Title:

Generic planar lattice patterns in liquid crystals

In this thesis we will be studying symmetries and pattern formation within a planar layer of liquid crystal. Some of the generic equilibrium patterns (steady states) on a square or hexagonal lattice that bifurcate from a homeotropic or planar isotropic state were calculated in Chillingworth and Golubitsky 2003 J. Mathematical Physics 44(9) 42014219. Continuing this work we calculate a second set of steady states and go on to calculate the time periodic solutions resulting from Hopf bifurcations in the same planar layer of liquid crystal. We describe the possible symmetries of the system by the group ΓL x S1, (or just ΓL in the steady states), ΓL = (H n T2) x Z2, where H is the holohedry of the chosen lattice L, that is the finite group of rotations and reflections that preserve the lattice, T2 = R2/L is the torus group representing translations on the lattice, Z2 represents the reflection in the xy plane, and S1 is the circle group representing time periodicity. We find the equilibrium solutions by applying the Equivariant Branching Lemma and finding isotropy subgroups of ΓL with fixedpoint subspaces of dimension 1. We then find the time periodic solutions using the Equivariant Hopf Theorem, finding isotropy subgroups of Γ x S1 with fixedpoint subspaces of dimension 2 by using the group theory methods shown in Dionne et al 1995 Phil. Trans. Physical Sciences and Engineering 352(1698) 125168.
