Date: Wed Sep 19 12:10:17 2007

Author: Richard Berg

Subject: Re: Chaotic Pendulum source

Post:

Sorry. I wrote that while i was half asleep and didn't proof read. It is
true that the characteristics of the device, such as friction and actual
machining errors, do play a part in how the motion develops. However,
that is not what the device is designed to show. It IS dewsigned to show
that two setups, started with virtually the same initial conditions, will
nevertheles diverge due to the minute differences in their starting
conditions. You need two pendula to show this. In fact, a nice way to
use the apparatus is to ask someone (perhaps a student) to attempt to
start the two pendula so that their motion will continue the same for all
time. This is impossible, due to the concept of chaos, irrespective of
other issues such as friction or mechanical errors.

Dick

On Wed, 19 Sep 2007, Bernard Cleyet wrote:

> 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.
>
> bc
>
> 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:
>>>
>>>
>>> http://en.wikipedia.org/wiki/Chaos_theory
>>>
>>>
>>> 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
>>>>
> cut
>
>>
>>
>

***********************************************************************
Dr. Richard E. Berg, Professor of the Practice
Director, Physics Lecture-Demonstration Facility