Bifurcations in the regularized Ericksen bar model

TL;DR

This paper analyzes bifurcations in a regularized Ericksen elastic bar model, revealing the structure of bifurcations, supporting a conjecture on symmetry of minimizers, and exploring complex bifurcation patterns.

Contribution

It provides a detailed bifurcation analysis of the regularized Ericksen model, supporting Muller's conjecture and introducing new conjectures based on local analysis and numerical results.

Findings

Evidence supporting Muller's conjecture on symmetry of minimizers

Detailed description of primary branch connections in bifurcations

Analysis of a loop structure in $(k,3k)$ bifurcations

Abstract

We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods, the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of M\"uller's conjecture \cite{Muller} concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of the primary branch connections that occur in this problem. We give a reformulation of M\"uller's conjecture and suggest two further conjectures based on the local analysis and numerical observations. We conclude by analysing a ``loop'' structure that characterizes $(k,3k)$ bifurcations.