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In this paper, the analytical solution that describes the evolution in time of the fractal damped Duffing equation subjected to external forces of elliptic type is derived using He's two-scale fractal transform and the elliptic balance method (EBM). This solution predicts the evolution in time of the Duffing equation and unveils qualitative and quantitative system behavior when the values of the fractal parameter varies, and how these affect the frequency, the wavelength, and the oscillation amplitude from the start of the motion. Comparison of the amplitude–time response curves over the selected time-interval with those obtained from numerical simulations confirms the accuracy of the derived analytical solution. [ABSTRACT FROM AUTHOR]