The main focus of this study is the development of an adapted complex variable method with respect to the equilibrium point in bistable nonlinear energy sink (NES), which is mainly investigated in the vicinity of 1:1 resonance. A simplified chaos trigger model is established to describe the distance between the stable phase cycle and the pseudo-separatrix. An analytical expression can predict the excitation threshold for chaos occurrence. The relative positions between the chaos trigger threshold line and the slow invariant manifold structure can interpret the distribution of response regimes under growing harmonic excitation. The degeneration of the response regimes can be demonstrated by the qualitative analysis method, which helps to classify the bistable NES. The experiment confirms the analytical result of intra-well oscillation in the frequency domain. The characteristic response regimes of weak, modest, and strong bistable NES are identified by the experimental results.