Abstract: The periodic solution of the ordinary differential equations is discretized using the finite difference method and the nonlinear algebraic equations with a parameter are obtained. The equations are solved continuously using the DERPAR algorithm and the stable and unstable periodic solutions are calculated. The feasibility of the method is verified by calculating the van de Pol equation and Lorenz equations. A railway passenger car model with 17 degrees of freedom is set up and the periodic solutions of the system are obtained. The solution diagram of the vehicle system in a large area is obtained which includes the stable and unstable periodic solutions. The Hopf bifurcation point and the nonlinear critical speed of the vehicle system are determined. The relations of vehicle system period and lateral displacement amplitude of wheelset with respect to the vehicle speed are investigated.