Cubic swap-map two-cycle factorization ($(x+y)(axy/3-1)=0$)

For $F(x,y)=(y,a(x-x^3/3)-y)$, a genuine two-cycle obeys $F(x,y)=(y,x)$ with $x\ne y$. Subtracting its two component equations gives $x^2+xy+y^2=3$, then substitution gives the displayed factorization. Cancelling $x+y$ loses the antipodal cycle $(\sqrt{3},-\sqrt{3})\leftrightarrow (-\sqrt{3},\sqrt{3})$, present for every $a>0$. The nonzero-sum family instead has $xy=3/a$ and $x+y=\pm \sqrt{3+3/a}$, requiring $a>3$. Retaining all vanishing factors is essential in polynomial cycle classification.

The nonzero-sum two-cycles of the cubic swap map exist for $a>3$. Their two-step characteristic polynomial has trace $T$ and determinant $D$ as displayed. Since $1-T+D=-2(a-3)(a+1)<0$, one periodic-orbit multiplier exceeds one, proving instability throughout. The other crosses minus one at $a=(1+\sqrt{29})/2$; the cycles are saddles before that threshold and repelling afterwards. At $a=3$ they meet the fixed branches in an unstable subcritical-flip branch.

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.