Reflection-symmetric one-to-one Hopf resonance

Two complex Hopf amplitudes of even and odd spatial parity transform under reflection as $(A,B)\mapsto (A,-B)$. In a nonresonant double Hopf bifurcation, the cubic terms are $A|A{|}^2,A|B{|}^2$ and their $B$ counterparts. At equal temporal frequencies, common phase covariance also permits $B^2\overline{A}$ and $A^2\overline{B}$. Distinct frequencies alone do not guarantee nonresonance: a ratio of two allows the quadratic terms $B^2$ and $A\overline{B}$. The frequency assumptions must therefore be included when deriving the normal form of a dynamical system.

In the symmetric resonant cubic amplitude equation, put $A=\sqrt{r}{e}^{i\Omega t}$, $B=C{e}^{i\Omega t}$, with $d_R>0$, ${\beta }_R<0$. The transverse equation is $\dot{C}=-(d+\beta r)C-\beta r\overline{C}$. Its real matrix has trace $-2(d_R+{\beta }_{R}r)$ and determinant $|d{|}^2+2rQ$, where $Q=\mathrm{Re}(\beta \overline{d})$. A positive steady threshold requires $Q<0$; a simple zero eigenvalue also requires $Q\ne Q_{\ast }={\beta }_R|d{|}^2/(2d_R)$. A Hopf bifurcation requires $Q>Q_{\ast }$, giving a positive determinant at $r_H$. For $Q<Q_{\ast }$ the first instability is steady, producing reflection-related mixed phase-locked periodic branches. For $Q>Q_{\ast }$ it is Hopf, producing modulated oscillations, generically quasiperiodic in the original variables. Equality is a double-zero degeneracy.