Attractor's capture probability in nonlinear argumental oscillators

The behavior of a space-modulated, so-called “argumental” Duffing oscillator, is studied. Starting from a known analytic implicit solution to the equations of motion, and using a Van der Pol representation in the (amplitude, phase)-space, the shape and distribution of the attractors' upstream basins are discussed, and various capture probabilities by the attractors are assessed under symbolic form. The expressions obtained can help in the design of structures in mechanical engineering, where most often the argumental phenomenon is to be avoided.