We investigate the behavior of a mesoscopic 1D ring or 2D torus in an external magnetic field by simulating the Time Dependant Ginzburg Landau (TDGL) equations with periodic boundary conditions. In 1D, we analyze the stability and the different possible evolutions for the phase-slip phenomena starting from a metastable state. We used those results to observe the dynamics of a vortex-antivortex pair dissociation in 2D.