Analysis of the dynamic structure factors of strongly coupled classical, symmetric electronic bilayers obtained from molecular dynamics simulation confirms the existence of two longitudinal collective modes. The in-phase mode resembles the optical mode seen in single-layer two-dimensional systems, while the out-of-phase mode shows an energy gap at wave numbers q→0 for interlayer separations d less than about 1.5 (in units of the Wigner-Seitz radius). These dispersion results corroborate recent molecular dynamics data obtained using a different algorithm and analysis, and for a lower value of the plasma parameter Γ. The energy gap is seen not to be very sensitive to the value of Γ. The analysis of our data has brought out an interesting feature: a bifurcation of the out-of-phase dispersion curve for a high value of Γ(=80) for intermediate values of d. This feature is not seen for lower values of Γ(=40) and has not been predicted by any theoretical model.