Multiplicity of periodic solutions in bistable equations

Gregory Berkolaiko, Michael Grinfeld

Research output: Contribution to journalArticle

2 Citations (Scopus)

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.
LanguageEnglish
Pages1001-1019
Number of pages18
JournalProceedings A: Mathematical, Physical and Engineering Sciences
Volume462
Issue number2067
DOIs
Publication statusPublished - 8 Mar 2006

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Magnetization
Periodic Solution
Multiplicity
Bifurcation
Magnetic fields
External Field
Magnetic Field
Magnets
Nonautonomous Differential Equations
magnetization
Differential equations
First order differential equation
magnetic fields
Ising
Thing
Time-varying
differential equations
magnets
Model
Similarity

Keywords

  • bistability
  • suzuki-kubo equation
  • mathematics
  • bistable systems

Cite this

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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.",
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Multiplicity of periodic solutions in bistable equations. / Berkolaiko, Gregory; Grinfeld, Michael.

In: Proceedings A: Mathematical, Physical and Engineering Sciences, Vol. 462, No. 2067, 08.03.2006, p. 1001-1019.

Research output: Contribution to journalArticle

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AU - Berkolaiko, Gregory

AU - Grinfeld, Michael

PY - 2006/3/8

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AB - 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.

KW - bistability

KW - suzuki-kubo equation

KW - mathematics

KW - bistable systems

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