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From: brian@hpcvlx.cv.hp.com (Brian Phillips)
Newsgroups: comp.theory.dynamic-sys
Subject: Pendulum Help
Message-ID: <110770003@hpcvlx.cv.hp.com>
Date: 31 May 91 18:56:18 GMT
Organization: Hewlett-Packard Co., Corvallis, OR, USA
Lines: 34



       An old classical mechanics problem... 

           I've been toying with a mathematical simulation of an
       inverted pendulum ( a rigid bar rotating in the plane about
       a pivot subjected to an up-down forcing function ).  The forcing
       function is sinusoidal and the pendulum is initially ajar at some
       small angle from top-dead-center (unstable equilibrium).  As I
       understand it, such a system can maintain itself without rotation
       if the correct conditions are established.  Is this correct?
           Can such a pendulum remain upright?  Does it periodically rotate
       to a new semi-stable angle?  What constitutes stability in such a
       system?  While I am interested in examining system behavior with
       regard to its transition from stability to (chaos?) instability, I
       am most curious about the parameter domains for values creating
       these transitions.  ie.  If I vary the length of the pendulum slightly
       and a previously stable pendulum simulation becomes otherwise, what 
       will I see if I refine such changes?   What does a region about a stable
       choice that includes unstable selections look like?  Are the boundaries
       well-behaved or are they more interesting?  What if I look at the 
       frequency of the forcing function much the same way?  
           I'm not particularly interested in a purely mathematical exercise
       where the results are merely artifacts derived from the simulation
       method and its accuracy.  Any guidance would be appreciated.

       Brian Phillips
       brian@hpcvxbjp.cv.hp.com