= Signed gluing-map reduction
{title2=$z=-xy,\quad z'=\mu+A\operatorname{sgn}(z)|z|^\delta$}
For $x'=-\mu\operatorname{sgn}(x)+A\operatorname{sgn}(y)|x|^\delta$, $y'=\operatorname{sgn}(x)$, the invariant sign sheet has $y=\pm1$ after one iterate. There $z=-xy$ obeys the displayed one-dimensional map. A negative fixed $z$ lifts to two symmetry-related <fixed points>, while a positive fixed $z$ lifts to one <period-two orbit>. The origin of the <return map> represents the <separatrix>, where the flow return time diverges, and is not an ordinary finite-period <orbit of a group action>.
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