### Abstract

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 language | English |
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Pages (from-to) | 57-71 |

Number of pages | 15 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 16 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 2011 |

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### Keywords

- bifurcations
- integro-diﬀerential equations
- steady state solutions

### Cite this

*Discrete and Continuous Dynamical Systems - Series B*,

*16*(1), 57-71. https://doi.org/10.3934/dcdsb.2011.16.57

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*Discrete and Continuous Dynamical Systems - Series B*, vol. 16, no. 1, pp. 57-71. https://doi.org/10.3934/dcdsb.2011.16.57

**Finite to infinite steady state solutions, bifurcations of an integro-differential equation.** / Bhowmik, S. K.; Duncan, D. B.; Grinfeld, M.; Lord, G. J.

Research output: Contribution to journal › Article

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

M3 - Article

VL - 16

SP - 57

EP - 71

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 1

ER -