= Unstable asymmetric cycles of a cubic swap map
{title2=$T=4-a,\quad D=-2a^2+3a+9$}
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.
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