Damped Mathieu Equation with a Modulation Property of the Homotopy Perturbation Method Yusry O. El-Dib and Nasser S. Elgazery * Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt * Corresponding Author: Nasser S. Elgazery. Email: nasser522000@gmail.com Received: 06 September 2020 Accepted: 25 June 2021 ABSTRACT In this article, the main objective is to employ the homotopy perturbation method (HPM) as an alternative to classical perturbation methods for solving nonlinear equations having periodic coefficients. As a simple example, the nonlinear damping Mathieu equation has been investigated. In this investigation, two nonlinear solvability conditions are imposed. One of them was imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance cases. The next level of the perturbation approaches another solvability condition and is applied to obtain the unknowns become clear in the solution for the first- order solvability condition. The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science. KEYWORDS Damped Mathieu Equation; parametric nonlinear oscillator; resonance instability; homotopy perturbation method (HPM) 1 Introduction In wide engineering and physical applications, the nonlinear oscillators exist. Also, the parametric excitation takes place when a modifying physical parameter, such as a moment of stiffness or inertia, acts in a forcing model. This excitement yields a variable time coefficient, commonly an oscillation, in the governing system of motion. On the other hand, an external excitation outcome acting as an inhomogeneous part in the model of motion. Furthermore, minor parametric excitement produces a major response when the frequency of the excitement is far from the fundamental resonance, as shown in [1–5]. A classical example of parametric excitation is the swinging pendulum with oscillating support. The equation of motion describing the model is the well-known Mathieu equation. In 1868 Mathieu studied the vibration of elliptical membranes [6]. Consequently, he introduced the Mathieu equation that is an example of a linear differential equation (LDE) with parametric excitation. The Mathieu equation has application to the dynamics of passive towed arrays in submarines, as well as serving as a useful model for many interesting problems in physics, biology, applied mathematics, and engineering mechanics fields [7]. For some of the non-linear variations of the Mathieu, the equation has been presented in [8,9]. Moreover, the oscillations of the mechanical systems under the action of an oscillatory external force may reveal a Duffing problem, for instance, see references [10–14]. Recently, Moatimid [15] attempted to This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. DOI: 10.32604/sv.2022.014166 ARTICLE ech T Press Science