| Summary: | In the present work, the version of homotopy perturbation included time-scales is applied to the governing equation of time-periodic delay Mathieu equation. Periodical structure for the amplitude of the zero-order perturbation is constructed. The stability analysis is accompanied by considering three-time-scales. Approximate periodic solutions are derived to the second accuracy of perturbations at the harmonic resonance case as well as at the non-harmonic resonance case. Stability conditions are derived in both cases. Numerical calculations have been done to illustrate the stability behavior at both resonance and non-resonance case. It is shown that the time-delay has a destabilizing influence. We note that the delayed of the parametric excitation has a great interested and application to the design of nuclear accelerators. Keywords: Homotopy perturbation method, Multiple-scales perturbation, Periodic delay Mathieu equation, Parametric resonance, Stability analysis
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