= Antipodal cycle of a cubic swap map
{title2=$0<a<1/2$}
Along the antipodal <two-cycle> of $F(x,y)=(y,a(x-x^3/3)-y)$, the <Jacobian matrix> is $B=\left(\begin{smallmatrix}0&1\\-2a&-1\end{smallmatrix}\right)$ at both points. Its two-step <periodic-orbit multipliers> are squares of the roots of $\lambda^2+\lambda+2a=0$. The Jury criterion makes the cycle attracting precisely in the displayed interval, and repelling for $a>1/2$. This stable family is distinct from the always unstable nonzero-sum cycles.
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