The Instability of Axisymmetric Solutions in Problems with Spherical Symmetry

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 once the equivariant structure of the vector field is known. In this paper, using this idea, it is shown that two eigenvalues, namely those with eigendirections given by $m = 2$ and $m = 3$ in the basis of spherical harmonics, are simply related and have opposite signs whatever l.