Dynamics of forced escape from asymmetric truncated parabolic well

This study presents an analytic method for the estimation of safe basins in the plane of the initial conditions of the escape of a classical particle from an asymmetrically truncated quadratic potential well. For this purpose, an analytic method to estimate the global optimum of the sum of two harmonic functions is proposed. This approach is based on the mapping of the arguments of the two harmonic terms to the surface of the unit torus, where a surrogate optimization problem obtained by the Taylor expansion of the original objective function is solved. Applying the proposed method to the aforementioned escape problem helps predict safe basins for any value of the excitation frequency provided that the exciting force is not too strong, generating essentially non-linear effects on potential boundaries. Specifically, interesting effects with regard to the shape of safe basins occur when the natural frequency of the potential well and frequency of excitation represent the ratio of two small integers.