= 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>.
Back to article page
