There has been a substantial amount of well-mixing epidemic models devoted to characterizing the observed complex phenomena (such as bistability, hysteresis, oscillations, etc.) during the transmission of many infectious diseases. A comprehensive explanation of these phenomena by epidemic models on complex networks is still lacking. In this talk, we study epidemic dynamics in an adaptive network model proposed by Gross et al, where the susceptibles are able to avoid contact with the infected by rewiring their network connections. Such rewiring of the local connections changes the topology of the network and inevitably has a profound effect on the transmission of infectious diseases, which in turn influences the rewiring process. We rigorously prove that such an adaptive epidemic network model exhibits degenerate Hopf bifurcation, homoclinic bifurcation, and Bogdanov-Takens bifurcation. Our study shows that human adaptive behaviors to the emergence of an epidemic may induce complex dynamics of disease transmission, including bistability, and transient and sustained oscillations, which contrast sharply with the dynamics of classic network models. Our results yield deeper insights into the interplay between the topology of networks and the dynamics of disease transmission on networks.