Summary
We provide a theoretical analysis of a Hopf bifurcation that can occur in systems with spherical geometry, based on the general theory of Hopf bifurcation in the presence of symmetry. In this particular bifurcation the imaginary eigenspace is a direct sum of two copies of the 5-dimensional irreducible representation of the groupSO(3). The same bifurcation has been studied by looss and Rossi (1988), using extensive computer-assisted calculations. Here we describe a simpler and more conceptual approach in which the representation ofSO(3) is realised as its conjugation action on the space of symmetric traceless 3 × 3 matrices. We prove the generic existence of five types of symmetry-breaking oscillation: two rotating waves and three standing waves. We analyse the stabilities of the bifurcating branches, describe the restrictions of the dynamics to various fixed-point spaces of subgroups ofSO(3), and discuss possible degeneracies in the stability conditions.
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Haaf, H., Roberts, M. & Stewart, I. A Hopf bifurcation with spherical symmetry. Z. angew. Math. Phys. 43, 793–826 (1992). https://doi.org/10.1007/BF00913409
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DOI: https://doi.org/10.1007/BF00913409