Electron. J. Diff. Eqns., Vol. 2004(2004), No. 36, pp. 1-14.

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 $-1$ 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.

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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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