In the present work, the investigation of the dynamics of three dimensions Mathieu equations containing periodic terms as well as the delayed parameters. Mathieu equations included the influence of damping terms and the coupled system involves both delayed and non-delay terms. The system is proposed as an extension of delayed two coupled Mathieu-type equations to higher dimensions, with emphasis on how resonance between the internal frequencies leads to a loss of stability. The method of multiple scales is used to examine the islands of stability near the resonance cases. The transition curves are analyzed using the method of harmonic balance, and we find we can use this method to easily predict the `resonance curves' from which bands of instability emanate. We note that the delayed higher dimension of the parametric excitation has a great interest and application to the design of nuclear accelerators.
Keywords: Three dimensions Mathieu equations; resonance curves; parametric excitation.