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

Let a finite or compact group $G$ act linearly on a real space $V$, and let $\dot{x}=F(x,\mu )$ be a smooth equivariant dynamical system: $F(gx,\mu )=gF(x,\mu )$. Assume $F(0,\mu )=0$. The representation is an absolutely irreducible real representation when its complexification is irreducible; equivalently, every real linear operator commuting with $G$ is a scalar multiple of the identity. Consequently the linearization at the trivial branch is $D_{x}F(0,\mu )=\alpha (\mu )I$. A generic steady-state crossing has $\alpha (0)=0$ and ${\alpha }^{\prime }(0)\ne 0$, with stable noncritical directions removed by a centre manifold if this is a reduction of a larger system.

The isotropy group of a vector $x$ is ${\Sigma }_x=\{g\in G:gx=x\}$. For a subgroup $\Sigma$, its fixed-point subspace of a group action is $\mathrm{Fix}(\Sigma )=\{x:\sigma x=x\text{ for every }\sigma \in \Sigma \}$. Its normaliser isThe normaliser preserves $\mathrm{Fix}(\Sigma )$, and $N_G(\Sigma )/\Sigma$ describes its residual symmetry. A steady-state axial subgroup is an actual isotropy subgroup with ${\dim }_{\mathbb{R}}\mathrm{Fix}(\Sigma )=1$.

The equivariant branching lemma states that, under the smoothness, equivariance, absolute irreducibility, and transverse scalar-eigenvalue crossing hypotheses above, each conjugacy class of axial isotropy subgroup gives a branch of nonzero equilibria in its fixed-point line through $(0,0)$. The generic leading nonlinear coefficient determines which parameter side the branch occupies and whether it is transcritical-like or pitchfork-like; the theorem does not assert that these branches are stable or exhaust every possible branch. The usual nondegeneracy condition excludes a vanishing leading nonlinear coefficient. Conjugate subgroups produce symmetry-related copies of a branch, rather than new isotropy types.

To explain the dynamical role of the fixed-point subspaces of a group action, if $x$ is fixed by $\Sigma$, then $\sigma F(x,\mu )=F(\sigma x,\mu )=F(x,\mu )$ for every $\sigma \in \Sigma$. Thus the vector field is tangent to that subspace, and uniqueness makes it flow invariant. On a fixed-point line one solves a scalar equilibrium equation and determines radial stability. Stability in the full space additionally requires all transverse eigenvalues to be stable. Higher-dimensional fixed-point spaces support their own restricted dynamics and may contain symmetry-constrained connecting orbits.

Finally, the normalizer action determines axial branch parity: if $N_G(\Sigma )/\Sigma$ contains an element acting as $x\mapsto -x$ on the fixed line, its scalar vector field must be odd, so a generic cubic term gives a pitchfork bifurcation. If that residual sign reversal is absent, a quadratic term is permitted and can give a transcritical-like branch. This is why fixed-space dimension alone does not determine either the branch type or its full stability.

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.