Patterns, Nonlinear Dynamics and Applications - PANDA

Paul Matthews

Pattern formation on a sphere

 
Abstract: Pattern formation on the surface of a sphere is described by equations involving interactions of spherical harmonics of degree l. When l is even, the leading-order equations are determined uniquely by the symmetry, regardless of the physical context. New general results are derived regarding the existence of solutions with dihedral symmetry and the stability of the axisymmetric state. Existence and stability results are found for even l up to l=12. Using either a variational or eigenvalue criterion, the preferred solution has icosahedral symmetry for l=6, l=10 and l=12.