•In this study, nonlinear dynamical equations are established for spring-loaded valves with bypass outlets.•Under constant inlet flow rate conditions, the valve disc's unstable flow interval is characterized by small flow.•The valve disc displays complex behavior under inlet time-varying flow conditions. A distinct frequency lock-in phenomenon occurs within the flow range of disk instability for fluctuating flows, while oscillations during the opening process are more pronounced compared to those in the closing process.•The introduction of a bypass outlet and an effective feedback control strategy substantially enhances the stability of the valve disc at low flow rates, showcasing the potential for improved valve performance in such conditions. The stability and control strategies of a spring-loaded valve with bypass outlet have been studied by combining numerical and theoretical analysis. First, a nonlinear dynamical model and a CFD model of a spring-loaded valve with bypass outlet are developed. The theoretical analysis requires parameters such as discharge coefficients and fluid forces obtained from CFD simulations. The equivalent area of the fluid forces and discharge coefficients are similar at different pressure differences, leading to a reduced-order formulation that serves the theoretical analysis of nonlinear dynamics. Second, the valve disc oscillates at small constant flow rates by nonlinear dynamical analysis. The oscillation frequency is a superposition of the Helmholtz cavity and the mass spring system, which increases as the valve disc impacts the seat or the upper limiter. More importantly, there is a frequency lock-in phenomenon at small fluctuating flow rates, which vanishes at large flow rates. The oscillations are not symmetric during the opening and closing processes, and the oscillations are more severe during the opening processes. Finally, the bypass outlet has a great effect on the stability of valve disc, which can appropriately kill the nonlinear component in the dynamic equation. A reasonable bypass sleeve design can reduce the unstable flow range and oscillation amplitude. This paper provides a reference for the design of spring-loaded valves.