Plotting Strange Attractors

Eric Weeks - personal pages - research - time series analysis

My Adventures in Chaotic Time Series Analysis

weeks@physics.emory.edu

1. Meet the time series
2. Fourier Transforms
3. Mutual information to find delay coordinates
4. Plotting attractors
5. Attractors in 3-D (you are here)
6. Autocorrelation functions
7. Poincare sections
8. 1-D Maps
9. More later. Perhaps fractal dimensions.

A0. Links and related information

For an explanation of what these pages are all about, select topic 1 above.

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 3-D. 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.

I use my delay coordinates program (not available yet) to turn the 1-D time series into 3-D points; the delay was chosen as discussed previously.
I rotate the data in 3-D using a program I just wrote called rotate. Click here to learn about this program. This program puts a box around the data to make the point of view clearer.
I then pipe the rotated data through sort -n +2 -3 (to sort on what is currently the z-data) and pipe the sorted data into psdraw. psdraw converts this into a pretty 3-D plot, which is what is displayed below. Click here to learn about the psdraw program.

Lorenz attractor:

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

Rössler attractor:

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 3-dimensional.

Hénon attractor:

As the map is a 2-D map, the full structure of the Hénon attractor can be seen in a 2-D plot; it does not need to be displayed in 3-D. But, to satisfy my curiosity, I plotted it in 3-D.

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

Experimental data: periodic

This data can be reasonably displayed in only 2 dimensions, but here is a 3D plot for comparison.

Experimental data: quasi-periodic-2

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)

Experimental data: quasi-periodic-3

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

Experimental data: chaotic (?)

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:

System Period Mutual info 1st minimum Mutual info 1st maximum

Lorenz Attractor ? 0.16 0.22

Rössler Attractor 6.07 1.52 3.10

Hénon Map 2.2 "1" none

Expt-periodic 6.9 0.9 2.1

Expt-q-periodic-2 13.0 2.5 6.8

Expt-q-periodic-3 6.1 2.0 3.2

Expt-chaotic 6.5 2.2 3.5

System	Period	Mutual info 1st minimum	Mutual info 1st maximum

Lorenz Attractor	?	0.16	0.22
Rössler Attractor	6.07	1.52	3.10
Hénon Map	2.2	"1"	none
Expt-periodic	6.9	0.9	2.1
Expt-q-periodic-2	13.0	2.5	6.8
Expt-q-periodic-3	6.1	2.0	3.2
Expt-chaotic	6.5	2.2	3.5

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: 2-D plots of strange attractors
Next page: autocorrelation functions

Links...

Graphs were made with a program I wrote, psdraw. The program makes PostScript output which I then converted to GIF. Rotations & 3-D boxes done with rotate, another program I wrote. Both psdraw and rotate are public domain.
If you have any questions or comments, send me email; email address below.

Current address:
Eric R. Weeks
weeks@physics.emory.edu
Department of Physics
Emory University
Atlanta, GA 30322-2430