On the periodically driven inverted pendulum

We study the solution properties of a family of inverted pendulum systems driven by odd periodic forcing. Using the Schauder fixed point theorem, we show that the inverted pendulum with an odd periodic driving acceleration at the pivot always possesses an odd periodic solution. Fundamental to the production of good estimates is the development of a Green's function for an unstable harmonic oscillator with Dirichlet boundary conditions. We also show that it is sometimes possible to use the Banach fixed point theorem to ensure that there is a unique solution within an invariant region of the space of possible solution curves. Using these results, we characterize the solutions of periodically driven inverted pendulum systems such as that given by ¿¿ = ¿² sin ¿ + ß sin (¿ - ¿t), which describes a pendubot with constant inner arm velocity. These results are important as the driven inverted pendulum is a common subsystem in systems ranging from motorcycles and bicycles to rockets and aircraft.