On the generalized averaging method of a class of strongly nonlinear forced oscillators

We present a modified version of the generalized averaging method for studying the periodic solutions of a class of strongly nonlinear forced oscillators of the form ẍ + ω2x + εf(x) P(Ωt) = 0, where f(x) is a nonlinear function of x, P(Ωt) is a periodic function of t, ε need not be small, and ω is a constant parameter. This equation can be used to describe, e.g., a pendulum with a vibrating length or the displacements of colliding particle beams in high energy accelerators. The new version is based on defining a new parameter α = α(ε) and a linear transformation of the time. This version is applied for the cases f(x) = x3 and f(x) = x4 with P(Ωt) = cos t and excellent agreement is found with the results of numerical experiments, for large values of ε.