In this paper, a magnetic gear two-speed transmission system is proposed. The system has the advantages of isolated vibration and overload protection. A nonlinear dynamics model including nonlinear magnetic coupling torque, torque ripple, time-varying stiffness of mechanical gears, tooth-side clearance, and integrated transmission error of magnetic gears is established. The dimensionless vibrational differential equations of the system are derived, and the bifurcation and chaos characteristics of the system are analyzed by using the bifurcation diagram and the Poincaré cross-section diagram. The results show that the system has multi-frequency narrowband bifurcation intervals accompanied by amplitude jumps. The inverse doubling period bifurcation, the inverse Hough bifurcation, which restores stability to the system, can be observed in this interval. Mechanical gears in the system are more prone to chaotic vibration than magnetic gears, but by improving the intermediate shaft damping can effectively reduce the multi-frequency jumping phenomenon of the system and avoid chaotic vibration of the system.