A homotopy perturbation method for a class of truly nonlinear oscillators

We apply a homotopy perturbation method to a class of nonlinear oscillators with a restoring force proportional to the odd powers p of the displacement. To derive the amplitude-frequency relations indexed by p, a Lindstedt-Poincare procedure is applied. Unlike in the case when p is a fixed numerical value, for general p, deriving formulas via traditional symbolic manipulation will result in plethora of hypergeometric and trigonometric expressions, and the technique of killing the secular terms becomes unmanageable. In this paper, we propose a functional analytic framework so that these difficulties can be overcome. We introduce a Volterra integral representation of the displacement, and the annihilation of the secular terms is replaced by enforcing an orthogonal solvability condition. All computations are in terms of inner product operations. We demonstrate by numerical examples that the first two or three approximates are sufficiently accurate for this class of truly nonlinear oscillators.