Bifurcations analysis of the twist-Fr\'eedericksz transition in a nematic liquid-crystal cell with pre-twist boundary conditions: the asymmetric case

TL;DR

This paper extends the analysis of the twist-Freedericksz transition in nematic liquid crystals to asymmetric boundary conditions, using phase-plane methods to derive bifurcation diagrams and stability results.

Contribution

It generalizes previous symmetric boundary condition studies by analyzing asymmetric conditions and provides detailed bifurcation and stability analysis.

Findings

Bifurcation diagrams for asymmetric boundary conditions are derived.

Stability of solutions is characterized using Maginu's method.

Comparison with symmetric case highlights effects of asymmetry.

Abstract

In the paper [Eur. J. of Appl. Math. \textbf{20}, (2009) 269--287] by da Costa et al. the twist-Fr\'eedericksz transition in a nematic liquid crystal one-dimensional cell of lenght $L$ was studied imposing an antisymmetric net twist Dirichlet condition at the cell boundaries. In the present paper we extend that study to the more general case of net twist Dirichlet conditions without any kind of symmetry restrictions. We use phase-plane analysis tools and appropriately defined time-maps to obtain the bifurcation diagrams of the model when $L$ is the bifurcation parameter, and related these diagrams with the one in the symmetric situation. The stability of the bifurcating solutions is investigated by applying the method of Kenjiro Maginu [J. Math. Anal. Appl. \textbf{63}, (1978) 224--243].