When both ends of an elastic continuous rotor are supported simply by double-row self-aligning ball bearings, the geometrical nonlinearity appears due to the stiffening effect in the elongation of the rotor if the movement of the bearings in the longitudinal direction is restricted. As the rotor becomes more slender, the geometrical nonlinearity becomes stronger. In this paper, we study on unique nonlinear phenomena caused by both of the nonlinear spring characteristics and an initial axial force in the vicinity of the major critical speed and twice in a very slender continuous rotor. When the rotor is supported horizontally, the difference in support stiffness and the asymmetrical nonlinearity appear as a result of shifting from the equilibrium position. By the influences of the internal resonance and the initial axial force, the nonlinear resonance phenomena become very complicated. For example, the peak resonance splits into two peaks, and these two peaks leave each other and then one becomes a hard spring type while the other becomes a soft spring type, respectively. Moreover, almost periodic motions and chaotic vibrations appear. In this paper, we prove the above phenomena theoretically and experimentally.