The ball-screw of machine tool feed system is simplified as Timoshenko beam. The forced state of the ball-screw is analyzed and the differential equation of motion for the ball-screw is built. To obtain the ball-Screw axial vibration equation, the differential equation of motion was simplified by using the assumed mode method. Axial vibration equation was satisfied of the form of duffing equation with the characteristics of nonlinearity. The numerical simulation of Duffing equation was proceeded by MATLAB. The effect of screw length, exciting force and damping coefficient on vibration were researched, and the axial vibration phase track diagram and Poincare section were obtained. The stability and periodic of the axial vibration were analyzed. The limit cycle of phase track diagram was enclosed. Axial vibration had two type-center singularity distributions on both sides of the origin. The singularity was attracted of vibration so as to reach a stable state, and Poincare section was shown that axial vibration appeared chaotic motion, quasi periodic motion or periodic motion. Singularity position was changed with the vibration system parameters, but not changed in distribution. The Periodic of the vibration was enhanced with increasing of exciting frequency and damping coefficient. Test of the feed system ball-screw axial vibration exists chaos movement, which provides a certain theoretical basis for the dynamic characteristic analysis of machine feed system ball-screw and optimization of structural parameters.