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 equilibrium states. These motions can be oscillatory, rotational or a combination of these. The asymptotic solution depends crucially upon the initial conditions imparted to the system for a given frequency and amplitude of forcing, used here as parameters. The existence of a large chaotic attractor has been numerically and experimentally verified, which persists for a reasonably broad range of the parameters. This chaotic solution is referred to as a tumbling motion since it includes rotations in both clockwise and anticlockwise directions, as well as oscillations about the hanging position. Embedded within the corresponding attractor is an infinite number of unstable periodic solutions which may be classified according to the number of oscillations or rotations within a given number of periods of the periodic driving force. In this paper, the topological theory of dynamical systems is used to pinpoint the location in parameter and phase space of desired orbits. Numerical procedures can then be readily applied to refine this information and a simple control algorithm applied to stabilize this unstable orbit. 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.