[sdiy] slightly ot: Chua circuit and chaos

[Grant Richter]

Hello All,

It's interesting to consider where analogue computation may be inherently
superior to digital methods.

There is a thing called the "cardinality" of infinite sets. In other words,
you can quantify how infinite an "infinite" set is.

Whole numbers have the cardinality of "Aleph Null" which is the lowest
cardinal order. Real numbers have the cardinality of "c" or the continuum,
the next larger cardinal number.

It can be argued that digital computation, by its nature, has a cardinality
equivalent to the whole numbers (the problem of quantization) whereas
analogue computation has a cardinality equivalent to the real numbers.

So analogue computation has a much larger set of available solutions than
digital methods. This is very handy for recursive chaotic systems where
digital methods tend to saturate.

>From a mathematical perspective, the soft saturation of an op-amp at the
rails still contains a larger set of solutions than all the IEEE Floating
Point numbers. In this case, soft saturation and thermal noise is what
allows the systems behavior to still be described at relative "infinity"
(the supply rails) whereas a digital system can return only a single
solution (largest number).

> 4. Hype: "Chaos" was a great hype in the 80s-90s. Suddenly everything
> was chaotic, a new "world concept" was promised.
> People from other fields of science or even people who
> had no clue about math at all wrote papers and books
> that can not withstand even 10 seconds of careful though.

If you ask the question "why does any moment in time know anything about any
other moment" there would seem to be a mechanism whereby information
persists over time. If this mechanism is recursive (a good way to cause
persistence) then Chaotic and Fractal behavior would be a natural
consequence of a recursive time mechanism.

Mathematics has simply discovered natural behavior (that's a joke, OK).

So it is possible that so called "Chaos" is the normal state of things and
our several hundred year illusion of predictability is just a sensor ghost.