The system

We've seen that the Yorke Curry system equations contained a specific "delinearization" operator which caused the system to behave chaotically.
This operator was : Y(n+2) = Y(n+1) +Y (n+1) ²

Now, it can be interesting to change the operator in order to create another system with a possibly different behaviour.
We choose Y(n+2) = Y(n+1) + abs(Y(n+1)*X(n+1))

The system is practically the same except that one Y term was replaced by a X term.
T0 remains the same, and the initial conditions are the same also.

Here is the system's behaviour as e evolves :

The windows of order can exhibit different behaviours :
- harmonic, period 2
- harmonic, period 3
- almost harmonic with patches of noise like between 1.84 to 1.85

After all, it seems that the intuition that made us change the delinearization operator was good.
It is true that we've lost the interesting inharmonic behaviour the original system used to exhibit, but we've gained another interesting behaviour : the intermittent patches of noise - which are not unlike the ones we came across during the Lorenz system study, though they sound much "softer".

The following two movies show the system's behaviour for all the relevant e values.

( false 2D attractor, (X(2i+2),X(2i)), with e from 1.740 to 2.165, step of 0.005, along with the audio samples corresponding to X(2i)
Quicktime movie, MPEG4 audio 64kbs, 2.2MB )
NB : in the following movie, the graphics show the system's stablilized state, though the audio extracts include the transients

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

The same samples, but with a spectrum display :

( X(2i) spectrum, with e from 1.740 to 2.165, step of 0.005, along with the audio samples corresponding to X(2i)
Quicktime movie, MPEG4 audio 64kbs, 2.8MB )
NB : in the following movie, both graphics and audio include the transients

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

If we look back at the system's behaviour table, we see a lot of things happening before getting into the properly chaotic zone :

It may be interesting to see more precisely what happens between 1.84 to 1.865.

That's the object of the two following movies.

( false 2D attractor, (X(2i+2),X(2i)), with e from 1.840 to 1.8649, step of 0.0003, along with the audio samples corresponding to X(2i)
Quicktime movie, MPEG4 audio 64kbs, 1.31MB )
NB : in the following movie, the graphics show the system's stablilized state, though the audio extracts include the transients

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

( X(2i) spectrum, with e from 1.840 to 1.8649, step of 0.0003, along with the audio samples corresponding to X(2i)
Quicktime movie, MPEG4 audio 64kbs, 1.60MB )
NB : in the following movie, both graphics and audio include the transients

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

to be continued