
The sinusoidally driven damped pendulum as well as the Lorenz system is a system with continuous time and a threedimensional state space. Its motion is ruled by the following set of differential equations: Contrary to the Lorenz system here is one degree of freedom which does not depend on the others, i.e. the driving (the third equation). This is because the system is driven by an external regular sinusoidal torque whose phase grows proportional to the time. Systems with this property are called nonautonomous. Another property that cannot be found at the Lorenz system is the periodicity of the state space. In order to determine the pendulum’s state uniquely it is sufficient to reduce both, the phase and the deflection x to the interval [ ... ] since it does not matter how many driving periods are passed and how many times the pendulum has rotated to the one or the other side. Depending on the driving amplitude and the driving period attractors of totally different kinds can exist. For a vanishing driving torque the neutral position (x = 0, v = 0 ) is a stable fixed point (in this limit the state space is two dimensional). For small driving amplitudes the motion of the pendulum follows the driving torque with some phase shift. The attractor now is a stable periodic orbit to which the trajectory of the pendulum converges no matter what the initial conditions of the motion may have been. For some ranges of the driving parameters, however, the pendulum can exhibit seemingly totally irregular motions. In this chaotic regime the trajectory follows a strange attractor.
A driven pendulum has been realized experimentally at the Physikalisches Institut der Uni Frankfurt about ten years ago and is still used for demonstrations in lectures [Heng94], [Hübinger94], [Doerner94]. At this pendulum the predictability of lowdimensional chaotic motions has been investigated [Doerner93], [Doerner94b], [Doerner99], [Doerner99b] and the local control method has been applied [Hübinger93], [Hübinger94b], [Doerner95]
A Poincaré section of the pendulum’s attractor with the plane , where the time grows continously causing a movielike effect. The section is plotted several times in order to visualize the periodicity of the attractor in the direction of the angular deflection.
The effective Lyapunov exponent measures the short time predictability of the motion. Here it is displayed on the Poincaré section for a forecasting interval lasting one driving period. Again the time grows continuosly and again the section is plotted several times in order to visualize the periodicity of the attractor in the direction of the angular deflection.
appendix: how to derive the dimensionless equations from the physical pendulum A physical pendulum is characterized by its moment of inertia I (unit of measurement kg m^{2}), its damping rate (unit of measurement kg m^{2}s^{1}), the repelling moment M resulting from a deflection by 90° (unit of measurement N m = kg m^{2}s^{2 }) and the sinusoidally driving torque with the amplitude A (unit of measurement N m = kg m^{2}s^{2}) and the frequency (unit of measurement s^{1}). Then the equation of motion reads:




