Universality of chaos as both problem and opportunity

[gogins]

I have proved that iterated function systems are effectively compositionally universal.

The are two problems: the parameter space is too big to efficiently use, and the compositions are distributed in way that is not musically intuitive, but rather is determined by the period doubling and consequent universality in the underlying chaotic dynamical systems (the elements of the Hutchinson operator in the IFS, which has a fixed point due to its contractivity).

These same problems apply as well to simpler dynamical systems used to generate pieces as time series of notes or chords.

I'm not sure my thinking is coherent here, but I have to come up with something, the compositional universality is obviously there, and is too valuable to let go.

Filtering?

Windowing?

[gogins]

Leon Chua strikes again. I hope this turns out to be as relevant as my first impression indicates.

Having skimmed the proof of the local activity theorem in this book, I can say that these results apply to systems of coupled cells, each with internal state and connections to other cells (ports). A cell is locally active iff when some port absorbs energy, another port emits more energy (a transistor is an example). A homogeneous lattice of cells with one locally active cell can exhibit globally chaotic behavior. When such behavior is neither periodic nor chaotic, it is active at the edge of chaos. This corresponds to the type of rule 110 one-dimensional cellular automaton that is proved capable of universal computation.

Although the local activity theorem is a valid proof as far as I can see, I don't see that it quantitatively defines the edge of chaos, and in fact such a definition may not exist, and whether a dynamical system lies at the edge of chaos may be undecidable in general.

A useful program running in such an array would have to halt with a useful output, and such programs would be included in any concept of the edge of chaos, so what I conjectured must be the case. A program that did not halt would be either chaotic or periodic.

[gogins]

Chua goes on (making me a bit nervous) [here](Chua, L.O. Memristors on ‘edge of chaos’. Nat Rev Electr Eng 1, 614–627 (2024). https://doi.org/10.1038/s44287-024-00082-1):

Rather than echoing the vision and perspectives proffered by numerous previous publications, this Review focuses on the recent resolution of four unsolved classic problems — Galvani’s ‘irritability’, the Hodgkin–Huxley ‘all-or-none’ mystery, the Turing instability and the Smale paradox — the oldest dating back 243 years to Galvani in 1781. Unlike advances reported previously, which tend to be ephemeral, our resolution of these problems is timeless, because they are a manifestation of a new law of nature, called the ‘principle of local activity’, which, within a certain relatively small parameter space, could harbour a physical state dubbed the ‘edge of chaos’. In this Review, we provide an explicit formula for calculating, via matrix algebra, the precise parameter range where a nonlinear device, or system, is locally active or operating on the edge of chaos. Unlike numerous unsuccessful attempts by luminaries, such as Boltzmann’s assay for decreasing entropy, Schrödinger’s futile search for negentropy, Prigogine’s quest for the ‘instability of the homogeneous’ and Gell-Mann’s musing on ‘amplification of fluctuations’, the principle of local activity provides an explicit formula to identify the parameter space where the edge of chaos reigns supreme.

I feel that here Chua is grandiose. However, his contribution of the memristor is an undeniable major result.

It's the final phrase of this quotation that interests me.

[gogins]

I think my question is becoming a little bit clearer -- not the answer, but the question, which is actually more important than the answer.

The question is, given that a system is universal, or at least has musically interesting aspects, what makes it not only intelligibly but also efficiently explorable? My original Mandelbrot set model is intelligibly explorable, but it is not efficiently explorable.