In this Chapter, analytical solutionsAnalytical solutions for period-m motionsPeriod-m motion in a buckled, nonlinear Jeffcott rotor system are obtained. This rotor system with two-degrees of freedom is a simple rotor dynamical system and periodic excitationsPeriodic excitation are from the rotor eccentricity. The analytical expressionsAnalytical expression of periodic solutions are developed. The corresponding stabilityStability and bifurcation analyses of period-m motionsPeriod-m motion are completed. Analytical bifurcation treesAnalytical bifurcation trees of period-1 motions to chaosChaos are presented. TheHopf bifurcations Hopf bifurcationsBifurcation of periodic motions can cause not only the bifurcation tree but quasi-periodic motionsQuasi-periodic motions. Displacement orbits of periodic motions in nonlinear rotor systems show motion complexity, and harmonic amplitude spectrumsHarmonic amplitude spectrums gives harmonic effects on periodic motions of the nonlinear rotor systems.