Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 78 / 4 / a / Solution

The equivariant Hopf theorem states that, under the usual nondegeneracy and transverse-crossing assumptions at an equivariant Hopf point, each complex axial isotropy subgroup of the spatial symmetry group combined with temporal phase produces a branch of periodic solutions with that spatiotemporal symmetry. Its common fixed-point space must have real dimension two, or complex dimension one. Restricting the Hopf normal form of a dynamical system to that plane gives a scalar complex amplitude equation; a nonzero real cubic coefficient provides a small periodic branch on the corresponding parameter side. The theorem supplies existence and symmetry, not stability.

Put $\zeta =e^{2\pi i/3}$ and write $T_{\theta }z=e^{i\theta }z$, so $T_{\theta }={\tau }_{\theta /{\omega }_0}$. Three complex axial fixed lines areFor the first generator, its eigenvalues are $1,{\zeta }^2$, so the fixed line is exactly $z_2=0$. Reflection supplies the conjugate line $z_1=0$, representing the opposite rotation direction. For the other two generators, swapping coordinates, with or without the common minus sign, gives the displayed one-complex-dimensional fixed spaces.

These subgroups are inequivalent: the rotating-wave spatial projection has order three, while the standing-wave projections have order two; the two reflection types differ by the central phase $T_{\pi }$, which conjugation cannot remove. Rotating the two standing lines by $\rho$ gives three orientations of each type, not new symmetry types. Therefore there are three distinct generic periodic branch types: rotating waves and two inequivalent standing-wave families. Their members have a period near $2\pi /{\omega }_0$, with the combined spatial–temporal operations determining their relative phase relations.