We consider the symmetry of discrete and continuous crystal structures which are compatible with a given choice of dislocation density tensor. By introducing the notion of a 'defective point group' (determined by the dislocation density tensor), we generalize the notion of Ericksen-Pitteri neighbourhoods to this context.

Spiral patterns on the surface of a sphere have been seen in laboratory experiments and in numerical simulations of reaction-diffusion equations and convection. We classify the possible symmetries of spirals on spheres, which are quite different from the planar case since spirals typically have tips at opposite points on the sphere. We concentrate on the case where the system has an additional sign-change symmetry, in which case the resulting spiral patterns do not rotate. Spiral patterns arise through a mode interaction between spherical harmonics degree ℓ and ℓ+1. Using the methods of equivariant bifurcation theory, possible symmetry types are determined for each ℓ. For small values of ℓ, the centre manifold equations are constructed and spiral solutions are found explicitly. Bifurcation diagrams are obtained showing how spiral states can appear at secondary bifurcations from primary solutions, or tertiary bifurcations. The results are consistent with numerical simulations of a model pattern-forming system.

Using the general theory of Hopf bifurcation with symmetry we study here the example where the group of symmetries is O(3), the rotations and reflections of a sphere. We make some amendments to previously published lists of C-axial isotropy subgroups of O(3) × S 1 and list the isotropy subgroups with fourdimensional fixed-point subspaces. We then study the particular example where O(3)×S 1 acts on the space V 3 ⊕V 3 where V 3 is the space of spherical harmonics of degree three. We find that in this case there are six C-axial isotropy subgroups of O(3)×S 1 . The equivariant Hopf theorem guarantees the existence of periodic solutions with each of these symmetries in O(3) × S 1 equivariant differential equations. Three of the solutions are found to be standing waves and the other three are travelling waves. We compute conditions for each of these solution branches to be stable and by restricting the O(3) × S 1 equivariant differential equations to four-dimensional invariant subspaces we are able to find additional periodic and quasiperiodic solutions.