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

Bhowmik, S. K. and Duncan, D. B. and Grinfeld, M. and Lord, G. J. (2011) Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete and Continuous Dynamical Systems - Series B, 16 (1). pp. 57-71. ISSN 1531-3492

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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.