= Signed gluing-map reduction
{title2=$z=-xy,\quad z'=\mu+A\operatorname{sgn}(z)|z|^\delta$}
Consider the branch return map
$$
x'=-\mu\operatorname{sgn}(x)+A\operatorname{sgn}(y)|x|^\delta,\qquad
y'=\operatorname{sgn}(x),
$$
with $A>0$, $\delta>0$, and nonzero return coordinates. After one iterate, $y\in\{-1,1\}$. On these two sign sheets, define $z=-xy$. Then $|z|=|x|$ and $\operatorname{sgn}(z)=-\operatorname{sgn}(x)\operatorname{sgn}(y)$, giving
$$
z'=-x'y'=\mu+A\operatorname{sgn}(z)|z|^\delta.
$$
This scalar equation describes the <signed gluing-map reduction>.
A negative scalar <fixed point> has $xy>0$, so $\operatorname{sgn}(x)=y$ and the next $y$ is unchanged. The two sign choices therefore lift it to two symmetry-related fixed points of the original return map. A positive scalar fixed point instead has $\operatorname{sgn}(x)=-y$; both $x$ and $y$ change sign on each step, giving one <period-two orbit>.
The value zero needs separate interpretation. In the underlying <homoclinic orbit> model it is the <separatrix> limit, where the flow's return time diverges. A formal continuous extension of the scalar formula at zero does not supply an ordinary finite-period <periodic orbit>. All branch iterations used to model finite flow returns must remain in the valid nonzero local domain.
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