The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators

In engineering, a fast estimation of the periodic property of a nonlinear oscillator is much needed. This paper reviews some simplest methods for nonlinear oscillators, including He's frequency formulation, the max-min approach and the homotopy perturbation method. A mathematical insight into He's frequency formulation is given, and the weighted average is introduced to further improve the estimated accuracy of the frequency. Fractional oscillators are also discussed.