A 2D discrete system.
X(n+1) = 1 - aX(n)² + bY(n)
Y(n+1) = X(n)
With b fixed at 0.4, here is how the system behaves depending on the a parameter :
| a values | behaviour |
|---|---|
0 to 0.975 |
harmonic, period 2 |
around 0.993 |
doubling period cascade |
0.993 to ≈ 1.26 |
chaotic with patches of order |
> 1.26 |
to ∞ |
Audio samples : X(n), near complete map - from 0.95 to 1.07
The first two movies consist in a map of the system, covering all the "interesting" values of a, b being fixed at 0.4.
As usual, the spectrum display is totally inefficient when it comes to represent chaos, though it works nicely when the system behaves "normally".
In chaotic areas, there is no visible relationship between what's heard and what's seen.
( X(n)'s spectrum, with a from 0.95 to 1.07, step of 0.075, along with the corresponding audio samples.
Quicktime movie, MPEG4 audio 64kbs, 7.8MB. )
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Here are the same samples, but presented with the attractor drawn by the consecutive (X(n+1), X(n)) points.
( False 2D attractor, (X(n+1),X(n)), with a from 0.95 to 1.07, step of 0.075, along with the audio samples corresponding to X(n)
Quicktime movie, MPEG4 audio 64kbs, 2.4MB. )
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The attractor display is much correlated to what's actually heard.
Audio samples : window of order in chaos section, 1.02035 to 1.020645
It's possible to find windows of order for certain values of a which correspond to a chaos zone.
Unlike what happened with the Lorenz system, here the transition between chaos and order is abrupt. There is no transition.
Instead, there is another phenomemon taking place : in the windows of order, the system is chaotic for a few 1/10 seconds, then it stabilizes to an harmonic behaviour. This transitional time is variable, and doesn't seem to obey any simple law.
Following is the movie featuring the behaviour of the system during one of these windows of order, after it's stabilized - the transitional parts of each sample have been removed.
( False 2D attractor, (X(n+1),X(n)), with a from 1.02035 to 1.020645, step of 0.000005, along with the audio samples corresponding to X(n).
No transitions.
Quicktime movie, MPEG4 audio 64kbs, 624kB. )
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Now here is the movie corresponding to the same values of a, but the transitions have not been edited out.
The attractor shows the chaotic transitions merged with the harmonic part, so it looks "half chaotic".
The longer the chaotic transition, the more "continous" or "noisy" the attractor.
( False 2D attractor, (X(n+1),X(n)), with a from 1.02035 to 1.020645, step of 0.000005, along with the audio samples corresponding to X(n).
With transitions.
Quicktime movie, MPEG4 audio 64kbs, 640kB. )
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Click the image to open the movie - Right click to save. |