Chaotic orbits of a pendulum with variable length

Date

2004-03-14

Authors

Furi, Massimo
Martelli, Mario
O'Neill, Mike
Staples, Carolyn

Journal Title

Journal ISSN

Volume Title

Publisher

Southwest Texas State University, Department of Mathematics

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 ω, 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.

Description

Keywords

Pendulum, Orbit, Chaotic, Separatrix

Citation

Furi, M., Martelli, M., O'Neill, M., & Staples, C. (2004). Chaotic orbits of a pendulum with variable length. <i>Electronic Journal of Differential Equations, 2004</i>(36), pp. 1-14.

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Attribution 4.0 International

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