 Distinction between linear resonance and nonlinear periodic bifurcation

Roots of unity among the linearization multipliers give periodic linearized perturbations but do not by themselves prove a branch of nonlinear periodic solutions. For the normalized delayed recurrence $F_r(x,y)=(y,r(1+y)/(1+(r-1)(1+x{)}^2)-1)$, the multipliers at $r=2$ are $e^{\pm i\pi /3}$. Nevertheless, with $Q=x^2-xy+y^2$, its sixth iterate has leading displacement $(r-2)Bv+QDv$, where $B=\left(\begin{array}{cc}1& 1\\ -1& 2\end{array}\right)$ and $D=\left(\begin{array}{cc}-1/2& -1\\ 1& -3/2\end{array}\right)$. The identity $\det (Bv,QDv)=Q^2/2$ obstructs cancellation for small nonzero $v$. Indeed any putative local six-cycle requires $r-2=O(|v{|}^2)$ by the norm equation, and then its determinant equation would read $0=Q^2/2+O(|v{|}^5)$, impossible. This is a concrete caution against promoting a linear resonance to a nonlinear bifurcation assertion.