Eric Weeks - personal pages - research - time series analysis

## My Adventures in Chaotic Time Series Analysis

weeks@physics.emory.edu

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

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

This page: top | lorenz | rossler | henon | expt: periodic | qperiodic-2 | qperiodic-3 | chaotic | bottom

This is a side project of mine, learning more about chaotic time series analysis. My motivation is that I have some time series from my experiment which I would like to analyze. I think some of them are chaotic, whereas others are clearly periodic.

I'll add to this web page as I have time and as I try different things. My hope is to link to whatever software I use and/or provide source code if necessary. For example, you can download the software that I used to make the graphs on this page.

## Download the time series

I have seven time series. You can download them in several forms:

timeser.zip: pkzip format; on UNIX systems, use unzip (303k)

timeser.tar.gz: use tar -zxvf to extract (305k)

timeser.tar.Z: use tar -Zxvf to extract (328k)

Each time series is an ASCII file.

## Meet the time series

To start with, I have three sets of data which I know are chaotic and for which I can probably look up information about.

### Lorenz attractor:

This page: top | lorenz | rossler | henon | expt: periodic | qperiodic-2 | qperiodic-3 | chaotic | bottom
Lorenz: time series | power spectrum | mutual information | attractor | attractor 3D | autocorrelation | poincare | 1-D maps

This was created by Runge-Kutta integration of the Lorenz equations. I used the subroutine rkdumb() taken from Numerical Recipes, with a step size of 0.01. The Lorenz equations are given by:
dx/dt = sigma * (y - x)
dy/dt = r * x - y - x * z
dz/dt = x * y - b * z
I use the standard values sigma=10.0, r = 28.0, b = 8/3.

### Rössler attractor:

This page: top | lorenz | rossler | henon | expt: periodic | qperiodic-2 | qperiodic-3 | chaotic | bottom
Rössler: time series | power spectrum | mutual information | attractor | attractor 3D | autocorrelation | poincare | 1-D maps

This was created by Runge-Kutta integration of the Rossler equations. I used the subroutine rkdumb() taken from Numerical Recipes, with a step size of 0.01. The Rössler equations are given by:
dx/dt = -z - y
dy/dt = x + a * y
dz/dt = b + z * (x - c)
I use the standard values a=0.15, b=0.20, c=10.0.

This was first proposed in the article: O. E. Rössler, Phys. Lett. 57A, 397 (1976).

### Hénon attractor:

This page: top | lorenz | rossler | henon | expt: periodic | qperiodic-2 | qperiodic-3 | chaotic | bottom
Hénon: time series | power spectrum | mutual information | attractor | attractor 3D | autocorrelation | poincare | 1-D maps

The Hénon map is given by:
x' = a + b * y - x^2
y' = x
I use a=1.4, b=0.3. I generated the map with the following command: ``` gawk 'BEGIN {x=1;y=1;for (t=1;t < 26384;t++) {xx=1.4+0.3*y-x*x;y=x;x=xx;print x}}' | tail -16384 ```

The map was suggested in the article "A two-dimensional mapping with a strange attractor," M. Hénon, Commun. Math. Phys. 50, 69-77 (1976).

### Experimental data: periodic

This page: top | lorenz | rossler | henon | expt: periodic | qperiodic-2 | qperiodic-3 | chaotic | bottom
Experimental/periodic: time series | power spectrum | mutual information | attractor | attractor 3D | autocorrelation | poincare | 1-D maps

Time series taken from my experiment. For more information on the experimental setup, see my research page. These are velocity time series taken with a hot film probe in my experiment. If you'd like more information email me.

### Experimental data: quasi-periodic-2

This page: top | lorenz | rossler | henon | expt: periodic | qperiodic-2 | qperiodic-3 | chaotic | bottom
Experimental/quasiperiodic-2: time series | power spectrum | mutual information | attractor | attractor 3D | autocorrelation | poincare | 1-D maps

### Experimental data: quasi-periodic-3

This page: top | lorenz | rossler | henon | expt: periodic | qperiodic-2 | qperiodic-3 | chaotic | bottom
Experimental/quasiperiodic-3: time series | power spectrum | mutual information | attractor | attractor 3D | autocorrelation | poincare | 1-D maps

### Experimental data: chaotic (?)

This page: top | lorenz | rossler | henon | expt: periodic | qperiodic-2 | qperiodic-3 | chaotic | bottom
Experimental/chaotic: time series | power spectrum | mutual information | attractor | attractor 3D | autocorrelation | poincare | 1-D maps

How can I tell the difference between the above data (quasiperiodic with three fundamental frequencies) and the data below (possibly chaotic)? From the power spectra; this will be on the next page. Also hopefully from trying to reconstruct a strange attractor from the chaotic data. To some extent, distinguishing these two time series is why I am working on this project.

This page: top | lorenz | rossler | henon | expt: periodic | qperiodic-2 | qperiodic-3 | chaotic | bottom

### 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