 Signed gluing-map reduction

Consider the branch return mapwith $A>0$, $\delta >0$, and nonzero return coordinates. After one iterate, $y\in \{-1,1\}$. On these two sign sheets, define $z=-xy$. Then $|z|=|x|$ and $\mathrm{sgn}(z)=-\mathrm{sgn}(x)\mathrm{sgn}(y)$, givingThis scalar equation describes the signed gluing-map reduction.

A negative scalar fixed point has $xy>0$, so $\mathrm{sgn}(x)=y$ 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 $\mathrm{sgn}(x)=-y$; 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.