In the talk, I will  introduce the Ginzburg- Landau system of superconductivity. When lowering the temperature, superconductors become the superconducting state from the normal state. In order to study this phase transition phenomena, we present a bifurcation and stability analysis on the Ginzburg- Landau system of superconductivity with an applied magnetic field and the de Gennes boundary condition. It is proved there are two different kinds of phase transition from the normal state to the superconducting state: one is jump transition and the other is continuous transition. In particular, we analyse the behavior of solution when the domain is a cylinder and the applied field is parallel to the axis.