Two Modifications of the Homotopy Perturbation Method for ‎Nonlinear Oscillators

Nonlinear vibration arises in engineering and physics, and the periodic motion of these nonlinear oscillatory systems have rich dynamics. An estimation of amplitude-frequency relationship of a nonlinear oscillator is much needed, therefore, well-known homotopy perturbation method is employed for this purpose. In this paper, two last modifications of the homotopy perturbation method are briefly reviewed, which couples with either the parameter-expansion technology or the enhanced perturbation method. Both modifications are extremely effective for nonlinear oscillators, and the cubic-quintic-septic Duffing oscillator is used as an example to elucidate the solution processes.