In this paper, we focus on determining the safe operational domain of a coupled actuator–valve configuration. The so-called “smart valves” system has increasingly been used in critical applications and missions including municipal piping networks, oil and gas fields, petrochemical plants, and more importantly, the U.S. Navy ships. A comprehensive dynamic analysis is hence needed to be carried out for capturing dangerous behaviors observed repeatedly in practice. Using some powerful tools of nonlinear dynamic analysis including Lyapunov exponents and Poincaré map, a comprehensive stability map is provided in order to determine the safe operational domain of the network in addition to characterizing the responses obtained. Coupled chaotic and hyperchaotic dynamics of two coupled solenoid-actuated butterfly valves are captured by running the network for some critical values through interconnected flow loads affected by the coupled actuators' variables. The significant effect of an unstable configuration of the valve–actuator on another set is thoroughly investigated to discuss the expected stability issues of a remote set due to others and vice versa.