A Simple Expansion-Based HPM for Cubic–Quintic Damped Nonlinear Oscillator

In the present article, a new amplitude expansion–based homotopy perturbation method (AE-HPM) is used to study the nonlinear behavior of a damped oscillator. The traditional homotopy perturbation method is extended, considering a simple amplitude expansion to determine the solution and amplitude frequency relationship for the damped nonlinear system, which could not be solved by the traditional approach. The simplicity, efficiency, and validity of the present AE-HPM are verified by applying it to the cubic–quintic oscillator and pendulum equation with linear damping. The analytical results obtained by the present method show that the oscillation amplitude decays exponentially with the damping parameter, while the frequency response is very much influenced by the system’s nonlinearity. The comparison of the results obtained by AE-HPM with numerical results and He’s frequency formulation (HFF) solution shows that the present method is accurate, converges faster, and can be used in a wider range of problems, while the traditional HPM fails. Therefore, the proposed method serves as a simple and dependable analytical approach for analyzing nonlinear damped oscillatory systems.