chaos in the ear

All-

a snip from an exchange between Breed & Erlich
>> Therefore, assuming the
>> parameter value varies smoothly with time, and at some times takes on
>> values corresponding to simple period-1 vibration, the only subharmonics
>> which can exist without chaos ever occuring are the subharmonics
>> corresponding to powers of 2. Period-3 oscillation (or, to a lesser
>> extent, periods of higher odd numbers) can be relatively common but
>> cannot smoothly connect with simpler behavior.
>
>It's unlikely that a period 3 cascade could be picked out from a chaotic
>region with one parameter. 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. I think there
>might even be a period 3, 9, 27, ... cascade in the Mandelbrot set.

Remember, Li and Yorke's "Period Three Implies Chaos" results, like those
of Feigenbaum, only apply to unimodal maps of the interval. The "Feigenbaum
constant" is not the same for all systems, and there are maps and ODEs that
period triple from simple behavior. In fact, just a couple of weeks ago, one
of my students found a period tripling sequence in a physical parametrically
driven pendulum.

John Starrett
http://www-math.cudenver.edu/~jstarret