<p>Bifurcations of chaotic mappings have been widely applied in computer sciences, such as digital image processing and communication systems. In this paper, we theoretically study the dynamic properties of a discrete chaotic system with rational fraction. Firstly, we use <i>complete discrimination system</i> theory to give a topological classification for the fixed points of the system. Secondly, by employing the center manifold theorem and bifurcation theory, we study all codimension 1 bifurcations and prove that the system produces pitchfork bifurcations, flip bifurcations and Neimark–Sacker bifurcations at different fixed points. Thirdly, we prove that the system can process codimension 2 bifurcations, including 1:2 and 1:3 resonances. Furthermore, using Takens’s theorem and the equivalence between the mappings and the time 1 mappings of flows, we present all the bifurcation phenomena when the parameters vary near the 1:2 and 1:3 resonance points, respectively. Finally, a numerical simulation to all the codimension 1 and 2 bifurcations of the system is given, which shows that there is a chaotic attractor near the 1:3 resonance point.</p>