Incidentally, local linear control works just fine on the inverted
pendulum, and the textbook controllers so constructed are not chaotic.
(Haim Bau and collaborators suppress chaos with a NON-chaotic
controller perturbation, BTW: see Phys Rev Lett 66:1123.) Moreover, a
sinusoidally forced pendulum need not just "fall over." The driven
pendulum on my desk, for instance, balances easily at the inverted
point if the drive frequency is high enough - a phenomenon called
parametric resonance. See the papers listed below if you're
interested.
P. J. Bryant and J. W. Miles, "On a Periodically Forced, Weakly Damped
Pendulum. {Part II}: Horizontal Forcing", Journal of the Australian
Mathematical Society", 1990.
D. D'Humieres, M. R. Beasley, B. Huberman, and A. Libchaber, "Chaotic
States and Routes to Chaos in the Forced Pendulum", Physical Review A,
1982.
> So how do you apply this to dancers ? Find some chaotic dynamic
> systems and map them to dance.
Ah, but there's the hard part. Can you prove to me how I can do that,
mathematically and IN GENERAL? (I sure couldn't, and I've been doing
nonlinear dynamics for many years.)
We should probably take this discussion off line, lest we bore
everyone with all this mathematics.
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