The system

The "delinearization" operator for the original Yorke-Curry system was :
Y(n+2) = Y(n+1) +Y (n+1) ²

The delinearization operator we chose for the first variant was :
Y(n+2) = Y(n+1) + abs(Y(n+1)*X(n+1))
( simple replacement of a Y term by a X term )

After a very simple modification, let's try a rather complicated one to see what happens :
Y(n+2) = Y(n+1) + X(n+1) ² + X(n+1) + abs(Y(n+1)*X(n+1))


.... that means we're studying the variable xX(2i) from the following system :

We start from R(n) and T(n).
R(n+1) = e * log ( 1+R(n) )
T(n+1) = T(n) + T0

Then R(n+1) and T(n+1) are converted into rectangular coordinates, which are X(n+1) and Y(n+1).
Y(n+2) = Y(n+1) + X(n+1) ² + X(n+1) + abs(Y(n+1)*X(n+1))
X(n+2) = X(n)

And then we get back to polar coordinates, R(n+2) and T(n+2).

The system's behaviour consists in the even indices only: [R(2i), T(2i)] or [X(2i), Y(2i)].

Here is the system's behaviour as e evolves :

The inharmonic part between 1 and 1.880 can be compared to the Yorke Curry original system's behaviour.

The novelty here is a high number of consecutive doubling period cascades between 2.160 to 2.445.
What's more, the doubling period phenomenon seems to come along with a light background noise which can be interesting.

Just above e=1, nice inharmonic sounds are generated.
Harmonics appear below and above the main note.

( X(2i) spectrum, with e from 1 to 1.2, step of 0.005, along with the audio samples corresponding to X(2i)
Quicktime movie, MPEG4 audio 64kbs, 1.2MB. )

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

Between e ranging from 2.160 to 2.445, no less than 7 period doubling cascades can be observed.

( X(2i) spectrum, with e from 2.160 to 2.445, step of 0.005, along with the audio samples corresponding to X(2i)
Quicktime movie, MPEG4 audio 64kbs, 2.2MB. )

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

It is very possible that, would the step be smaller, we would observe even more doubling period cascades.

to be continued