= 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 $B=\left(\begin{smallmatrix}1&1\\-1&2\end{smallmatrix}\right)$ and $D=\left(\begin{smallmatrix}-1/2&-1\\1&-3/2\end{smallmatrix}\right)$. 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.
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