The Instability of Axisymmetric Solutions in Problems with Spherical Symmetry

Abstract

Among all possible equilibria that may bifurcate from the trivial state for one-parameter vector fields with $O(3)$-symmetry, one generically exists, whatever the (absolutely irreducible) representation of $O(3)$ is. This state is characterized by its group of symmetry, which includes rotations about a fixed axis, and for that reason is called “axisymmetric.” Recall that invariant spaces under irreducible representations of $O(3)$ have dimension $2l + 1$ and are generated by spherical harmonics $Y_m^l (\theta ,\phi )$, $ - l \leqq m \leqq l$. If l is even, the instability of the axisymmetric solutions follows from a theorem of Ihrig and Golubitsky [Phys. D (1984), pp. 1–33]. If l is odd, this theorem fails because it requires a condition on the quadratic part of the Taylor expansion of the equivariant vector field, but in that case it must have a zero quadratic part. However, the linearized vector field along an axisymmetric solution is diagonal in this basis and the computation of its eigenvalues is easy...