In this dissertation, the mechanical models of 2-DOF and 4-DOF nonlinear journal bearing rotor systems are established. A more accurate model of oil film forces is derived from Reynolds equations. The periodic motions in such nonlinear journal bearing systems are obtained through discrete mapping method. Such a semi-analytical method constructs an implicit discrete mapping structure for periodic motions by discretization of the continuous journal bearing rotor differential equations. Stable and unstable periodic solutions of periodic motions are obtained with prescribed accuracy. The bifurcation tree of periodic motions in rotor system without oil film forces is demonstrated through the route from period-1 motion to period-8 motion. Stable period-2 and unstable period-1 motion are presented for 2 DOF journal bearing rotor system. Possibly infinite periodic solutions are found in 4 DOF journal bearing rotor system. For the rotor systems, the stability and bifurcations of periodic motions are analyzed through eigenvalue analysis of the corresponding Jacobian matrix of the discretized nonlinear systems. The frequency amplitude characteristics of periodic motions in 2 DOF journal bearing system are presented for a good understanding of the nonlinear dynamics of journal bearing rotor system in frequency domain . The rich dynamics of the journal bearing systems are discovered. The numerical illustrations of stable periodic motions are brought out with the initial conditions from analytical prediction.