 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.