 Solution

For $\delta =1+\epsilon >1$, the graph of $h(z)$ has a positive maximum at $z=r$ and a negative minimum at $z=-r$. The central branch $|z|<r$ is stable, while the formal outer branches $|z|>r$ are unstable. Near the gluing point only the stable central branch is relevant: it crosses from negative $z$ to positive $z$, representing the stable two-to-one gluing described in part (b). The formal full power-law diagram has three fixed points between its two large saddle-node bifurcations.

For $\delta =1+\epsilon <1$, the positive branch has its minimum at $(\mu ,z)=(-C,r)$, while the negative branch has its maximum at $(C,-r)$. The inner pieces $|z|<r$ are unstable and outer pieces stable. In the gluing-map cusp interval there is a stable negative branch, a stable positive branch, and an unstable central branch; the unstable branch passes through the singular gluing point with opposite sign to $\mu$. Outside the gluing-map cusp just one stable outer branch remains. The lower panels of the figure plot these diagrams in fold-scaled variables; dashed lines denote instability and solid lines dynamical stability. Neither diagram has a flip bifurcation, since the scalar derivative is positive wherever defined.