The dynamics of one dimensional discontinuous map is investigated by the complex network approach. The complex networks are built for corresponding time series of different dynamical states. There are P independent full-linked networks for an arbitrary P-period attractor and many independent complex networks of different sizes for a chaotic attractor. Meanwhile, the control parameter dependence of the link density, the clustering coefficient and the average length of the shortest paths for the complex networks are calculated and discussed. The correlation of these characteristic quantities with the dynamics of the system is analyzed. The result shows that the link density shows a discontinuous change for different periodic attractor and a nonsmooth change at the transition point between a periodic attractor and a chaotic attractor; the nonsmooth changes of the clustering coefficient and the average length of the shortest paths appear at the transition point between periodic and chaotic attractors, and the merging point of chaotic attractors, respectively. These phenomena imply that the characteristic quantities might be proper indexes to describe the dynamical states and exhibit their transitions. Moreover, they may help to check out the periodic attractors when they are coexistence with some other attractors.

recurrence plot; complex network; discontinuous map; dynamical transitions