The non-linear dynamics of a rigid rotor levitated by active magnetic bearings isinvestigated. The vibrations in the horizontal and vertical directions are analyzed on thecenter manifold near the double-zero degenerate point by using normal-form method. Theresulting normal forms in the horizontal and vertical directions are di!erent due to the e!ectof rotor weight. It is shown that the vibratory behavior in the vertical direction can bereduced on the center manifold to the Bogdanov}Takens form. For the autonomous case,there exist saddle-node bifurcation and Hopf bifurcation for local analysis, anda saddle-connection bifurcation for global analysis. For non-autonomous case, theMelnikov technique is used to determine the critical parameter at which the homoclinicorbits intersect transversally. For the vibrations in the horizontal direction, the essentialnon-linear terms of the truncated normal form are third order. The behaviors of zerosolutions are given. Finally, numerical simulations are performed to verify the analyticalpredictions.