RE: TUNING digest 1409

>Remember, Li and Yorke's "Period Three Implies Chaos" results, like those
>of Feigenbaum, only apply to unimodal maps of the interval.

Don't these occur far more often (in non-contrived physical situations)
that any other maps? (Clearly, a subjective question.)

>The "Feigenbaum
>constant" is not the same for all systems,

Right. It is 4.6692 . . . only for the unimodal, finite second
derivative maps that seem to occur most often. It is -2.8336 . . . for
the Fibonacci cascade of the zero-slope cubic nonlinearity, for example.

>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.

I would love to learn more about this. Was this period tripling sequence
(presumably meaning a trifurcation cascade) discovered by accident, or
was the pendulum specifically rigged to produce it?