Primary resonant optimal control for homoclinic bifurcations in single-degree-of-freedom nonlinear oscillators

A primary resonant optimal control method (PROCM) is presented based on the adjustable role played by the phase shift in a general single-degree-of-freedom nonlinear oscillator. By Melnikov's method and rigorous mathematical deductions, the optimization solutions for the amplitude coefficients to be used as the control parameters can be obtained, and the force term as the controller can be designed. The main novelty of this PROCM is able to enlarge to the largest possible degree the control region where homoclinic transversal intersections embedding in systems dynamics do not occur, and this is accomplished at lowest cost. This method is confirmed by the hardening Helmholtz-Duffing oscillator, in which the various homoclinic bifurcations can be efficiently controlled locally and globally.