Equivariant Hopf theorem

At a nondegenerate equivariant Hopf bifurcation, each complex axial isotropy subgroup of the spatial group combined with temporal phase gives a branch of periodic solutions with that spatiotemporal symmetry. Assumptions include an isolated semisimple imaginary critical pair of the required representation type, transverse parameter crossing, and a nonzero cubic coefficient on the fixed-point plane. The statement guarantees existence and symmetry type, not stability or which parameter side contains every branch.

A primary periodic branch whose phase advance is compensated by a spatial rotation. For the natural plane representation, the spatial pattern rotates rather than remaining on a fixed reflection axis. Opposite chiralities are exchanged by reflection.

// Target: dynamical-systems.bigb

A primary periodic branch retaining a spatial reflection at every time. In the natural plane representation, each real-copy component oscillates along a fixed reflection axis. The fourfold dihedral case has distinct axial and diagonal standing-wave types.

// Target: dynamical-systems.bigb

In Hopf coordinates where the rotation acts as $(i{q}_1,-i{q}_2)$, reflection swaps components, and temporal phase multiplies both equally, cubic equivariance permits precisely the three displayed monomials in the first component. Swapping indices gives the second equation, with the same complex coefficients. The rotating, axial standing, and diagonal standing branches are $q_2=0$, $q_2=q_1$, and $q_2=i{q}_1$. Their squared component amplitudes are $\mu /\mathrm{Re}a$, $\mu /\mathrm{Re}(a+b+c)$, and $\mu /\mathrm{Re}(a+b-c)$, respectively, when positive.

// Target: dynamical-systems.bigb

Spatially complex coordinates on two real copies need not both have the same temporal phase weight. If their time-circle action is $(z_1,z_2)\mapsto (e^{-i\varphi }{z}_1,e^{i\varphi }{z}_2)$, conjugating the second coordinate gives a common Hopf phase. The linear terms then have the same imaginary frequency in $q$ coordinates but opposite imaginary frequencies in the original $z$ coordinates. Failing to make this conjugation produces amplitude equations inconsistent with the specified symmetries.

// Target: dynamical-systems.bigb

A spatial group element combined with a time shift may preserve a periodic orbit even if the spatial element alone does not fix it at each instant. The temporal circle acts complex linearly on Hopf amplitudes. Complex axial isotropy subgroups identify the primary branches guaranteed by the equivariant Hopf theorem.

// Target: dynamical-systems.bigb

For the $D_3$ action $\rho (z_1,z_2)=({\zeta }^{-1}{z}_1,\zeta z_2)$, $m(z_1,z_2)=(z_2,z_1)$ and common phase multiplication, the fifth-order normal form has first component $z_1[\alpha +a|z_1{|}^2+b|z_2{|}^2+c|z_1{|}^4+d|z_1{|}^2|z_2{|}^2+e|z_2{|}^4]+f{\bar{z}}_1^2{z}_2^3$ and second component obtained by swapping indices. Here all coefficients may be complex. Its three primary symmetry types are a one-component rotating wave and the two standing-wave lines $z_1=z_2$ and $z_1=-z_2$, together with their group conjugates. The two standing amplitudes split at fifth order through $\mathrm{Re}f$.

If only reflection and common temporal phase remain, the linear complex matrix is $\left(\begin{array}{cc}\alpha & \eta \\ \eta & \alpha \end{array}\right)$. The symmetric and antisymmetric modes have eigenvalues $\alpha +\eta$ and $\alpha -\eta$, generically separating their Hopf thresholds. One-component rotating-wave axes are no longer invariant. Their hyperbolic finite-amplitude periodic solutions can persist as mixed-mode solutions, often through secondary symmetry-breaking bifurcations, while the double primary onset splits into two simple Hopf onsets. Coefficients determine the detailed secondary diagram and stability.