A homotopy perturbation method for a class of truly nonlinear oscillators (Q2046716)

In this paper, the authors study a family of nonlinear oscillators governed by the equation \[ \frac{d^{2}x}{dt^{2}}= -\beta x^{p},\beta>0, \] under the initial conditions \(x(0)=A\) and \(\frac{dx}{dt}(0)=0\). Here \(\beta >0\) is a constant and \(p=1,3,5,\dots\). The purpose of the authors is to find approximate formulas for the amplitude-frequency relation \(\omega = \varpi(A)\) of the oscillator and its trajectory for each \(p\) by the homotopy perturbation method. To derive the amplitude-frequency relations indexed by \(p\), they apply a Lindstedt-Poincare procedure. A functional analytic framework is introduced to overcome certain difficulties. Here all computations are made in terms of inner product operations. Certain numerical examples are provided to demonstrate that the first two or three approximates are sufficiently accurate for the study of this type of nonlinear oscillators.