Control of Fractal Erosion of Safe Basins in a Holmes–Duffing System via Delayed Position Feedback

A linear delayed position feedback control is applied to control the erosion of safe basins in a Holmes–Duffing system. The conditions of fractal erosion of the safe basin of the controlled system on the basis that the range of time delay leading to good control is obtained by the Melnikov method. It is found that the increasing time delay can reduce the basin erosion under a weak and positive feedback gain. Then the evolutions of safe basins with time delay are presented in detail by the fourth Runge-Kutta and Monte-Carlo methods, which shows that the safe basin of the controlled Holmes–Duffing system can be expanded, and its fractal can be reduced by the increasing time delay. These results suggest that delayed position feedbacks can be used as a good approach to control the erosion of safe basins.