Adaptive network is characterized by feedback loop between states of nodes and topology of the network. In this paper, for adaptive epidemic spreading model, epidemic spreading dynamics is studied by using a nonlinear differential dynamic system. The local stability and bifurcation behavior of the equilibrium in this network model are investigated and all kinds of bifurcation point formula are obtained by analyzing its corresponding characteristic equation of Jacobian matrix of the nonlinear system. It is shown that, when the epidemic threshold is less than epidemic persistence threshold R00c, the disease always dies out and the disease-free equilibrium is asymptotically locally stable. If R0c01, a backward bifurcation leading to bistability possibly occurs, and there are possibly three equilibria: a stable disease-free equilibrium, a larger stable endemic equilibrium, and a smaller unstable endemic equilibrium. If R01, the disease is uniformly persistent and only one endemic equilibrium is asymptotically locally stable. It is also found that the system has saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Numerical simulations are given to verify the results of theoretical analysis.