This paper is based on the quantity competition for two heterogeneous and bounded rational firms in the oligopoly market. A nonlinear dynamic Cournot duopoly model is constructed in discrete time, and the existence and stability of the boundary equilibrium point and the Cournot-Nash equilibrium are analyzed. The stability region of the Cournot-Nash equilibrium is given. Numerical simulation and bifurcation diagrams are used to study the complex dynamic behaviour of the system with the varying parameters chosen in the parameter space. It is conclude that the CournotNash equilibrium loses its stability, and the chaotic attractor or other complex dynamic behaviours arise when the speed of adjustment is too high. Furthermore,a tiny variation of the initial conditions causes the drastic fluctuations in the output. The heterogeneous cost function of the system also gives rise to different bifurcation at the same time.