The system
A two variable, discrete system.
X(n+1) = 1 + Y(n) - a|X(n)|
Y(n+1) = bX(n)
b is fixed at 0.5
a is taken as the evolving parameter
Here is a synopsis of the system behaviour depending on a ( a ranging from 1 to 2)
| a values | behaviour |
|---|---|
< 1.5 |
harmonic, period 2 |
~ 1.5 to ~ 1.553 |
noise appears |
~ 1.553 to ~ 1.65 |
grains of higher noise appear |
~ 1.65 to ~ 1.75 |
very noisy |
> 1.75 |
to + ∞ |
Audio samples : X(n), complete map from 1.4490 to 1.5730
The spectrum display being not very interesting, only the attractor display is presented.
( false 2D attractor, [X(n+1), X(n), with a from 1.4490 to 1.730, arbitrary steps, along with the corresponding audio samples.
Quicktime movie, MPEG4 audio 64kbs, 1MB )
 |
Click the image to open the movie - Right click to save. |
The different behaviours appear quite clearly both on the attractor display and on the audio track.
Two interesting transitions will be studied more carefully : the first transition to noise, and the "noise grain" apparition near 1.553.
This study will be made in the "optimization" section.