 Signed gluing-map reduction

For $x^{\prime }=-\mu \mathrm{sgn}(x)+A\mathrm{sgn}(y)|x{|}^{\delta }$, $y^{\prime }=\mathrm{sgn}(x)$, the invariant sign sheet has $y=\pm 1$ 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.