= Solution
For a <fixed point> write $\mu=h(z)=z-A\operatorname{sgn}(z)|z|^\delta$. A <saddle-node bifurcation> requires both $f(z)=z$ and $f'(z)=1$. For $\delta\ne1$ these conditions give
$$
\boxed{r=(A\delta)^{-1/(\delta-1)},\qquad
z=\pm r,\qquad\mu=\pm\frac{\delta-1}{\delta}r.}
$$
The second derivative is nonzero at either nonzero <saddle-node bifurcation>, making it an ordinary <saddle-node bifurcation>. For $\delta<1$, define
$$
C(\delta)=\frac{1-\delta}{\delta}(A\delta)^{1/(1-\delta)}.
$$
The <saddle-node bifurcations> occur at $(\mu,z)=(-C,r)$ and $(C,-r)$, and \b[three scalar <fixed points> coexist for $|\mu|<C$]. The two outer branches have <fixed-point multiplier> below one and are stable, while the middle branch is unstable. Negative points lift to two symmetry-related <limit cycles>, so these scalar branch counts need not equal the number of flow <limit cycles>. The cusp records coexistence of distinct return-map branches; even outside it a negative branch represents a symmetry-related pair of flow cycles.
If $\delta=1-\eta$, $\eta\downarrow0$, then
$$
\boxed{C(1-\eta)\sim\frac\eta e A^{1/\eta}.}
$$
This exponentially narrow region ends in a <gluing-map cusp> at $(\mu,\delta)=(0,1)$. At its boundaries the graph of $f$ is tangent to the diagonal at $z=+r$ or $-r$, as shown below. At exactly $\delta=1$, $f=\mu+Az$ and $A<1$ gives one stable <fixed point> for nonzero $\mu$.
For $\delta=1+\eta$, the formal <saddle-node bifurcation> radius is $(A\delta)^{-1/\eta}\to\infty$, and the corresponding <saddle-node bifurcation> parameters also diverge. Thus the near-origin <gluing-map cusp> lies on the $\delta<1$ side; the formal large <saddle-node bifurcations> of the power-law formula on the other side are outside the local gluing-map approximation. They must not be drawn as a second small <gluing-map cusp> near the origin.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-60-gluing.png]
{title=Exponentially narrow gluing-map cusp, tangencies at its two fold boundaries, and stable and unstable fixed-point diagrams on either side of delta one}
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