Homotopy perturbation method for a conservative x1=3 force nonlinear oscillator

The homotopy perturbation method is used to obtain the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to x1=3. We find this method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 0.60% for small and large values of oscillation amplitude, while this relative error is as low as 0.050% for the second iteration. Comparison of the results obtained using this method with those obtained by different harmonic balance methods reveals that the former is more effective and convenient for these types of nonlinear oscillators.