Nonlinear dynamics, Hopf bifurcation, and optimal control in a reduced-order Doi–Hess model for shear-driven nematic liquid crystals

We investigate the nonlinear dynamics of a shear-driven nematic liquid crystal using a reduced-order formulation derived from Doi–Hess kinetic theory. Starting from the orientation distribution function, the macroscopic alignment tensor is introduced as the second-moment order parameter, which is symmetric and traceless and therefore admits a five-dimensional representation in terms of a linearly independent, nonorthogonal tensor basis. By projecting the full tensor evolution equation, comprising Landau–de Gennes relaxation and shear-induced reorientation, onto this invariant subspace, we obtain a closed system of five coupled nonlinear ordinary differential equations. The resulting model retains the essential physics of nematic ordering, including cubic nonlinear saturation and flow-alignment effects governed by the imposed shear rate. Numerical continuation using MatCont reveals a Hopf bifurcation and an associated periodic-orbit branch as the imposed shear rate is varied. The first Lyapunov coefficient confirms the nature of the bifurcation and the transition to oscillatory dynamics. In addition, the system is embedded within an optimal control framework, where anisotropy is minimized subject to dynamical constraints, both with and without explicit consideration of the Hopf bifurcation structure. Results demonstrate a significant reduction in anisotropy under bifurcation-aware control, highlighting the importance of instability mechanisms in determining optimal performance. The study provides a unified framework linking tensor kinetic theory, nonlinear dynamical systems, and optimal control in complex fluid dynamics.