The system
Whereas the standard logistic equation was...
U(n+1) = (R+1) U(n) - R U(n)²
or
U(n+1) - U(n) = U(n) * R * ( 1 - U(n) )
...this one is
U(n+1) = U(n) * exp ( R* (1-U(n)) ).
It exhibits the same kind of behaviour ( doubling periode cascades, "windows of order"... ).
Here is how the system behaves depending on the R parameter :
| R values | behaviour |
|---|---|
1 to ~1.8 |
constant value |
~ 1.8 to 2.5 |
periodic, period 2 |
~ 2.526 |
period 4 |
~ 2.697 |
end of doublic period cascade |
~ 2.697 to ~ 3.5 |
chaotic, steadily noisier and noisier numerous windows of order ( ex 2.71, 2.775, 2.86, 2.92, 2.965 ) |
above 3.5 |
less and less pseudo periodic then stable (0) |
Audio samples : complete map from 2.65
The map is divided into two movies.
The first movie begins at 2.65 (period 4) and goes to 3.05, with a step of 0.00125.
As the step is quite small (the movie includes 321 values of Par), several windows of order are visible and audible.
As usual, would the step have been even smaller, more windows of order would have been visible.
( False 2D attractor, (X(n+1),X(n)), with R from 2.65 to 3.05, step of 0.00125, along with the audio samples corresponding to X(n)
Quicktime movie, MPEG4 audio 64kbs, 5.5MB. )
 |
Click the image to open the movie - Right click to save. |
The second movie begins at 3.05, where the other one had ended, and goes up to 13.05 in 41 values, large step of 0.25 ( 200 times the other movie's step ).
( False 2D attractor, (X(n+1),X(n)), with R from 3.05 to 13.05, step of 0.25, along with the audio samples corresponding to X(n)
Quicktime movie, MPEG4 audio 64kbs, 664kB. )
 |
Click the image to open the movie - Right click to save. |
The behaviour tends to consists in enormously high single sample values, with other sample values getting close to zero.
Those "outbursts" tend to get fewer and fewer, until the system is "silent" (0 only)
to be continued.