Techniques of equivariant bifurcation theory are used to study the Hopf bifurcation problem on a square lattice where the group Γ = D4 ⋉ T2 acts Γ-simply on ℂ8. This enables the analysis of the stability of solutions found in a previous analysis (Silber and Knobloch 1991 Nonlinearity 4 1063-106) of the Γ-simple representation on ℂ4 to both solutions which are spatially periodic on a rhombic lattice, and to a countably infinite number of oscillatory 'superlattice' solutions. The normal form for the bifurcation is computed, and conditions for the stability of all 17 ℂ-axial branches are given. Numerical investigations indicate that there exist open regions of coefficient space where the dynamics of the cubic order truncation of the normal form are chaotic. Chaotic dynamics have not previously been found in simpler Hopf bifurcation problems in normal form.
ASJC Scopus subject areas
- Applied Mathematics
- Mathematical Physics
- Statistical and Nonlinear Physics