Discrete-time partial control of transiently chaotic systems under stochastic perturbations

Stochastic perturbations can steer the response of a nonlinear system to specific regions of the system state space, including equilibrium points, periodic solutions, and aperiodic (including chaotic) solutions. This study constructs a discrete partial control scheme to confine the response trajectories of a forced Duffing oscillator, showing transient chaos, within a chaotic attractor, thereby avoiding undesired behavior. The system is analyzed in the presence of white noise, using a minimally bounded control input. The primary motivation is to evaluate the viability of applying bounded control inputs at discrete time instants over intervals differing from the forcing period and to examine the required control input magnitude for response confinement. Results demonstrate that the system can be effectively controlled with relatively small magnitude control inputs applied at discrete time instants. A careful selection of time steps is crucial to ensure that the response trajectory remains within the desired region. The scheme's formulation and the associated algorithm are independent of the system dimension.