Delayed feedbacks to control the fractal erosion of safe basins in a parametrically excited system

Three types of delayed feedbacks, namely, the delayed position feedback (DPF), delayed velocity feedback (DVF) and delayed state feedback (DSF), are proposed to control the erosion of safe basins in a parametrically excited system. The time delay and the gain in the feedbacks are chosen as control parameters. The safe basin erosion is measured by the size and nature of the safe basin. The nature of the safe basin, fractal or smooth, is investigated by evaluating the fractal dimension. The critical values of the control parameters are obtained for any points in the safe basin not to be eroded. It is concluded that delayed feedbacks can be applied to control the extent of the erosion. Moreover, the comparison among the effects of the three types of delayed feedbacks reveals that DPF is the best strategy to reduce the fractal erosion of safe basins at small values of the delay and the gain, while DSF becomes the best when the delay or the gain is large.