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A pure shear mechanical model of low gear of six-degree-of-freedom two-speed transmission system is established by using lumped parameter method. The Runge-Kutta method is used to numerically solve the aforementioned nonlinear system. The variation of transmission error between gears is analyzed by using global bifurcation, time domain diagram, phase diagram and Poincare cross section. Moreover, the transfer error bifurcation characteristics of the solar wheel and the first planetary wheel under different gear moduli are investigated. The results show that: by taking the excitation frequency as the control parameter, the system includes period-1 motion, period-2 motion, quasi-periodic motion, multiperiodic motion, and chaotic motion. With the increase of gear modulus, the system mainly presents chaotic motion in the medium frequency range (0.5<ω_h≤2). It shows stable period-1 motion in the high frequency range (2<ω_h≤3), and the higher the modulus, the wider the high frequency range of period-1 motion. The research results can provide reference for the design and optimization of this kind of two-speed transmission system in the future.