Averaging methods for the phase shift of arbitrarily perturbed strongly nonlinear oscillators with an application to capture

Strongly nonlinear oscillators under slowly varying perturbations (not necessarily Hamiltonian) are analyzed by putting the equations into the standard form for the method of averaging. By using the usual near-identity transformations, energy-angle (and equivalent action-angle) equations are derived using the properties of strongly nonlinear oscillators. By introducing a perturbation expansion, a differential equation for the phase shift is derived and shown to agree with earlier results obtained by Bourland and Haberman using the multiple scale perturbation method. The slowly varying phase shift is used (by necessity) to determine the boundary of the basin of attraction for competing stable equilibria, even though these averaged equations are known not to be valid near a separatrix (unperturbed homoclinic orbit).