2021/01/24 by D. R. J. Chillingworth, David Chillingworth, M. Gregory Forest +2
Computer Science · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Angular velocity #Bifurcation #Classical mechanics #Equivariant map #Flocking (texture) #Geology #Geometry #Liquid crystal #Mathematics #Mechanics #Nonlinear Dynamics and Pattern Formation #Optics #Perturbation (astronomy) #Physics #Plane symmetry #Principal axis theorem #Quantum chaos and dynamical systems #Quantum mechanics #Rotation (mathematics) #Shear (geology) #Shear flow #Shear stress #Simple shear #Vortex #Vorticity #cond-mat.soft #math-ph #math.DS #math.MP #msc:34C14 #msc:34C23 #msc:34C25 #msc:34C29 #msc:37C27 #msc:37C81 #msc:37G15 #msc:37G40 #msc:37N10 #msc:76T99 #msc:92F05
paper · pdf · doi:10.1007/s00205-021-01703-x
published as Archive for Rational Mechanics and Analysis, 2021-09-07 · 49 pages. Version revised in response to referees' comments, exposition improved. Now published
openalex publication_date 2021/09/07 · arxiv created 2021/09/27 · arxiv updated 2021/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
Abstract We use geometric methods of equivariant dynamical systems to address a long-standing open problem in the theory of nematic liquid crystals, namely a proof of the existence and asymptotic stability of kayaking periodic orbits in response to steady shear flow. These are orbits for which the principal axis of orientation of the molecular field (the director) rotates out of the plane of shear and around the vorticity axis. With a small parameter attached to the symmetric part of the velocity gradient, the problem can be viewed as a symmetry-breaking bifurcation from an orbit of the rotation group SO(3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>SO</mml:mi><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> that contains both logrolling (equilibrium) and tumbling (periodic rotation of the director within the plane of shear) regimes as well as a continuum of neutrally stable kayaking orbits. The results turn out to require expansion to second order in the perturbation parameter.