Introduction (3)
Chaotic systems used for other purposes than oscillators

When we will be studying, for instance, the Henon system, we will see that for certain parameter values, the system begins to behave chaotically, then stabilizes to an harmonic behaviour.

This "transitional state" can last more or less time.
Since :
1. It's a noisy transition stabilizing to an harmonic behaviour
2. Several chaotic systems exhibit this property,
it would be quite possible to eventually get something like that, based on several systems :

The question remains, whether or not this would be useful.
At the least, it's less obviously interesting than the use of chaotic systems as base oscillators.

I already studied this topic very, very briefly when I was working for Ircam, and it was part of a research which wasn't specifically focused on chaos, but on what was called délinéarisation.

The online address is http://recherche.ircam.fr/equipes/design/activites/traitements_install/dec2000/chapitre1.html, but the page is in french only.

The principle was to study the use of the Lorenz attractor as an alternative modulation aspect - modulation, the MIDI parameter #1.
The study, however brief, and using only one parameter value for the Lorenz system, was actually interesting : it showed that the use of the Lorenz system as a modulation "shape" gave a better result than either a random shape or the usual sine shape.

Better, that is less noticeable as such, but more efficient when it comes to enrich the sound like a "real" vibrato would do.

This modulation aspect will also be examined in the present study.

The "modulation" use and the "oscillator" use should be treated quite diffently.

Let's look at the behaviour of the Lorenz oscillator for two different Par values :

	212.5, attractor	212.5, waveform
	214, attractor	214, waveform

The two associated samples actually sound very diffently, though the waveforms are extremely similar.

As one can see on the attractor ( read as explained in "methods, how to read attractors" ), sample 214 is harmonic, whereas sample 212.5 is noisy.
To actually listen to the samples, please refer to : "raw system outputs, the Lorenz system", and see the transition from 195 to 224.75 ( from disorder to order ).

But it will be extremely hard to know the difference if those waveforms are slowed down and used as modulation shapes !

Thus the whole "gradation" stuff makes much less sense here.

As far as modulation is concerned, instead of thinking about something like :

... we should rather think about something like :

... so the method is much more simple : the only thing to be done is to get interesting and distinct behaviours for particular parameter values in particular systems.
Those will be the modulation shapes.