The coupled-core fluxgate magnetometer (CCFM) is a magnetic sensor, which is an improvement upon the single fluxgate sensor in signal measurement precision and power efficiency. The proper operation of the CCFM requires oscillatory behavior in the dynamics of the device to drive it in and out of saturation. Though the dynamical equations well approximate the behavior of the actual device, a small time delay in the coupling is assumed due to finite signal transmission times. The simplest case of a CCFM system exhibiting oscillations (three coupled cores) is studied. The corresponding system of equations exhibits rich dynamical behavior in the region of operation of the device. It is determined in this region that a stable limit cycle and two synchronous equilibria have large basins of attraction. A center manifold reduction is performed in both instantaneous and delayed coupling cases at the Hopf bifurcations, showing that the Hopf bifurcations are subcritical. Calculation of numerical Floquet multipliers of periodic orbits emanating from the Hopf bifurcations demonstrates that these orbits are unstable, which verifies that the Hopf bifurcations are subcritical. Two-parameter Hopf bifurcation curves are computed, which show that the basin of attraction of the synchronous equilibria is reduced to nothing as the time delay is increased to the Hopf bifurcation curve. Analytical and numerical computations show that the size of the basins of attraction of the synchronous equilibria and the stable limit cycle vary according to the coupling strength and time delay.
