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This paper introduces a gating function that straightforwardly represents the fluxgate mechanism separately from the complications of hysteresis and demagnetization. The characteristics of the gating function are easily measurable and can be regarded as the performance indices of the sensor. The paper also presents a general-state equation that is valid not only for single sensors, but also for symmetric multiple sensors (e. g., gradiometers). Based on the theories of Floquet-Lyapunov and of Hsu, the instability criterion and the method of calculation are investigated. The generalized instability map illustrated on a parameter plane will be useful for the design of stable fluxgate sensor systems.