Abstract: The differential equation of slow dynamics system, that is, the Nonlinear Energy Sink (NES) system, with purely cubic stiffness under ground harmonic excitation has been derived based on complex-averaging method. Then, the boundaries for the system of Saddle-node bifurcation and Hopf bifurcation are obtained with the tool of multi-scale method. The numerical simulation results for the system under base harmonic excitation indicate that three solutions are within the region of the boundaries of Saddle-node bifurcation and the rest one is out of the region. The period solution is unstable in the region of the boundaries of Hopf bifurcation but it is stable out of the region. Only for lower mass ratio can the bifurcation boundaries of two excitation types be similar and it can give rise to larger distinction with the frequency detunning parameter ranging. The numerical results agree well with the theory analytical predictions and the same for the amplitude of slow dynamic system and the theory analytical predictions. It also turns out that weakly modulated response appears with certain conditions and multiple solutions can coexist.