 Antipodal cycle of a cubic swap map

Along the antipodal two-cycle of $F(x,y)=(y,a(x-x^3/3)-y)$, the Jacobian matrix is $B=\left(\begin{array}{cc}0& 1\\ -2a& -1\end{array}\right)$ 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.