as y and fitting a line very quickly through an eyeball estimate of the scaling region gave a slope of 1.97 (I've seen 2.0 reported) - so not bad for a quick eyeball fit. One thing: the upper region is incorrect - there must still be an error However the upper region is the nonlinear part which is never used anyway. So it should be ok for now. I'll have to track down the problem with the upper part. I've never seen that before, but my data sets are usually about 1/4 th that size.