The system
A 2D discrete system.
x(n+1)=x(n)+y(n+1) mod 2Pi
y(n+1)=y(n)+k sin x(n) mod 2Pi
"k" is the main parameter.
Obviously, "k sin x(n) mod 2Pi" is the delinearization factor.
Behaviour of the system depending on k :
| k values | behaviour |
|---|---|
0 |
harmonic, period 8 samples |
until ~ 0.7 |
mainly inharmonic, with a chaotic component |
around 0.8 |
really chaotic, with patches of harmonic order |
~ 0.8 to ~4 |
completely chaotic |
> 4 |
tends to be white noise |
X(n), k from 0 to 1 : overview
(Click on the thumbnails to get larger images)
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Notice how the different harmonics evolve : this leads to a great variety of inharmonic content.
It is also clearly visible that this inharmonic spectrum is noisy.
When k reaches a certain threshold (~0.8), the spectrum becomes less readable, the harmonics are drowned into noise : this correspond to a really chaotic behaviour.
Here is the waveform for such a behaviour (k=1) :
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close view |
general view |
Notice how the oscillator switches between different behaviours.
X(n), k from 0 to 1 : audio samples
The first movie shows 21 audio samples + their spectrum, with k ranging from 0 to 1, 21 values.
The samples have been transposed 2 octaves down so they are more understandable.
The number written at the top of the movie is k.
( X(n)'s spectrum, with k from 0 to 1, step of 0.05, along with the corresponding audio samples.
Quicktime movie, MPEG4 audio 64kbs, 1.2MB. )
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Click the image to open the movie - Right click to save. |
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 k from 0 to 1, step of 0.05, along with the audio samples corresponding to X(n)
Quicktime movie, MPEG4 audio 64kbs, 1.1MB. )
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Click the image to open the movie - Right click to save. |
Behaviour for k > 1
After 0.8, the oscillator tends to be more and more noisy, until it gets close to white noise :
k |
original pitch |
transposed 2 oct. down |
1 |
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1.5 |
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2 |
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2.5 |
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3 |
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3.5 |
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4 |
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4.5 |
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5 |
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5.5 |
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6 |
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... |
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11 |
The harmonic content tends to get lower and lower, and the patches of periodic signal disappear.
It almost sounds like white noise.
For k =11 , the signal looks like this :
... and the spectrum :
...whereas a white noise spectrum looks like this :
