Multiplicity of periodic solutions in bistable equations
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.
Cite
@article{arxiv.cond-mat/0310735,
title = {Multiplicity of periodic solutions in bistable equations},
author = {Gregory Berkolaiko and Michael Grinfeld},
journal= {arXiv preprint arXiv:cond-mat/0310735},
year = {2007}
}
Comments
Fixed typos; added and reordered figures. 18 pages, 6 figures. An animation of orbits is available at http://www.maths.strath.ac.uk/~aas02101/bistable/