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.