This paper is focused on investigating the bifurcation and vibration resonance problems of fractional double-damping Duffing time delay system driven by external excitation signal with two wildly different frequencies ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} and Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}. Firstly, the approximate expressions of the critical bifurcation point and response amplitude Q at low-frequency ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} are obtained by means of the direct separation of the slow and fast motions. And then corresponding numerical simulation is made to show that it is a good agreement with the theoretical analysis. Next, the influence of system parameters, including internal damping order α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, external damping order λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}, high-frequency amplitude F, and time delay size τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}, on the vibration resonance is discussed. Some significant results are obtained. If the fractional orders α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} and λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} are treated as a control parameter, then α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} and λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} can induce vibration resonance of the system in three different types when the response amplitude Q changes with the high-frequency amplitude F. If the high-frequency amplitude F is treated as a control parameter, then F can induce vibration resonance of the system as well at some particular points. If the time delay τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} is treated as a control parameter, not only can τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} induce three types of vibration resonance, but the response amplitude Q views periodically with τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}. In addition, the resonance behaviors of the considered system are more abundant than those in other similar systems since the internal damping order α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, external damping order λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}, time delay τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} and cubic term coefficient β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} are introduced into the system which changes the shapes of the effective potential function.
