Signed gluing-map reduction ($z=-xy,z^{\prime }=\mu +A\mathrm{sgn}(z)|z{|}^{\delta }$)

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.

For $f(z)=\mu +A\mathrm{sgn}(z)|z{|}^{\delta }$, a nonzero fixed point satisfiesAssume $A>0$, $\delta >0$ and $\delta \ne 1$. A saddle-node bifurcation has multiplier $f^{\prime }(z)=A\delta |z{|}^{\delta -1}=1$, so it occurs at $z=\pm r$ withAt either nonzero point, $f^{\prime \prime }$ is nonzero and the parameter enters additively, giving the ordinary nondegenerate fold conditions.

When $0<\delta <1$, define $C=(1-\delta )r/\delta$. The scalar extension has three fixed-point branches for $|\mu |<C$: two stable outer branches and an unstable inner branch. For nonzero fixed points, their derivatives are below and above one because $A\delta |z{|}^{\delta -1}$ decreases with $|z|$. At $\mu =0$, the inner branch lies at the singular separatrix value; the signed gluing-map reduction explains why it is excluded as a finite-time flow return.

For fixed $0<A<1$ and $\delta \uparrow 1$ from below, the fold magnitude has asymptotic value $C\sim (1-\delta )A^{1/(1-\delta )}/e$. This is the half-width; the total parameter interval has width $2C$. The exponentially narrow gluing-map cusp gives the exact rescaling. For $\delta >1$ approaching one, the formal fold coordinate instead tends to infinity, outside the local return-map region.

Fix $0<A<1$ in the signed gluing-map reduction and write $\delta =1-\eta$ with $0<\eta <1$. The saddle-node curves of a signed power gluing map occur at parameter values $\mu =\pm C$, whereThus $C$ is the half-width of the three-branch interval, and $2C$ is its total width.

To see the scale, use $(1-\eta {)}^{1/\eta }\to e^{-1}$ and $(1-\eta {)}^{-1}\to 1$. ThenSince $A^{1/\eta }=\exp (\log A/\eta )$ and $\log A<0$, the width decays faster than every positive power of $\eta$. A parameter scan with an algebraic step size can therefore miss the coexistence interval entirely.

This is an essential exponential scaling, rather than the power-law fold separation in the ordinary smooth cubic cusp. The three branches counted here are scalar return-map branches. Their physical lifts differ: a negative scalar fixed point represents two symmetry-related cycles, while a positive one represents one cycle visiting both lobes. The singular zero value is a separatrix limit with infinite return time.