J. Appl. Comput. Mech., 4(4) (2018) 260-274 DOI: 10.22055/JACM.2017.23591.1164 ISSN: 2383-4536 jacm.scu.ac.ir Published Online: Sep 20 2018 Stability Analysis of a Strongly Displacement Time-Delayed Duffing Oscillator Using Multiple Scales Homotopy Perturbation Method Yusry O. El-Dib Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt, yusryeldib52@hotmail.com Received October 04 2017; Revised December 13 2017; Accepted for publication December 15 2017. Corresponding author: Yusry O. El-Dib, yusryeldib52@hotmail.com Copyright © 2018 Shahid Chamran University of Ahvaz. All rights reserved. Abstract. In the present study, some perturbation methods are applied to Duffing equations having a displacement time-delayed variable to study the stability of such systems. Two approaches are considered to analyze Duffing oscillator having a strong delayed variable. The homotopy perturbation method is applied through the frequency analysis and nonlinear frequency is formulated as a function of all the problem’s parameters. Based on the multiple scales homotopy perturbation method, a uniform second-order periodic solution having a damping part is formulated. Comparing these two approaches reveals the accuracy of using the second approach and further allows studying the stability behavior. Numerical simulations are carried out to validate the analytical finding. Keywords: Homotopy perturbation method, Multiple scales method, Frequency analysis, Periodic nonlinear solution, Stability analysis, Displacement delay Duffing oscillator. 1. Introduction This paper concerns with the effect of a displacement time-delayed Duffing type oscillator. The purpose is to find nontrivial solutions and investigate the stability analysis. The corresponding system consists of a strong nonlinear time-delayed Duffing oscillator which is governed by the following second-order nonlinear differential equation:    2 2 3 2 , 0 1; (0) 1, dy dy y Qy yt y y dt dt             (1) where , , , Q   , and  are real physical quantities describing the damping, coefficient of delay, coefficient of nonlinearity, time delay, and the natural frequency, respectively. The Duffing equation is used to model the nonlinear dynamics of special types of electrical and mechanical systems. This differential equation, named after the studies of Duffing in 1918 [1], has a cubic nonlinearity and describes an oscillator. It has drawn extensive attention due to the richness of its chaotic behavior with a variety of interesting bifurcations. Some of its applications are in electronics, mechanics, electromechanics, and biology. On the other hand, the brain is full of oscillators at micro and macro levels [2]. In addition, it has many applications in neurology, ecology, secure communications, cryptography, and chaotic synchronization. Equation (1) may serve as the simplest model for describing the dynamic of various controlled physical and engineering systems [3–5]. Other works have been devoted to study the dynamic of a Duffing oscillator under a delayed feedback control [6]. In addition, Eq. (1) has been considered in a study [7] as a simple model for a vibration problem in turning machine for