Electron. J. Diff. Eqns., Vol. 2004(2004), No. 36, pp. 114.
Chaotic orbits of a pendulum with variable length
Massimo Furi, Mario Martelli, Mike O'Neill, & Carolyn Staples
Abstract:
The main purpose of this investigation is to show that a
pendulum, whose pivot oscillates vertically in a periodic fashion,
has uncountably many chaotic orbits. The attribute chaotic
is given according to the criterion we now describe. First, we
associate to any orbit a finite or infinite sequence as follows.
We write 1 or
every time the pendulum crosses the position
of unstable equilibrium with positive (counterclockwise) or negative
(clockwise) velocity, respectively. We write 0 whenever we find a
pair of consecutive zero's of the velocity separated only by a
crossing of the stable equilibrium, and with the understanding that
different pairs cannot share a common time of zero velocity.
Finally, the symbol
,
that is used only as the ending
symbol of a finite sequence, indicates that the orbit tends
asymptotically to the position of unstable equilibrium. Every
infinite sequence of the three symbols
represents a real
number of the interval
written in base 3 when
is replaced
with 2. An orbit is considered chaotic whenever the associated
sequence of the three symbols
is an irrational number of
.
Our main goal is to show that there are uncountably
many orbits of this type.
Submitted November 25, 2003. Published March 14, 2004.
Math Subject Classifications: 34C28.
Key Words: Pendulum, orbit, chaotic, separatrix.
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Massimo Furi
Dipartimento di Matematica Applicata "Giovanni Sansone"
Universita degli Studi di Firenze
Via S. Marta 3, 50139 Firenze, Italy
email: furi@dma.unifi.it 

Mario Martelli
Department of Mathematics,
Claremont McKenna College
Claremont, CA, 91711, USA
email: mmartelli@mckenna.edu 

Mike O'Neill
Department of Mathematics,
Claremont McKenna College
Claremont, CA, 91711, USA
email: moneill@mckenna.edu 

Carolyn Staples
Department of Mathematics,
Claremont McKenna College
Claremont, CA, 91711, USA
email: cstaples@mckenna.edu 
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