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

TL;DR

This paper analyzes a bistable integral equation modeling phase transitions on a circle, studying how the number and stability of steady states change with diffusion, revealing symmetry-driven bifurcations from infinite to finite solutions.

Contribution

It provides a detailed bifurcation analysis of a convolution model, highlighting the role of symmetry in the emergence and stabilization of equilibria.

Findings

Transition from infinite to three steady states as diffusion varies

Symmetry influences the generation and stability of solutions

Connection between continuity and stability is complex and noteworthy

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.