Introduction (2)
Chaotic systems used as audio oscillators - method

So, we want tu use chaotic systems as audio synthesis oscillators.
There are other uses ( attack generation, LFO use... ), but they're not the main point. See "introduction, page 3" for those subjects.

So, if we deal with oscillators, ultimately we want to get something like that :

This so called "perceptively linear evolution" can be many different things, eg :

- amount of "noise grains" in an harmonic context (Lorenz system)
- "noisy perturbation" amount in an harmonic context (Lorenz system)
- amount of harmonics / sub harmonics adding themselves to a sine wave (logistic equation)
- "noisy perturbation" amount, evolving along with the number of harmonics ( logistic equation )
- spectral gravity center ( centre de gravité spectral ) correlated with the aspect of a high noisy part ( Duffing system )
- "noise grains" turning into inharmonicity ( Chirikov system )
- intermittent "noise grains" appearing in a treble noisy aspect on top of an harmonic context ( Lozi system )

.... and there are a lot of other possible evolutions.

Very often, those evolutions include a "disorder <-> order" aspect.
To get back to the "order / disorder" axis, they can be represented the following way :

But this "order / disorder" aspect is not always obvious.
For instance, in the given examples, the Duffing and Chirikov systems don't feature such an "order gradation".

We have to break the problem into several distinct steps, and draw the method from the goals.

Let 's take the logistic equation, for instance :
U(n+1) = (R+1) U(n) - R U(n)²
( this system is also studied later in details, to begin with at "Raw system outputs : the logistic equation". )

Step 1a : Par

As there is only one, the parameter will be R.

When there are two or several parameters, experience shows that the parameter behaviours are extremely similar.

Step 1b : extraction of the relevant Par values

In this system, several insteresting gradations can be seen, even at first glance.

- there is the first doubling period cascade for R values between 2.54 and 2.57
- then, the progressive apparition of noise near 2.57
- just after, a reverse doubling cascade, this time with noise, between 2.57 and 2.58
- further, there is something that looks like some kind of "tripling period cascade", near 2.8.
- etc

( step 1 is addressed in the section "Section 1, Raw system outputs" )

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Step 2 : timbre optimisation

These interesting gradations are often more easily seen than heard - and for reasons which are generally quite simple.

For instance let's take the first doubling period cascade : lower harmonics appear gradually when R increases, but some of those sub harmonics are 40, 60, or even 70dB below the base frequency.

The solution here is a simple spectral compression, to reduce the level difference between harmonics, so the phenomenon can be properly heard.

( step 2 is addressed in the section "Section 2, Optimization" )

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Step 3 : perceptive linearization

This psychoacoustic problem is of a very common kind.

It is well known that a sound doesn't appear to be two times louder when its RMS power is doubled. Thus the use of the dB unit and its declinations.
The same way, if we take our subharmonic cascade, the system exhibits one subharmonic near R=2.47, and two near R= 2.555.
But the third one does not appear at R=2.64, but at R=2.565. And the fourth one at R=2.569, the fifth one at R=2.5698...

Thus, a perceptively linear evolution would correspond to a non linear parametric evolution.

This phenomenon happens for each and every system : in each case, a "transfer function" has to be found so the perceptive evolution is perceived as linear.

Obtaining this transfer function is not always a simple thing.
Being rigorous here would mean needing advanced mathematics knowledge, so the only way to do it is by closely testing the system's behaviour, getting meaningful Par values, and then deducing an approximation of the transfer function, which is not very difficult, but remains quite long.

( step 3 is addressed in the section "Section 3, Linearization" )