Chaotic Orbits of a Pendulum with Variable Length

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.