FLEXIBLE CONTROL OF THE PARAMETRICALLY EXCITED PENDULUM

The parametrically driven pendulum exhibits a large variety of stable periodic and chaotic motions, together with the hanging and inverted e quilibrium states. These motions can be oscillatory, rotational or a c ombination of these. The asymptotic solution depends crucially upon th e initial conditions imparted to the system for a given frequency and amplitude of forcing, used here as parameters. The existence of a larg e chaotic attractor has been numerically and experimentally verified, which persists for a reasonably broad range of the parameters. This ch aotic solution is referred to as a tumbling motion since it includes r otations in both clockwise and anticlockwise directions, as well as os cillations about the hanging position. Embedded within the correspondi ng attractor is an infinite number of unstable periodic solutions whic h may be classified according to the number of oscillations or rotatio ns within a given number of periods of the periodic driving force. In this paper, the topological theory of dynamical systems is used to pin point the location in parameter and phase space of desired orbits. Num erical procedures can then be readily applied to refine this informati on and a simple control algorithm applied to stabilize this unstable o rbit. The initial theoretical approach provides greater flexibility in enabling the system to achieve a variety of different periodic states by small adjustments of the driving frequency. Remarks are also made regarding experimental implementation.