Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 36, pp. 1-14.
Title: Chaotic orbits of a pendulum with variable length
Authors: Massimo Furi (Univ. degli Studi di Firenze, Italy)
Mario Martelli (Claremont McKenna College, CA, USA)
Mike O'Neill (Claremont McKenna College, CA, USA)
Carolyn Staples (Claremont McKenna College, CA, USA)
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 $-1$ 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 $\omega$, 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 $\{1,-1,0\}$ represents a real
number of the interval $[0,1]$ written in base 3 when $-1$ is replaced
with 2. An orbit is considered chaotic whenever the associated
sequence of the three symbols $\{1,2,0\}$ is an irrational number
of $[0,1]$. 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.