A ring network with fractional-order bistable oscillators is proposed, and the relationship between synchronization and parameters, such as coupling modes and the initial structural conditions, etc., is investigated. Based on the bistable characteristics of P-R oscillator, the effects of the coupling strength and the structures in initial conditions on the dynamic behaviors of the ring network are investigated by analyzing the largest conditional Lyapunov exponents, the largest Lyapunov exponents and the bifurcation diagrams, etc. Further investigation reveals that the ring network can be controlled to form chaotic synchronization, chaotic non-synchronization, synchronous amplitude death, synchronous non-amplitude death, etc. by changing the initial conditions and the coupling strength. Furthermore, the contours of the largest conditional Lyapunov exponents and the largest Lyapunov exponents also show how the dynamic behaviors of the network are influenced by the competition between couplings along directions of y and z, strongly relies on the initial structural conditions of network.