Abstract: The structure characteristic of a pressure relief valve embedded within a hydraulic circuit was analyzed. A nondimensional mathematical model was constructed by considering fluid compressibility,tube elasticity and energy loss when the valve poppet impacts its seat. Lyapunov exponent analyses were carried out. The aim is to capture the instability mechanism and chatter behavior of the relief valve. The non-smooth dynamical system theory and software MATLAB were used to draw one-parameter and two-parameter bifurcation diagrams. Grazing bifurcation which occurs at the poppet departure from the seat was explained. The results show that the flow rate into the valve and the setting pressure directly determine oscillation characters,and Hopf bifurcation,grazing bifurcation,periodic and chaotic phenomena appear. The measured bifurcation diagram at the spring pre-compression value x₀=5 mm was presented.The mathematical model of the relief valve was verified on a test platform.