Summary: We apply J.-H. He’s homotopy perturbation method [Int. J. Mod. Phys. B 20, No. 10, 1141–1199 (2006; Zbl 1102.34039)] to find improved approximate solutions to conservative truly nonlinear oscillators. This approach gives us not only a truly periodic solution but also the period of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of parameters in the case of the cubic oscillator, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed.
For the second order approximation we show that the relative error in the analytical approximate frequency is approximately 0.03% for any parameter values involved. We also compare the analytical approximate solutions and the Fourier series expansion of the exact solution. This allows us to compare the coefficients for the different harmonic terms in these solutions.
The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems.