The system

This is a system which is neither classical nor well documented.

Unlike the logistic equation, which is related to real situations, like population evolution over time, this one seems to have been specifically "built" in order to exhibit a chaotic behaviour. When giving a close look to the Curry-Yorke system equations, it's possible to clearly see "delinearization" and "stabilization" steps.

Here is the system :
( R and T are the polar coordinates, X and Y the rectangular coordinates, T0 and e are the parameters )

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) + Y(n+1) ²
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)].

It's quite clear that :
- Y(n+2) = Y(n+1) + Y(n+1) ² is the delinearization operator, which will cause a chaotic behaviour to emerge
- T(n+1) = T(n) + T0 will cause the system to be periodic or pseudo periodic
- R(n+1) = e * log ( 1+R(n) ) is the stabilization operator, which will prevent the system to go to infinite values

For our study, we take T0=2 (expressed in radians), and e will be the parameter.
Here is the system's behaviour as e evolves :

The following movie shows the system's behaviour for all the relevant e values.

( 2D attractor, (X(2i),Y(2i)), with e from 1.59 to 1.89, step of 0.01, along with the audio samples corresponding to X(2i)
Quicktime movie, MPEG4 audio 64kbs, 860kB. )

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

As usual, we've been studying the system's stabilized state - after the transition time related to the choice of the initial condition has disappeared.
Even so, for e = 1.88, the chaotic behaviour behaves as a transitional state, but which is very long.

The following movie focuses on the inharmonic part, and on the transition to chaos.

( X(2i) spectrum, with e from 1.592 to 1.704, step of 0.004, along with the audio samples corresponding to X(2i)
Quicktime movie, MPEG4 audio 64kbs, 680kB. )

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

Here inharmonicity takes several aspects :
- small symmetrical rays around the main harmonics (ex 1.604)
- two harmonic notes superimposed (ex 1.656)
- a main note with another inharmonic note (ex 1.644)

Chaos appers between 1.656 and 1.660.
There is already a "window of order" at 1.672.

to be continued