= Reflection-preserving splitting of a dihedral Hopf bifurcation
If only reflection and common temporal phase remain, the linear complex matrix is $\begin{pmatrix}\alpha&\eta\\\eta&\alpha\end{pmatrix}$. 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.
Back to article page
