The concept of symmetric bifurcation for symmetric wheel-rail system was defined. Thus, the time response can be achieved by the numerical integration method, and an unfixed and dynamic Poincare section and its symmetric section for symmetric wheel-rail system were established. Then the 'resultant bifurcation diagram' method was constructed. The method was used to study the symmetric/asymmetric bifurcation behaviors and chaotic motions of a two-axle railway bogie running on an ideal straight and perfect track. The results indicate that a lot of symmetric motions and plenty of asymmetric motions exist in the symmetric railway bogie system within the investigated speed range. In addition, the rule of symmetry breaking in the system is through a blue sky catastrophe in the beginning.