Date: Wed Sep 19 10:58:50 2007

Author: Bernard Cleyet

Subject: Re: Chaotic Pendulum source

I don't understand. My impression was the only cause of divergence was
initial condix. I objected to this. Either cause of divergence is
easily quantified w/ one apparatus, as I suggested. IIRC, Wiki didn't
discuss comparing mechanical apparatuses WRT slight differences in their
character. Also I don't understand a theoretical design demonstration.
What do you mean by your second sentence? Your first four sentences are
not self consistent.

One reason I suggested Wiki. is I didn't find in it the necessity of two
"identical" apparatuses.


Richard Berg wrote:

> Certainly the initial conditions have an impact on how the motion
> develops. My point is that the demonstration is not designed
> theoretically to show that. It is designed to demonstrate that the
> small differences in initial conditions lead to much larger deviations
> of the ensuing motion. And the double pendulum does that well. This
> is consistent with the discussion in Wikipedia.
> Dick
> On Tue, 18 Sep 2007, Bernard Cleyet wrote:
>> Unless one makes changes in the mechanism, I don't think one can
>> claim the divergence is solely due to initial conditions and not also
>> changes in the mechanism or environment. In a sense such changes are
>> changes in the initial conditions, they just come later!
>> Regarding the definition of chaos, I recommend Wiki's:
>> bc, sad the UCSC analogue lab that required examination of
>> Roessler's, Lorenz', et al. equations is no longer.
>> Richard Berg wrote:
>>> It is my understanding that Model 1 of the double chaos pendulums
>>> was made by Troy Shinbrot, who did his PhD at the University of
>>> Maryland. The device is described in the paper:
>>> Shinbrot, Grebogi, Wisdom, and Yorke, Chaos in a Double Pendulum,
>>> AJP 60, 491-499, (1992).
>>> and in a supplemental sheet that describes the construction of the
>>> pendulum:
>>> Supplemental Information Sheet, T. Shinbrot, (1989)
>>> The idea is to demonstrate that due to the chaotic nature of the
>>> devices even the most minute difference in the initial starting
>>> conditions will result in virtually total divergence of their
>>> motions. You can show "chaos" with a single pendulum, but not
>>> "chaos" as defined by the masters: Jim Yorke, et. al. You need
>>> two. Chaos refers to the gross deviation due to miniscule
>>> differences in the starting conditions. If you use one such
>>> pendulum, you show that the motion appears random, but chaos is more
>>> carefully defined in this case, I believe. Chaos is calculable, if
>>> you can model the system accurately enough.
>>> Dick


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