It is a critical issue to maintain stability in high-speed railway vehicles and to ensure comfortable and safe driving. Multi-body models of railway vehicles have non-linear properties originated from the wheel-rail contact and characteristics of the suspension systems. The critical speed values at which the unstable oscillations and the amplitudes of the limit cycle-type vibrations take place vary by adjusting the design parameters; therefore, these effects on non-linear railway dynamics must be evaluated with a higher precision by using numerical and/or analytical methods to determine the bifurcation behavior. The main objective of this paper is to examine the non-linear phenomena in a railway bogie from a broad perspective, concentrating on non-linear analysis methods. Thus, non-linear equations of motion of a 12-degrees of freedom railway bogie involving dual wheelsets, non-linear wheel flange contact, heuristic non-linear creep model, and suspension system are solved in the time domain with small time steps by using ode23s (stiff/Mod.Rosenbrock) method. The critical speeds were calculated with respect to the effects of various lateral stiffness and damping coefficients. The bifurcation diagrams of the maximum lateral displacement of the leading wheelset were depicted within a wide speed range. In the case of the suspension parameter set where the subcritical/supercritical Hopf bifurcation takes place, the phase portraits and the symmetric/asymmetric oscillations of the leading wheelset at the critical speed were represented. The type of the Hopf bifurcation can be transformed from the subcritical state to the supercritical state by increasing the given suspension ratio. The Lyapunov exponents of the lateral displacement, lateral velocity, yaw angular displacement, and yaw angular velocity of the leading wheelset were evaluated above the critical speeds to examine chaotic motion. The effect of the suspension parameters on the non-linear dynamical behavior of the railway bogie at the stability limit and on the bifurcation type has been proved.