Duffing-Holmes equation can be transformed into discrete nonlinear dynamical system by means of Euler′s method, and the standard Holmes map can be then obtained. The main purpose of this paper is to study the existence and stability of the fixed points of this Holmes map and to analyze the existence of Pitchfork bifurcation, Flip bifurcation and Hopf bifurcation of the map by using the center manifold theorem, and the conditions of the parameters corresponding to specific bifurcations are also given. In addition, it proves that the system is chaotic in the sense of Marotto. Finally, numerical simulations verify these theoretical results.