Abstract: 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.