More chaotic stuff

Ah yes, more on chaos theory shoehorned into a musical discussion.
Musicians can still be reassured that they don't need to know this
stuff.


Paul Erlich wrote on the version of Duffing's equation I posted:

>While most physical nonlinearities are characterized by a quadratic
>fixed point, this appears to be one with a zero-slope cubic
nonlinearity
>(inflection point).

"Most physical nonlinearities" is something of a generalisation. I
thought Duffing's equation is pretty much a classic for forced
oscillations, but maybe only in pure mathematics. I never did any chaos
theory in physics. Not important here, as I assume we're still thinking
of an artificial process in the brain.

>Instead of a period-doubling cascade, changes in the
>parameter value for this type of system lead to a Fibonacci sequence of
>period lengths. So changing the parameter in this case could decrease
>the period to 2 or increase it to 5. I don't see how this is more
>"stable" that a typical order 1/n^2 subharmonic for a
>quadratic-fixed-point nonlinearity.

I meant "stable" in the usual sense that, of you perturb the system
slightly, it will not tend towards an entirely different behaviour.
This is the minimum criterion for a limit cycle to be at all useful.
There are, in fact, four centres and three (unstable) saddles. Centres
are stable, but not asymptotically stable, and this is usually how limit
cycles arise, although I don't know if that's relevant in this case.
Slight damping means the centres become stable spirals, and their basins
of attraction are much reduced.

Here's some text:

"For the linear equation this[not the system above] periodic motion
appears to be merely an anomalous case of the usual almost-periodic
motion, depending on a precise relation between the forcing and natural
frequencies. Also, any damping will cause its disappearance. When the
equation is nonlinear, however, the generation of alien harmonics by the
nonlinear terms may cause a stable subharmonic to appear for a range of
the parameters, and in particular for a range of applied frequencies.
Also, the forcing amplitude plays a part in generating and sustaining
the subharmonic even in the presence of damping. Thus there will exist
the tolerance of slightly varying conditions necessary for the
consistent appearance of a subharmonic and its use in physical systems."

D.W.Jordan & P. Smith "Nonlinear Ordinary Differential Equations" 2nd
edition, Oxford, 1987, p.196.

The following general case (Duffing's equation without damping) is then
covered:

x'' + a*x + b*x^3 = G*cos(w*t)
w^2*x'' + a*x + b*x^3 = G*cos(tau)

In the first equation, differentiation is with respect to t, in the
second wrt to tau. The two are equivalent, and the second is used as
the frequency is not known before the parameters are chosen. There's a
lengthy calculation, that I haven't followed, the result being (p.200):

a =~ w^2/9

As "the calculations are valid without the restriction on b." I think b
should still be small, though. It looks like this is a good way of
generating a subharmonic of a particular value, if that's what you want
to do. Not all of these subharmonics are stable, though, so I quoted
one that is proved such as an example in the next section.

Whatever mechanism, some kind of nonlinear equation seems to be the way
to perform frequency reduction. The most useful subharmonic will still
be of order 1/2, but that is not inevitable. According to the text,
this sort of thing goes on in quartz clocks.

> The actual frequency at which periodicity pitch gives up is about 5-6
> kHz. Really high!

Yes, this is higher than I thought. I was expecting a few hundred Hz.

>>The latest book's I've read --
>>which, as they came from libraries, are a decade or three old -- say
that
>>the original experiments showing octave invariance were performed on
>>musically literate subjects.

> Are white rats musically literate?

No, obviously not. I'd still prefer humans, but please outline any
experiments you know of. Not having access to a university library, I
can't get hold of academic papers very easily.

>>More specifically, literate in Western
>>Classical Music, where the notation is octave based.

>Many other cultures use octave-based notation.

Whoa! Hold on there! When did I say anything else?

>>> As for the apparantly irregular "subharmonic" which Gary observed in
the
>>> bassoon waveform, this can easily be explained by assuming some
>>> parameter of non-linearity (perhaps lip pressure) was hovering
around a
>>> value at which an initial period doubling occurs. So the amplitude
of
>>> this period-2 subharmonic could have been changing, and it could
cease
>>> to exist for a while, returning again just as easily after either an
odd
>>> or even number of period-1 oscillations.

>>This assumes the system is teetering in a bifurcation point, which is
>>actually very unlikely.

>Not unlikely at all! Look again at a bifurcation diagram -- there is
>quite a large range of values where the period-2 behavior has a
>relatively constant amplitude, but at one end of this range the
>amplitude suddenly decreases and period-1 behavior ensues. If the
>bassoonist is trying to keep within this range, say because he is
trying
>to play as loud as he can and octave subharmonics don't bother him but
>chaos does, the parameter will likely take on just such a range of
>values.

If there's some reason to move toward a bifurcation, that would make a
difference. If "the period-2 behaviour has a relatively constant
amplitude" why not play right in the middle of the period 2 region,
instead of veering toward the bifurcation?

>>I'd guess the amplitude of the subharmonic is
>>chaotic. Or, the subharmonic is entirely chaotic but appears to be
order
>>1/2 at certain times

>>A period doubling cascade can be
>>linked to amplitude, though.

>These don't sound like examples of real-world dynamics.

On thinking about it, the chaotic subharmonic is unlikely.

It's obvious I'm relying on one book, but anyway ... The example is for
Duffing's equation again:

x'' + k*x' + x^3 = G*cos(w*t)

The bifurcation parameter is G (really Gamma) which is the forcing
amplitude. A period doubling cascade is demonstrated (pp336-345).

>>It's unlikely that a period 3 cascade could be picked out from a
chaotic
>>region with one parameter.
>
>Not unlikely at all, I've done it with my vocal cords.

Well done! How did you perform this experiment?

>>In the Mandelbrot set, though, there is a
>>fairly large period 3 region. Similar things might occur with
>>differential equations of 2 parameters for all I'd know.

>Actually, a differential equation needs to have at least 3 parameters
>for chaos to occur. However, the picture along whichever of the
>parameters is responsible for chaos will be the good old bifurcation
>diagram. The Mandelbrot set represents a discrete (not continuous)
>dynamical system, which only needs one parameter to exhibit chaos. If
>you restrict yourself to the real line, you see the usual bifurcation
>diagram again (proceeding from right to left).

I meant that the bifurcation parameter becomes a plane, like the
Mandelbrot set, rather than that the system has two variables. I can
see that isn't clear.

Minor correction: that should be "an autonomous differential
equation..."

>>I think there
>>might even be a period 3, 9, 27, ... cascade in the Mandelbrot set.
>
>Naah, the Mandelbrot set is clearly all about period doubling; the
>iteration of a point on the set's boundary doubles the angle the point
>makes on the unit circle (to which the boundary can be conformally
>mapped). Although you could probably find some contorted path in the
>complex plane to support your guess.

You bet! The way I would implement twelfth-reduction, though, is to use
a 1/3 order subharmonic generator, and repeat the process until the
frequency became suitably low.

Incidentally, Benoit Mandelbrot is a Polish mathematician, with some
links with France, working in the USA. I have no idea how he pronounces
his name, but everyone I know, including one Italian who's met him,
sounds the last 't' in 'Mandelbrot'.