  We study the number of periodic solutions in two first order non-autonomous
differential equations both of which have been used to describe, among other
things, the mean magnetization of an Ising magnet in the time-varying external
magnetic field. When the strength of the external field is varied, the set of
periodic solutions undergoes a bifurcation in both equations. We prove that
despite profound similarities between the equations, the character of the
bifurcation can be very different. This results in a different number of
coexisting stable periodic solutions in the vicinity of the bifurcation. As a
consequence, in one of the models, the Suzuki-Kubo equation, one can effect a
discontinuous change in magnetization by adiabatically varying the strength of
the magnetic field.
