Multiplicity of periodic solutions in bistable equations

TL;DR

This paper investigates the bifurcation behavior of periodic solutions in two related bistable differential equations modeling magnetization, revealing different bifurcation types and implications for magnetization control.

Contribution

It demonstrates that similar equations can exhibit fundamentally different bifurcation behaviors, affecting the number and stability of periodic solutions.

Findings

Different bifurcation types in the two equations.

Discontinuous magnetization change possible in Suzuki-Kubo model.

Variation in stable periodic solutions near bifurcation points.

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 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.