= Saddle-node curves of a signed power gluing map
{title2=$r=(A\delta)^{-1/(\delta-1)},\quad\mu=\pm\frac{\delta-1}{\delta}r$}
The <fixed point> relation is $\mu=z-A\operatorname{sgn}(z)|z|^\delta$. Setting the <fixed-point multiplier> $A\delta|z|^{\delta-1}$ equal to one gives the two <saddle-node bifurcations> at $z=\pm r$. For fixed $0<A<1$, as $\delta\uparrow1$ from below their parameter width tends to zero as $(1-\delta)A^{1/(1-\delta)}/e$, forming an exponentially narrow <gluing-map cusp>. Three scalar <fixed points> coexist inside this region: two stable outer branches and one unstable inner branch. For $\delta>1$ near one the formal <saddle-node bifurcations> recede to infinity, outside the local gluing-map domain.
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