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The possibility of mode switching from one pulse to another in a 6-cavity gigawatt magnetron with axial extraction of radiation through a horn antenna (such a magnetron is known as the MDO, i.e., magnetron with diffraction output) using a weak (200-300 kW), short (15-ns), and single-frequency RF signal was demonstrated using particle-in-cell simulations in our earlier work. This mode switching exploits the symmetric nature of the MDO that facilitates the use of any eigenmode as the operating mode. All scenarios of mode switching were considered using common properties of dynamical systems with two stable states separated by an unstable saddle point. In this paper, we continue to study the problem of mode switching, but this time for a 12-cavity MDO, for which we found splitting of the radiation frequency for each eigenmode owing to its different longitudinal distributions. Since splitting manifests as a bifurcation of frequency for definite values of the applied axial magnetic field, scenarios of frequency switching for this 12-cavity magnetron are considered.