To improve the operation characteristics of a direct-acting relief valve with chatter, a non-dimensional mathematical model is derived considering fluid compressibility, tubes elasticity and energy loss during the collision between the valve poppet and its seat. First, the stable equilibrium states of the system are learned from the phase and vector field diagrams drawn under four different pre-stress value of the spring. Then, one-parameter and two-parameter bifurcation diagrams, which are based on the theory of non-smooth dynamical systems drawn by MATLAB, reveal Hopf bifurcation, saddle-node bifurcation of cycles, generalized Hopf bifurcation and cusp bifurcation in the system. Finally, the spool displacement bifurcation diagram and spectra waterfall diagram are measured on the test platform for the verification of the mathematical model of the relief valve. The results show that the chaos or period impacting-scillation at low flow rate could turn into period non-pacting- scillation or stable equilibrium by increasing the flow rate. Thus, the operation characteristics of a direct-acting relief valve with chatter are improved.