Yusry El-Dib - Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

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.

Homotopy Perturbation Method, Multiple scales method, Frequency analysis, Periodic nonlinear solution, Stability analysis, Displacement delay Duffing oscillator