Global optimal control and system-dependent solutions in the hardening Helmholtz–Duffing oscillator

A method for controlling nonlinear dynamics and chaos based on avoiding homo/heteroclinic bifurcations is applied to the hardening Helmholtz–Duffing oscillator, which is an archetype of a class of asymmetric oscillators having distinct homoclinic bifurcations occurring for different values of the parameters. The Melnikovʹs method is applied to analytically detect these bifurcations, and, in the spirit of the control method developed by the authors, the results are used to select the optimal shape of the excitation permitting the maximum shift of the undesired bifurcations in parameter space. The two main novelties with respect to previous authorsʹ works consist in the occurrence (i) of a more involved control scenario, and (ii) of system-dependent optimal solutions. In particular, the possibility of having global control “with” or “without symmetrization”, related to the optimization of the critical amplitudes versus the critical gains, is illustrated. In the former case, two sub-cases are found, namely, “pursued” and “achieved” symmetrization, which take care of the possibly simultaneous occurrence of right and left homoclinic bifurcations under optimal excitation.