 Solution

For a fixed point write $\mu =h(z)=z-A\mathrm{sgn}(z)|z{|}^{\delta }$. A saddle-node bifurcation requires both $f(z)=z$ and $f^{\prime }(z)=1$. For $\delta \ne 1$ these conditions giveThe second derivative is nonzero at either nonzero saddle-node bifurcation, making it an ordinary saddle-node bifurcation. For $\delta <1$, defineThe saddle-node bifurcations occur at $(\mu ,z)=(-C,r)$ and $(C,-r)$, and 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 $\delta =1-\eta$, $\eta \downarrow 0$, thenThis 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.