Study on strongly odd nonlinear oscillations by means of the homotopy analysis method

An analytical technique, namely the homotopy analysis method (HAM), is applied to solve periodic solutions for free oscillations with strongly odd nonlinearities of 5 degree polynomials. Unlike perturbation methods such as the method of multiple scales which must depend on a small parameter, HAM does not depend on any small physical parameters at all. Thus, it is valid for both weakly and strongly nonlinear problems. Besides, different from all other analytic techniques, the HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter h . In this paper, periodic analytic approximations for free oscillations with odd nonlinearities of degree 5 polynomials are obtained by using the HAM for the first time, which agree well with numerical results. This article shows that the HAM is a powerful and effective technique for nonlinear dynamical systems.