Optimal control of nonregular dynamics in a duffing oscillator

A method for controlling nonlinear dynamics and chaos previously developed by the authors is applied to the classical Duffing oscillator. The method, which consists in choosing the best shape of external periodic excitations permitting to avoid the transverse intersection of the stable and unstable manifolds of the hilltop saddle, is first illustrated and then applied by using the Melnikov method for analytically detecting homoclinic bifurcations. Attention is focused on optimal excitations with a finite number of superharmonics, because they are theoretically performant and easy to reproduce. Extensive numerical investigations aimed at confirming the theoretical predictions and checking the effectiveness of the method are performed. In particular, the elimination of the homoclinic tangency and the regularization of fractal basins of attraction are numerically verified. The reduction of the erosion of the basins of attraction is also investigated in detail, and the paper ends with a study of the effects of control on delaying cross-well chaotic attractors.