Solution Landscape of the Onsager Model

We investigate critical points of the Onsager free-energy model on a sphere with different potential kernels, including the dipolar potential, the Maier-Saupe potential, the coupled dipolar/Maier-Saupe potential, and the Onsager potential. A uniform sampling method is implemented for the discretization of the Onsager model, and solution landscapes of the Onsager model are constructed using saddle dynamics coupled with downward/upward search algorithms. We first construct the solution landscapes with the dipolar and Maier-Saupe potentials, for which all critical points are axisymmetric. For the coupled dipolar/Maier-Saupe potential, the solution landscape shows a novel non-axisymmetric critical point, named tennis, which exists for a wide range of parameters. We further demonstrate various non-axisymmetric critical points in the Onsager model with the Onsager potential, including square, hexagon, octahedral, cubic, quarter, icosahedral, and dodecahedral states. The bifurcation diagram is presented to show the primary and secondary bifurcations of the isotropic state and reveal the emergence of the critical points. The solution landscape provides an efficient approach to show the global structure of the model system as well as the bifurcations of critical points, which can not only verify the previous analytic results but also propose several conjectures based on the numerical findings.