With　the　complex-averaging　method,　a　slow　dynamics　system　for　nonlinear　energy　sinks　(NES)　system　with　purely　cubic　stiffness　under　ground　harmonic　excitation　was　formulated　to　gain　the　necessary　condition　and　sufficient　condition　of　strongly　modulated　response　(SMR).　Analysis　of　the　phase　portrait　of　the　slow　invariant　manifold　(SIM)　shows　that　SMR　appears　when　the　excitation　amplitude　meets　the　condition　of　folded　singularity,　whereas　it　may　be　attracted　to　certain　stable　branches　of　SIM.　The　case　study　and　discussion　show　that　the　SMR　should　exist　if　the　response　of　the　slow　dynamics　system　surpass　the　extreme　value　of　the　SIM;　however,　it　has　not　attracted　to　the　stable　branches　of　SIM　and　can　also　form　the　continuous　closed　loops　without　any　local　circles.　Theoretical　analysis　of　the　slow　dynamics　system　agrees　with　real　numerical　simulation　results.　It　is　relatively　simple　and　easy　to　identify　for　numerical　curves　of　slow　dynamics　system　as　well　as　convenient　calculation.　©　2019,　Editorial　Department　of　Journal　of　Beijing　University　of　Technology.　All　right　reserved.