Time-Delay Control Strategies for the Nonlinear Dynamics of a Gravity Wave-Driven Buoyant Pendulum

This work examines how time-delayed feedback control (TDFC) can regulate the motion of a submerged buoyant pendulum interacting with surface gravity waves. The system is modeled through a nonlinear differential equation that captures lateral displacement, with external excitation arising from wave-induced drag under small slope approximations. By analyzing time-based signals and frequency characteristics, key phenomena such as primary resonances and subharmonic components are identified. The method of multiple scales is employed to derive analytical expressions, which are then confirmed with numerical simulations to reflect the nonlinear dynamics observed. The stability of the steady-state solutions of the system is evaluated by calculating the eigenvalues of the linearized system matrices. The impact of feedback control incorporating time delay is explored in depth—specifically, how fixed feedback gains paired with varying delay durations influence system behavior. Results show that modifying the delay can either suppress or amplify oscillations, offering a strategic means to fine-tune the response of the pendulum. This insight provides practical guidance for either reducing vibrations in marine structures or boosting energy extraction in wave-powered systems. The overall findings highlight TDFC as a versatile and effective approach to managing nonlinear motion in hydrodynamic environments.