Multiplicity of periodic solutions in bistable equations

Berkolaiko, Gregory and Grinfeld, Michael (2006) Multiplicity of periodic solutions in bistable equations. Proceedings A: Mathematical, Physical and Engineering Sciences, 462 (2067). pp. 1001-1019. ISSN 1471-2962

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Abstract

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 a time-varying external magnetic field. When the amplitude of the external field is increased, the set of periodic solutions undergoes a bifurcation in both equations. We prove that despite superficial 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 amplitude of the magnetic field.