Date: Wed Sep 19 16:50:14 2007

Author: Urs Lauterburg

Subject: Re: Chaotic Pendulum source


Very nice to see you on tap-l. Your set-up is
impressive and also shows how nice it is to have
LabVIEW to do the measures. I'll check on your
AJP article when going to the library next time.



Urs Lauterburg
Physics demonstrator
Physikalisches Institut
University of Bern

>You can see the divergence arising from small differences in initial
>conditions of a driven chaotic pendulum from
>phase space representations of the motion seen
>The video consists of 17 hours of real data
>taken with the apparatus described in:
>Robert DeSerio, Chaotic pendulum: The complete
>attractor, AJP 71, 250-257, (2003)
>The loop shows 50,000 cycles of the drive and
>each frame of the video loop shows 50,000
>points with the horizontal direction giving the
>angular position of the pendulum
>(middle of figure is with the pendulum inverted,
>i.e., in an unstable upside down position)
>and with the vertical direction giving the
>angular velocity (middle of figure is zero
>The different frames of the loop are from
>sequential angular positions of the rotating
>shaft that drives the
>pendulum. These three variable are the only
>time-dependent variables in the apparatus and
>thus any point in any frame represents a
>complete description of its current state.
>While points near each other in any frame may come from very different times
>in the 17-hour run, they may be considered as
>nearly identical starting conditions.
>Watching the pattern you can see regions where
>the points diverge more quickly--as they come
>close to
>the center of the figure. The data set is a
>finite representation of the strange attractor
>this system and can be used to calculate its
>Lyapunov exponents and fractal dimension.
>The apparatus is used in our upper division lab. See:
>for more info.
>Robert DeSerio, Ph.D. Ph. (352) 392-1690
>Director, Instructional Labs Fax (352) 392-0524
>University of Florida
>Department of Physics
>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.
>>Dr. Richard E. Berg, Professor of the Practice
>>Director, Physics Lecture-Demonstration Facility
>>U.S. mail address:
>>Department of Physics
>>University of Maryland
>>College Park, MD 20742-4111
>>Phone: (301) 405-5994
>>FAX: (301) 314-9525

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