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.