A unified control framework of the non-regular dynamics of mechanical oscillators

A method for controlling non-linear dynamics and chaos based on avoiding homo/heteroclinic bifurcations is considered in a unified framework. Various non-linear oscillators, both hardening and softening, symmetric and asymmetric, considered as archetypes of a more general class of single-d.o.f. systems, are analyzed. The Melnikovʹs method is applied to analytically detect the homo/heteroclinic bifurcations, and the results are used to select the optimal shape of the excitation permitting the maximum shift of the undesired bifurcations in parameter space. The generic character of the optimization problem is highlighted, and the problem itself is discussed in detail. Various control strategies are proposed, based on the elimination either of a single bifurcation (one-side controls) or of all bifurcations (global control). The optimization problems are solved, analytically and numerically, under various forms, by taking into account the physical admissibility of the related optimal excitation and the easiness of implementation in practical applications. It is shown that the solutions of one-side control problems are always system independent. In turn, the solutions of global control problems are system independent for the large class of (symmetric) systems considered in this work, although there are other systems whose solutions are system dependent. These considerations support the very generic nature of the optimal control.