The dimensionality reduction and bifurcation of some high-dimensional relative-rotation nonlinear dynamical system are studied. Considering the nonlinear influence factor of a relative-rotation nonlinear dynamic system, the high-dimensional relative-rotation torsional vibration global dynamical equation is established based on Lagrange equation. The equivalent low-dimensional bifurcation equation, which can reveal the low-dimensional equivalent bifurcation equation between the nonlinear dynamics and parameters, can be obtained by reducing the dimensionality system using the method of Lyapunov-Schmidt reduction. On this basis, the bifurcation characteristic is analyzed by taking universal unfolding on the bifurcation equation through using the singularity theory. The simulation is carried out with actual parameters. The parameter region of torsional vibration and the effect of the parameters on the vibration are discussed.