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.　©　2017,　Engineering　Mechanics　Press.　All　right　reserved.