This very famous sytem, which is generally used when showing pictures of so called "strange attractors", is a three variable, continuous system defined by the following equations :

D(X)(t)=10*(Y(t)-X(t))
D(Y)(t)=-X(t)*Z(t)+Par*X(t)-Y(t)
D(Z)(t)=X(t)*Y(t)-(8/3)*Z(t)

Here is a synopsis of its behaviour depending on the "Par" parameter ( Par ranging from 0 to 300 )

As it's usual in such systems, the "window of order " between 148 and 166 is not the only one that can be found, it's just the biggest.
In fact, there is an infinite number of such windows, from the biggest one (148 to 166) to infinitely small ones.
The order windows one can see depends on the scale at which one is looking to the system.

The first two movies display the whole "map" of the system, with Par's value increasing regularly from 0 to 300.

The first movie presents the audio samples along with their spectral analysis.
The spectrum shows clearly what's going on when the system has a periodic / ordered behaviour, but is much less efficient when it comes to chaotic behaviour.

( X(t)'s spectrum, with Par from 0 to 300, step of 2.5, along with the corresponding audio samples.
Quicktime movie, MPEG4 audio 64kbs , 7.9MB. )

Click the image to open the movie - Right click to save.

The second movie proposes the same samples, along with an "attractor" kind of display. It's a pretty efficient way to represent the system's behaviour
With a bit of training, it's possible to look at the display and imagine the aspect of the corresponding sample.

( 2D attractor, (X(t),Y(t)), with Par from 0 to 300, step of 2.5, along with the audio samples corresponding to X(t)
Quicktime movie, MPEG4 audio 64kbs , 2.9MB. )

Click the image to open the movie - Right click to save.

Since this is a three variable system, it can be interesting to know how the three variables sound comparately to each other.
Actually, the three timbres are very similar.

( 9 Quicktime movies, MPEG4 audio, 12kB each )

The differences between the three variables :
- Y(t) is the same as X(t) with some kind of a 6dB/oct low cut filter
- Z(t) is the same as Y(t), just one octave higher

Apart from those differences, they behave exactly the same way throughout the whole system evolution.

As seen above, there is a large "window of order" inside the chaotic area, roughly from 148 to 166.
It's interesting to hear how "chunks" or "grains of noise" slowly get in the way of the harmonic sound.

( 2D attractor, (X(t),Y(t)), with Par from 166.0604 to 167.8000, variable step, along with the audio samples corresponding to X(t)
Quicktime movie, MPEG4 audio 64kbs , 804kB. )

Click the image to open the movie - Right click to save.

At the end of the Par evolution, the system slowly drifts away from chaotic behaviour, to eventually become, and remain, harmonic.
From an audio point of view, the "grains of noise" slowly disappear, leaving only an harmonic sound meddled with noise, this noise also getting thinner and thinner.

( 2D attractor, (X(t),Y(t)), with Par from 195.00 to 224.75, step of 0.25, along with the audio samples corresponding to X(t)
Quicktime movie, MPEG4 audio 64kbs , 1.4MB. )

Click the image to open the movie - Right click to save.