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In order to reflect the dynamic response of the rubber vibration isolation system more accurately in medium and low frequency ranges, the nonlinear Zener model is used to characterize the mechanical properties of viscoelastic materials like rubber. The approximate analytical solution of the system response is obtained by harmonic balance method, and its correctness is verified through numerical method and Universal Mechanism (UM) software simulation. The quantitative relationship between mass displacement and node displacement is calculated, and a new method to obtain hysteresis characteristics of rubber vibration isolation system based on approximate analytical solution of the model is presented. Under the influence of symmetry and hysteresis, the system possesses bifurcation, chaos, polymorphic coexistence and other complex nonlinear dynamical behaviors. Then, with the help of global bifurcation diagram, phase diagram, Poincaré map and cell mapping, the formation mechanism of polymorphic coexisting motion induced by pitchfork bifurcation, saddle-node bifurcation, period-doubling bifurcation and boundary crisis is comprehensively analyzed. Based on this, the formation mechanism of bursting oscillation is revealed. Finally, the influence of nonlinear stiffness on the transition law of the system periodic motion is analyzed, and the distribution law of the coexisting attractors and the variation of the attracting domain in various of polymorphic coexisting motions are presented. The research results of the complex dynamic behavior of the nonlinear Zener vibration isolation system in this paper can provide some theoretical guidance for the optimal design of the viscoelastic vibration isolation system. [ABSTRACT FROM AUTHOR]