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, wellknown 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.