Influence of damping on the delta-kicked harmonic oscillator with Heaviside kick

A damped harmonic oscillator, kicked at a fixed frequency by a kick with a fixed strength but with a sign given by the sign of the velocity of the oscillator, is investigated. This system is encountered when investigating a water-filled U-tube in which bubbles are released at the bottom. A mapping similar to the one of Scott et al. [Physica D 155 (2001) 34] is used to describe the kicked oscillator, but here incorporating the effect of friction. Similar to Scott et al. [Physica D 155 (2001) 34] the stable orbits can be calculated, but now for the dampened case. The behavior of the oscillator is investigated numerically as a function of its period number and the strength of the friction parameter. In general, the higher the damping the simpler the trajectory of the oscillator and the quicker the steady state is reached. However, for certain values of the period number and the friction parameter complex trajectories are found, some of which may not have reached a steady state, due to slow relaxation. The oscillator responds sensitive to changes in both the period number and the friction parameter. Finally, the influence of the initial conditions on the final mode of the oscillator has been investigated. It is found that the system is sensitive to the initial conditions. In the initial conditions plain, four spiraling arms are found where the mode of the oscillator deviates from the rest of the plain.