Stabilization of unstable periodic orbits in a Duffing system using an optimal time-delayed feedback control and pseudospectral method

Delayed feedback control is one of the most important methods for stabilizing unstable periodic orbits (UPOs) embedded within chaotic attractors. This paper compares two delay control techniques commonly used to stabilize UPOs: time-delay feedback control (TDF) and extended time-delay feedback control (ETDF). These chaos control strategies are investigated using the nonlinear Duffing system. The Chebyshev pseudospectral collocation method is employed to convert a delay differential equation (DDE) into a system of ordinary differential equations (ODEs). The chaos control problem is then transformed into an optimization problem, where the cost function minimizes a combination of the maximum magnitude of the Floquet multiplier and the maximum Lyapunov exponent of the target UPO. A comparison of the TDF and ETDF methods, along with the pseudospectral Chebyshev method, is made in terms of the stability of the controlled orbit and control effort.