Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 85 / 3 / b / Solution

Use the natural dihedral group generators $\rho z=e^{2\pi i/n}z$ and $mz=\bar{z}$, and orient the unfolding parameter so the trivial equilibrium loses stability as $\mu$ increases through zero. Rotation equivariance requires a monomial $z^p{\bar{z}}^q$ in $\dot{z}$ to have $p-q\equiv 1(\mathrm{mod}n)$, while reflection makes its coefficient real. These constraints give the dihedral steady-state normal form. The reflection axes have angles $\theta =k\pi /n$ and one-dimensional fixed-point subspaces of a group action; their equilibria are the branches guaranteed by the equivariant branching lemma.

For $n=3$, a generic quadratic anisotropy dominates:In polar coordinates, $\dot{r}=\mu r+b{r}^2\cos 3\theta +O(r^3)$ and $\dot{\theta }=-br\sin 3\theta +O(r^2)$. At a nonzero equilibrium, $\chi =\cos 3\theta =\pm 1$ and $r=-\mu /(b\chi )+O({\mu }^2)>0$. There are three symmetry-related equilibria on each parameter side, on opposite half-axes as the sign of $\mu$ changes. The radial and angular eigenvalues are $-\mu +O({\mu }^2)$ and $3\mu +O({\mu }^2)$, respectively. Hence all three nonzero branches are saddles in the full plane, even though radial stability exchanges at the crossing. The origin is a sink for $\mu <0$ and a source for $\mu >0$. The reflection normaliser has no residual sign reversal in this odd dihedral case, so these are transcritical-like branches, not symmetry-forced pitchforks.

For $n=4$, both relevant nonlinearities are cubic:There are two conjugacy classes of reflection isotropy: the coordinate axes, with $\chi =\cos 4\theta =1$, and the diagonals, with $\chi =-1$. Generically $b\ne 0$ and $a\pm b\ne 0$. Each class has a pitchfork bifurcation, with four symmetry-related equilibria when its squared radius is positive:These formulas give all generic existence and stability cases. If $a<-|b|$, both classes emerge for $\mu >0$; the class with $b\chi >0$ consists of stable nodes and the other of saddles. If $a>|b|$, both classes exist for $\mu <0$; the class with $b\chi >0$ consists of saddles and the other of sources. If $|a|<|b|$, one class exists on each side and both consist of saddles. Equality $a=\pm b$ is nongeneric and requires higher-order terms. Unlike the odd dihedral cases, the residual normaliser quotient contains a sign reversal, enforcing exact paired pitchfork branches.

For $n=5$, cubic radial saturation occurs before the first angular anisotropy:For generic $a,b\ne 0$, ten equilibria lie on the reflection half-axes whenever $-\mu /a>0$, divided into two distinct five-point group orbits. Their radii and eigenvalues areIf $a<0$, they emerge for $\mu >0$: the five with $b\chi >0$ are stable nodes and the other five are saddles. If $a>0$, they exist for $\mu <0$: the five with $b\chi >0$ are saddles and the others are sources. The leading amplitude scaling is pitchfork-like, but the quartic term distinguishes opposite half-axes and generally gives different corrections to their radii; there is no exact sign-reversal symmetry on an individual fixed line. Thus $D_3$ has quadratic saddle branches, $D_4$ has two cubic axial branch types, and $D_5$ has cubic radial branches whose angular stability splits at quartic order.