Optimal control of homoclinic bifurcation: Theoretical treatment and practical reduction of safe basin erosion in the Helmholtz oscillator

A control method of the homoclinic bifurcation is developed and applied to the nonlinear dynamics of the Helmholtz oscillator. The method consists of choosing the shape of external and/or parametric periodic excitations, which permits us to avoid, in an optimal manner, the transverse intersection of the stable and unstable manifolds of the hilltop saddle. The homoclinic bifurcation is detected by the Melnikov method, and its dependence on the shape of the excitation is shown. We successively investigate the mathematical problem of optimization, which consists of determining the theoretical optimal excitation that maximizes the distance between stable and unstable manifolds for fixed excitation amplitude or, equivalently, the critical amplitude for homoclinic bifurcation. The optimal excitations in the reduced case with a finite number of superharmonic corrections are first determined, and then the optimization problem with infinite superharmonics is investigated and solved under a constraint on the relevant amplitudes, which is necessary to guarantee the physical admissibility of the mathematical solution. The mixed case of a finite number of constrained superharmonics is also considered. Some numerical simulations are then performed aimed at verifying the Melnikov's theoretical predictions of the homoclinic bifurcations and showing how the optimal excitations are indeed able to separate stable and unstable manifolds. Finally, we numerically investigate in detail the effectiveness of the control method with respect to the basin erosion and escape phenomena, which are the most important and dangerous practical aspects of the Helmholtz oscillator.