In order to obtain the necessary and sufficient conditions for the strongly modulated response (SMR) of a nonlinear energy sink (NES) system under harmonic excitation, the SMR study of the NES system is carried out. Firstly, the equation of systematic slow variation flow is derived by using the complex-averaging method. Secondly, the multi-scale method is used to realize the separation of the rapid variable manifold and the slow variable manifold. And the systematic slow invariant manifold under the different parameters and the global bifurcation property are ob? tained. Then, by building the one-dimensional mapping function of the rapid variable manifold, the sufficient and necessary conditions for the NES system in the SMR state are revealed. Finally, the simulation circuit of the NES sys? tem is constructed, and the response detection circuit test is carried out. The simulation and experimental results show that the SMR is caused by the saddle-node bifurcation of limit cycles in the slowly varying power flow of the coupling system, and it is an actual phenomenon in engineering. The system in which SMR state can appear must sat? isfy the following two conditions: the response of the NES system exceeds the amplitude of the extremum point on the slow invariant manifold, but it does not attract to a branch of the slow invariant manifold, and a continuous jump loop without falling into a local cycle is formed.