In rotating machines, the choice of suitable supports to reduce the vibrations has always been of great importance. The most common supports are rolling and sliding bearings. The presence of undesirable misalignments in the system can cause damage to roller bearings. Also, in sliding bearings, the oil whip causes different excitation frequencies that in some cases may increase the amplitude of the response and ultimately resonance in the system. The use of squeeze film dampers (SFDs) has been able to solve such defects and reduce the vibrational amplitude response. In this paper, the nonlinear behavior of a rotor system with SFD is investigated. The system includes a flexible shaft along with an unbalance rigid disc that runs on two SFDs. The main difference of the model used in this research with previous works is that in the previous studies, the shaft was modeled as a lumped mass on the bearings and disk, while in this research, the shaft is modeled using finite element. In this modeling, there are 6 elements for the shaft, so the total number of degrees of freedom of the system is 28. The governing equations of the shaft are extracted based on Timoshenko's beam theory and finally, the nonlinear equations of motion of the whole system are solved by the Newmark method. The analysis methods employed in this study is inclusive of the dynamic trajectories of the rotor center and bearing center, Poincare´ maps, frequency spectrum, time waveform and bifurcation diagrams. The analysis was performed by examining the unbalance parameter, excitation frequency, damper parameters and the stiffness parameter of the retainer spring. The results indicate periodic, subharmonic, quasi-periodic and chaotic behaviors in the system that changing these parameters can delay, reduce or eliminate non-synchronous regions. Application of the results can be useful to improve the vibration behavior of rotary systems and eliminating the destructive behaviors such as fatigue failure and bearing failure due to the unstable response of the system. Considering such measures will increase the system's efficiency and will improve the useful life of the components. Key words Squeeze film damper (SFD), Flexible rotor, Finite element method, Bifurcation.