Bifurcations and stability of the vertically forced n-pendulum as n→∞

Previous work has considered the configurations in which an n-pendulum stabilizes when forced by high-frequency open-loop periodic excitation of the pendulum base. In this paper, we study the bifurcations and stabilization of inverted equilibria of the vertically forced n-pendulum when n→∞, while total length and mass are held constant. We show that as n becomes large, the frequencies at which inverted equilibria stabilize also become large, and tend to infinity as n→∞. This example illustrates a fundamental difficulty in the synthesis of open-loop control laws from discretizations of infinite-dimensional systems