A brief introduction to the development of the homotopy perturbation method is given, and the main milestones are elucidated with more than 90 references. This paper further improves the method by constructing a homotopy equation with one or more auxiliary parameters embedding in the linear term with a clear advantage in accelerating and controlling the approximation convergence speed. Moreover, a revision of a recent amplitude-period approximation formula is presented providing an answer to an open problem related to the optimal approximation along with a new universal formula. Duffing equation is used as an example to illustrate the solution process for the homotopy perturbation method, and only one or few iterations are needed in practical applications, making the method much attractive. From the side of amplitude-period formulation, the nonlinear pendulum, the Duffing equation and an oscillator with discontinuity are analyzed providing an asymptotic exact equivalence for bigger parameter values in the case of Duffing's system. This mini review gives a tutorial guideline for practical applications of the homotopy perturbation method, the references are not exhaustive.
The simplest frequency formulation for conservative oscillators was proposed in 2019 (Results Phys 2019;15:102546). However, it becomes invalid for non-conservative oscillators. This work suggests the simplest amplitude-period formulation for non-conservative oscillators. The existence of a periodic solution of a second-order ordinary differential equation is given, and the periodic orbits are easily obtained. To the best of the authors’ knowledge, such a powerful result is not available in the literature, providing a tool to determining periodic orbits/limit cycles in the most general scenario.
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