Finite to infinite steady state solutions, bifurcations of an integro-differential equation. (English) Zbl 1221.37106

Bhowmik, Samir K.; Duncan, Dugald B.; Grinfeld, Michael; Lord, Gabriel J.

Discrete Contin. Dyn. Syst., Ser. B 16, No. 1, 57-71 (2011).

Summary: 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 increased to examine the transition from an uncountably 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.

MSC:

37G40	Dynamical aspects of symmetries, equivariant bifurcation theory
45K05	Integro-partial differential equations

Keywords:

integro-differential equation; bifurcations; continuum limit; semi-discrete; regularity