In this paper, the semi-analytical solutions of period-1 and period-2 motions in a nonlinear Jeffcott rotor system are presented through the discrete mapping method. The periodic motions in the nonlinear Jeffcott rotor system are obtained through specific mapping structures with a certain accuracy. A bifurcation tree of period-1 to period-2 motion is achieved, and the corresponding stability and bifurcations of periodic motions are analyzed. For verification of semi-analytical solutions, numerical simulations are carried out by the mid-point scheme.

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