The problem of avoiding the homoclinic bifurcation of the hilltop saddle of the Helmholtz equation by a shrewd choice of the shape of the external excitation is considered. The distance between the perturbed manifolds is computed by means of the Melnikov’s method, and its dependence on the shape of the excitation is emphasized. Successively, it is shown how it is possible to determine a theoretical optimal excitation which maximizes the distance between stable and unstable manifolds for a fixed excitation amplitude or, equivalently, which maximizes the critical amplitude for homoclinic bifurcation.
The practical case of a finite number of subharmonics is considered in detail. The corresponding optimal problems are solved numerically and the related optimal excitations are given. It is shown that when the number of subharmonics increases, the critical threshold for homoclinic bifurcation tends to double with respect to the reference case of harmonic excitation. Some numerical simulations are finally performed to verify the theoretical predictions and the effectiveness of the control procedure.