This paper is devoted to exploring the complex dynamics of contest model, where two agents compete for some object with asymmetric valuations by simultaneously choosing efforts at each step. We build the nonlinear discrete system to describe the dynamic contest with bounded rationality, and discuss the stability conditions of the Nash equilibrium theoretically. Meanwhile, our numerical simulation experiments also reveal that the model can exhibit very complex dynamical behaviors. In particular, there exist two different routes to chaos for the system: the period-doubling (flip) bifurcation which leads to periodic cycles and chaos, and the Neimark-Sacker bifurcation which derives an attractive invariant closed curve. These two routes are significantly different for economic views. In addition, the stability of Nash equilibrium point is badly affected by the system parameters, such as the adjustment speeds, the values of the object, and so on. Therefore, the parameter adjustment method could be properly applied to make the system return to its stable state.