Eric Weeks
 personal pages  research
 time series analysis
My Adventures in Chaotic Time Series Analysis 
weeks@physics.emory.edu 
A0. Links and related information
For an explanation of what these pages are all about, select topic 1 above.
This page: top  lorenz  rossler  henon  expt: periodic  qperiodic2  qperiodic3  chaotic  bottom
For many of these attractors, they properly should be embedded in three dimensions (or perhaps even more). So, I have crudely attempted to reproduce them in 3D. This was entirely with homemade crude software.
Brief description of software: I wanted to use my psdraw program to plot the data. In the end I added a little bit of code to psdraw in order to handle this.
The Lorenz attractor is a strange attractor, a geometrical object with fractal dimension. Sometime later I may try to find the dimension. Note that these views don't look like the "standard" views of the Lorenz attractor, as they are made from delay coordinates rather than the actual system coordinates.
These were generated with the commands: rotate s 80 t 15 p 15 lorenz.3d  sort n +2 3  psdraw zOc0.04 X 30 30 30 30 > atlorr1.ps
rotate s 80 t 115 p 15 lorenz.3d  sort n +2 3  psdraw zOc0.04 X 30 30 30 30 > atlorr2.ps
These were made with:
rotate t 155 p 40 s 70 rossler.3d  sort n +2 3 
psdraw zOc0.04 X 25 25 25 25 > atros3d1.ps
rotate t 170 p 310 s 70 rossler.3d  sort n +2 3 
psdraw zOc0.04 X 25 25 25 25 > atros3d2.ps
Recall that the equation describing the Rössler attractor are
3dimensional.
This page:
top 
lorenz 
rossler 
henon 
expt: periodic 
qperiodic2 
qperiodic3 
chaotic 
bottom
Hénon:
time series

power spectrum

mutual information

attractor

attractor 3D

autocorrelation

poincare

1D maps
As the map is a 2D map, the full structure of the Hénon attractor can be seen in a 2D plot; it does not need to be displayed in 3D. But, to satisfy my curiosity, I plotted it in 3D.
This was produced with the command: rotate s 70 t 120 p 30 henon.3d  sort n +2 3  psdraw zOc 0.04 X 2.8 2.8 2.8 2.8 > henon3d.ps
This page:
top 
lorenz 
rossler 
henon 
expt: periodic 
qperiodic2 
qperiodic3 
chaotic 
bottom
Experimental/periodic:
time series

power spectrum

mutual information

attractor

attractor 3D

autocorrelation

poincare

1D maps
This data can be reasonably displayed in only 2 dimensions, but here is a 3D plot for comparison.
This page:
top 
lorenz 
rossler 
henon 
expt: periodic 
qperiodic2 
qperiodic3 
chaotic 
bottom
Experimental/quasiperiodic2:
time series

power spectrum

mutual information

attractor

attractor 3D

autocorrelation

poincare

1D maps
The second picture was made with: rotate s 70 t 40 p 20 r 180 expqp2.3d  sort n +2 3  psdraw zOc0.05 X 1.5 1.5 1.5 1.5 > atqp2.1.ps
I have made 9 small pictures, which show a gradual change in the point of view of this particular attractor. Click here to see them. (109 k of pictures total)
This page:
top 
lorenz 
rossler 
henon 
expt: periodic 
qperiodic2 
qperiodic3 
chaotic 
bottom
Experimental/quasiperiodic3:
time series

power spectrum

mutual information

attractor

attractor 3D

autocorrelation

poincare

1D maps
This attractor looks like a tube with a little bit of fuzz inside it. The pictures below show the view looking down the length of the tube, and at the side of the tube. The individual trajectories are harder to see than the other attractors shown on this page, as the data is sampled 4 times slower, so the points don't look as continuous as the other experimental data sets.
The first was made with the command: rotate s 80 t 55 p 35 exptqp3.3d  sort n +2 3  psdraw zOc0.04 X 2.2 5 2.2 5 > pic1.ps
The second was made with the command: rotate s 80 t 130 p 10 exptqp3.3d  sort n +2 3  psdraw zOc0.04 X 2.2 5 2.2 5 > pic2.ps
This page:
top 
lorenz 
rossler 
henon 
expt: periodic 
qperiodic2 
qperiodic3 
chaotic 
bottom
Experimental/chaotic:
time series

power spectrum

mutual information

attractor

attractor 3D

autocorrelation

poincare

1D maps
These three pictures were made with the commands:
rotate s 70 t 15 p 200 exptchao.3d  sort n +2 3  psdraw zOc0.04 X 1.6 5.7 1.6 5.7 > atchar1.ps
rotate s 70 t 125 p 20 r 180 exptchao.3d  sort n +2 3  psdraw zOc0.04 X 1.6 5.7 1.6 5.7 > atchar2.ps
rotate s 70 t 100 p 15 exptchao.3d  sort n +2 3  psdraw zOc0.04 X 1.6 5.7 1.6 5.7 > atchar3.ps
So far the analysis has given me several different times. From the Fourier transform I have certain characteristic periods. From the mutual information calculation I have the first minimum. Thus, a table:
Note that all times are in natural units (seconds for experimental data, iterations for Hénon map, the time step of the equations for the Rössler and Lorenz systems). Links in the table will take you to the appropriate location where the information was found.
Previous page: 2D plots of strange attractors
Next page: autocorrelation functions
This page: top  lorenz  rossler  henon  expt: periodic  qperiodic2  qperiodic3  chaotic  bottom