Optimal numerical control of single-well to cross-well chaos transition in mechanical systems

The optimal numerical control of nonlinear dynamics and chaos is investigated by means of a technique based on removal of the relevant homo/heteroclinic bifurcations, to be obtained by modifying the shape of the excitation. To highlight how the procedure works, the analysis is accomplished by referring to the Duffing equation, although the method is general and holds, at least in principle, for whatever nonlinear system. Attention is focused on the single-well to cross-well chaos transition due to a homoclinic bifurcation of an appropriate period 3 saddle [Int. J. Bifur. Chaos 4 (1994) 933]. It is shown how it is possible to eliminate this bifurcation simply by adding a single superharmonic correction to the basic harmonic excitation. Successively, the problem of the optimal choice of the superharmonic is addressed and solved numerically. The optimal solutions are determined in the two cases of symmetric (odd) and asymmetric (even) excitations, and it is shown how they entail practical, though variable, effectiveness of control in terms of confinement and regularization of system dynamics.