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

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.

Common temporal phase requires a monomial in the first component to contain one more unbarred factor than barred factors. If the exponent differences for the two coordinates are $m_1,m_2$, then $m_1+m_2=1$. Equivariance under $\rho$ requires its spatial weight $-m_1+m_2$ to be $-1$ modulo three, hence $m_1\equiv 1(\mathrm{mod}3)$. Up to degree five, $m_1=1,m_2=0$ gives $z_1$ times all quadratic and quartic polynomials in the intensities $|z_1{|}^2,|z_2{|}^2$. The other possibility is $m_1=-2,m_2=3$, giving ${\bar{z}}_1^2{z}_2^3$ at degree five. No even-degree monomials satisfy the temporal weight.

Reflection determines the second component by exchanging indices. Thus the dihedral threefold Hopf normal form, through fifth order, iswith complex coefficients, $\alpha =\mu +i{\omega }_0$ at leading parameter order, and higher spatial-degree terms omitted. Coefficients may vary smoothly with the bifurcation parameter. The phase-sensitive quintic is allowed by the discrete threefold rotation, even though continuous independent rotations of the two coordinates would forbid it.

For a rotating wave, substitute $(z_1,z_2)=(r{e}^{i\Omega t},0)$. Its component amplitude and frequency obeyHere subscripts denote real and imaginary parts. For a standing wave, substitute $(z_1,z_2)=(r{e}^{i\Omega t},\sigma r{e}^{i\Omega t})$, with $\sigma =\pm 1$. The resonant term becomes $\sigma f{r}^5{e}^{i\Omega t}$ in the first component, and reflection ensures the second component agrees. ThereforeSet $A_s=a_R+b_R$ and $B_{\sigma }=c_R+d_R+e_R+\sigma f_R$. For constant leading coefficients, solving for $r^2$ givesSmooth parameter dependence adds common higher corrections but does not remove the generic difference between the two standing families. The rotating and standing component amplitudes differ already at cubic order unless their coefficient combinations accidentally agree. The standing waves have the same cubic amplitude, butwhich is generically nonzero. Thus all three branch types generically have different amplitudes, with the distinction between the standing families first visible at fifth order. In total Euclidean norm the rotating wave has amplitude $r_{\mathrm{rot}}$, whereas each standing wave has amplitude $\sqrt{2}{r}_{\sigma }$. The relevant branches exist on the parameter sides making their small squared amplitudes positive; nonzero cubic coefficients are part of the generic Hopf assumptions.

Without the threefold rotation, a complex linear map need only commute with coordinate swapping and common phase. Commuting with $m_x$ makes its two diagonal entries equal and its two off-diagonal entries equal; common phase still forbids conjugate-linear terms. Thus the additional leading coupling iswhere $\eta$ is a small complex symmetry-breaking coefficient and a small common diagonal shift can be absorbed into $\alpha$. The old rotation forbade these off-diagonal entries because the two coordinates have different rotation weights. Further symmetry-breaking cubic and quintic terms are also allowed; retaining only the new linear terms describes the leading onset splitting.

In symmetric and antisymmetric coordinates $z_{+}=(z_1+z_2)/\sqrt{2}$, $z_{-}=(z_1-z_2)/\sqrt{2}$, the eigenvalues are $\alpha +\eta$ and $\alpha -\eta$. With $\alpha =\mu +i{\omega }_0$, the two primary Hopf thresholds and frequencies are thereforeThe reflection-preserving standing line $z_1=z_2$ remains invariant. The line $z_1=-z_2$ also remains invariant because it is fixed by reflection combined with phase $\pi$. Their generic periodic branches survive, now starting at separate thresholds when $\mathrm{Re}\eta \ne 0$. If that real part vanishes, threshold coincidence is a nongeneric additional condition; frequency splitting may still occur.

The pure rotating-wave axes are not invariant: at $z_2=0$ the second equation has ${\dot{z}}_2=\eta z_1$. Hence the old pure rotating-wave branches cease to be primary one-component branches from the origin. At fixed nonzero symmetry breaking, a simple Hopf onset has just the appropriate symmetric or antisymmetric mode. Away from the degenerate onset, normally hyperbolic rotating-wave periodic orbits may persist as mixed-mode periodic states with both components nonzero, paired by reflection. In common unfoldings such a pair is reached through a secondary reflection-breaking pitchfork bifurcation of periodic orbits from a standing branch. Its precise attachment, any folds, and stability depend on the nonlinear coefficients and cannot be fixed by symmetry alone.

The three rotationally conjugate orientations within each original standing family are no longer equivalent. Only the lines aligned with the surviving reflection symmetries are forced primary branches; other members can deform, pin to preferred orientations, or connect through secondary bifurcations. The generic bifurcation diagram thus replaces the coincident multi-branch Hopf onset by two separated simple Hopf onsets, removes the pure traveling-wave invariance, and unfolds the rotational degeneracies. This is the reflection-preserving splitting of a dihedral Hopf bifurcation; it does not by itself prescribe a unique global diagram.