Finite to infinite steady state solutions, bifurcations of an integro-differential equation

S. K. Bhowmik, D. B. Duncan, M. Grinfeld, G. J. Lord

Research output: Contribution to journalArticle

3 Citations (Scopus)
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Abstract

We consider a bistable integral equation which governs the stationary solutions of a
convolution model of solid–solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is varied to examine the transition from an infinite number of steady states to three for the continuum limit of the semi–discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.
Original languageEnglish
Pages (from-to)57-71
Number of pages15
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume16
Issue number1
DOIs
Publication statusPublished - Jul 2011

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Integrodifferential equations
Bifurcation (mathematics)
Steady-state Solution
Stationary Solutions
Integro-differential Equation
Integral equations
Bifurcation
Stabilization
Phase transitions
Continuum Limit
Integral Equations
Circle
Phase Transition
Symmetry
Model

Keywords

  • bifurcations
  • integro-differential equations
  • steady state solutions

Cite this

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title = "Finite to infinite steady state solutions, bifurcations of an integro-differential equation",
abstract = "We consider a bistable integral equation which governs the stationary solutions of aconvolution model of solid–solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is varied to examine the transition from an infinite number of steady states to three for the continuum limit of the semi–discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.",
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language = "English",
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Finite to infinite steady state solutions, bifurcations of an integro-differential equation. / Bhowmik, S. K.; Duncan, D. B.; Grinfeld, M.; Lord, G. J.

In: Discrete and Continuous Dynamical Systems - Series B, Vol. 16, No. 1, 07.2011, p. 57-71.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Finite to infinite steady state solutions, bifurcations of an integro-differential equation

AU - Bhowmik, S. K.

AU - Duncan, D. B.

AU - Grinfeld, M.

AU - Lord, G. J.

PY - 2011/7

Y1 - 2011/7

N2 - We consider a bistable integral equation which governs the stationary solutions of aconvolution model of solid–solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is varied to examine the transition from an infinite number of steady states to three for the continuum limit of the semi–discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.

AB - We consider a bistable integral equation which governs the stationary solutions of aconvolution model of solid–solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is varied to examine the transition from an infinite number of steady states to three for the continuum limit of the semi–discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.

KW - bifurcations

KW - integro-differential equations

KW - steady state solutions

UR - http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.5136v1.pdf

U2 - 10.3934/dcdsb.2011.16.57

DO - 10.3934/dcdsb.2011.16.57

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EP - 71

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

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