 Unstable asymmetric cycles of a cubic swap map

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.