Sub-harmonic resonances of nonlinear oscillations with parametric excitation by means of the homotopy analysis method

An analytical technique, namely the homotopy analysis method (HAM), is applied to solve periodic solutions for sub-harmonic resonances of nonlinear oscillations with parametric excitation. Unlike perturbation methods, 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 Letter, periodic analytic approximations for sub-harmonic resonances of nonlinear oscillations with parametric excitation are obtained by using the HAM for the first time, which agree well with numerical results. This Letter shows that the HAM is a powerful and effective technique for nonlinear dynamical systems.