Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 85 / 4 / b / Solution

Let a smooth $G$-equivariant vector field have a trivial equilibrium branch and a critical real eigenspace $V\oplus V$, where $V$ is an absolutely irreducible real representation of the finite group $G$. Assume the critical linearization is semisimple with eigenvalues $\pm i{\omega }_0$, ${\omega }_0\ne 0$, no other imaginary-axis eigenvalues are present, and the real part of the critical eigenvalue crosses zero transversely as the parameter varies. The equivariant Hopf theorem then gives, generically, a branch of small-amplitude periodic solutions for each conjugacy class of complex axial isotropy subgroup of $G\times S^1$ in the critical representation. The restricted Hopf bifurcation on its two-dimensional fixed space must be nondegenerate, with nonzero leading radial cubic coefficient. Each branch has the associated spatiotemporal symmetry of a periodic orbit and period tending to $2\pi /|{\omega }_0|$.

The spatial multiplicity means this is not the ordinary simple-eigenvalue Hopf hypothesis: the required critical representation is the whole two-copy isotypic component, and the semisimple pair has the corresponding symmetry multiplicity. The theorem guarantees primary branch existence and symmetry, but neither their stability nor the side of the parameter on which they exist. Those require the nonlinear coefficients and transverse perturbation analysis. A centre manifold and a symmetry-compatible normal form of a dynamical system isolate the critical dynamics and supply those coefficients.