Solution landscape of the Onsager model identifies non-axisymmetric critical points

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