Regular and chaotic states in a local map description of sheared nematic liquid crystals

We propose and study a local map capable of describing the full variety of dynamical states, ranging from regular to chaotic, obtained when a nematic liquid crystal is subjected to a steady shear flow. The map is formulated in terms of a quaternion parametrization of rotations of the local frame described by the axes of the nematic director, subdirector, and the joint normal to these, with two additional scalars describing the strength of ordering. Our model yields kayaking, wagging, tumbling, aligned, and coexistence states, accommodated in a phase diagram which closely resembles phase diagrams obtained using representations of the dynamics which are based on ordinary differential equations. We also study the behavior of the map under periodic perturbations of the shear rate. Such a map can serve as a building block for the construction of lattice models of the complex spatiotemporal states predicted for sheared nematics.