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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.

With reflection symmetry, choose $\mu =\nu =m$ and $A=B$. The two primary homoclinic orbit curves are then encountered simultaneously at $m=0$. For $m<0$ the return has two attracting single-lobe cycles; for $m>0$ it has one attracting two-lobe cycle. This is the symmetric homoclinic gluing bifurcation: two cycles meet the two saddle loops and are replaced by a cycle traversing both lobes. Since $\delta >1$, the saddle equilibrium contracts strongly enough for the nearby cycles to be stable.

One can also use the signed gluing-map reduction $z=-xy$. In the symmetric return map it gives $z^{\prime }=m+A\mathrm{sgn}(z)|z{|}^{\delta }$. A negative fixed point lifts to two reflection-related flow cycles; a positive fixed point lifts to one period-two orbit of the two-sheet section map, hence one two-lobe flow cycle. This distinction prevents counting the two section intersections as two different flow cycles.

Without reflection symmetry the two splitting parameters are independent. The primary loops occur separately, and the two compound curves bound small coexistence wedges where a single-lobe and a two-lobe cycle are both stable. Thus the six-region picture is the generic two-parameter unfolding of the symmetric gluing event. The simultaneous encounter of both primary loops is codimension two for a general planar flow but codimension one in the reflection-symmetric family. The contraction argument applies to the small return domain; it makes no assertion that a globally arbitrary $P,Q$ system has only these cycles everywhere.

After one iteration $y_n\in \{1,-1\}$, so work on that invariant sign sheet. For $x_n\ne 0$,since $z_n=-x_n{y}_n$ and $|y_n|=1$. This is the signed gluing-map reduction. If an arbitrary initial value of $y$ is admitted, the formula applies after that first iterate, not before it. The point $z=0$ represents a separatrix at which the original flow return is singular, even though the scalar map can be extended continuously there.

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.

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.

A symmetric homoclinic gluing bifurcation joins two symmetry-related periodic orbits into one orbit that traverses both lobes of a figure-eight separatrix. At the transition, the two loops are homoclinic orbits to the same saddle equilibrium. The two original cycles spend increasingly long near that saddle before the connection is reached.

A signed transverse coordinate distinguishes the lobes. Reflection can make the corresponding return map odd, so fixed points $x_{\ast }$ and $-x_{\ast }$ represent the separate cycles, while a two-cycle passing between positive and negative coordinates can represent the glued orbit. The signed gluing-map reduction describes the local power-law passage and the global reinjection; its coefficients and saddle eigenvalue ratio determine the existence and stability of these branches.

The geometry alone does not guarantee attracting cycles. For example, the exact Hamiltonian double-well homoclinic orbits form a figure-eight but are surrounded by conserved-energy families, rather than isolated attracting cycles. Dissipation, a suitable parameter unfolding and the return-map conditions must supply the gluing scenario. A damped double-well model with symmetry-related Hopf cycles is one setting in which those conditions can be investigated.