Nonlinear phase-space structure, bifurcation, and topological solitons in nematic liquid crystal reorientation.

We investigate field-induced director reorientation in nematic liquid crystals within the Frank-Oseen continuum framework through a global phase-space formulation that reveals the nonlinear geometric structure of equilibrium director configurations and their connection to the classical Fréedericksz transition. Reformulating the governing second-order equation as an autonomous dynamical system using a dimensionless spatial coordinate, we characterize the global organization of equilibrium states as a function of dimensionless control parameter |λ| = (L/ξH)2, which quantifies the competition between magnetic and elastic torques. The analysis reveals a supercritical pitchfork bifurcation of the bulk spatial dynamical system at λ = 0, marking the onset of symmetry breaking and generating an alternating sequence of stable centers and unstable saddles. The separatrix trajectories connecting neighboring saddles are identified as static topological solitons (domain walls), providing a natural geometric framework for characterizing their spatial width, energy, and stability. An associated three-dimensional energy-landscape representation provides a direct connection between phase-space topology and physical multistability, demonstrating how finite-cell boundary anchoring unfolds the idealized bulk bifurcation into the experimentally observed Fréedericksz transition. These results establish a unified geometric framework connecting nonlinear dynamical systems theory with continuum liquid-crystal physics, providing new physical insight into symmetry breaking, multistability, and localized topological structures in anisotropic soft matter.