Different from the usual approach in the inertial frame, the stability problem of fictitious Saturn Trojans is studied in the synodic frame in this paper. First, some numerical facts are shown to allow us to simplify the force model. Then, motion equations centred at the geometrical triangular libration points for the planar S-JS model are derived. Using these equations, the resonance mechanism that causes the instability is studied. We confirm the opinion that the secular resonances and the near-commensurability between the libration frequency and the great inequality are the reasons to cause the instability of motions close to the triangular libration point. By studying the survivability of the long period family and the short period family in the S-JSUN model, the planar stable region far away from the triangular libration point is studied. By frequency analysis of the orbits in the stable region, we are able to find two secular resonances associated with the boundary of the stable region. Three-dimensional motion is also discussed, by starting with the survivability of the vertical period family in the S-JSUN model. The secular resonance that causes the orbit inclination restriction on the Trojans is qualitatively discussed. Lastly, the effects of planetary migrations are briefly studied. With the contribution in this paper, a global picture of the dynamics around the triangular libration points in the Sun-Saturn system perturbed by Jupiter is presented.