= Solution
For $\mu>0$, the left <equilibrium point> $(-\sqrt\mu,0)$ is a <stable node>, the right <equilibrium point> $(\sqrt\mu,0)$ a <saddle equilibrium>, and the <saddle equilibrium>'s left unstable branch flows directly towards the node. The right branch takes the prescribed global excursion. Its landing relative to the two <equilibrium points> gives the six local portraits in the figure.
If $\mu>\nu^2$, then $-\sqrt\mu<\nu<\sqrt\mu$ for either sign of $\nu$. The global branch lands between the <equilibrium points> and flows left into the node. There is no nearby <Poincaré return map> cycle, and the landing approaches the node from its right.
If $0<\mu<\nu^2$ and $\nu>0$, the branch lands to the right of the <saddle equilibrium>. The contracting <Poincaré return map> has an attracting <fixed point>, giving a stable <periodic orbit>; the node also remains as a separate attractor with its own basin. If instead $\nu<0$, the branch lands to the left of the node and approaches it from the left. There is no nearby <periodic orbit> in that case.
If $\mu<0$, the two <equilibrium points> have vanished and a stable returning cycle exists for either sign of $\nu$. For negative $\nu$ the cycle traverses the full bottleneck and has the divergent period in part (c). For fixed positive $\nu$ it enters to the right of the bottleneck; its local period approaches $1/\nu-1/h$, so it need not diverge when $\mu\to0^-$. The outer arcs in the figure encode only the given return; the local equations do not specify the full <vector field> away from the origin.
The boundaries explain the codimension-two organization. On $\nu=\sqrt\mu>0$ the <saddle equilibrium> has a <homoclinic orbit>. On $\mu=0$, $\nu<0$, the return passes through the semistable <equilibrium point> at a <saddle-node bifurcation>, producing a <saddle-node bifurcation on an invariant circle>. On $\mu=0$, $\nu>0$, the node and <saddle equilibrium> instead collide away from the persistent cycle. At $\mu=\nu=0$, the returning <separatrix> hits the <equilibrium point>'s strong <stable manifold> at the <saddle-node bifurcation>: this is the <saddle-node separatrix-loop bifurcation>. The curve $\nu=-\sqrt\mu$ changes which side of the node receives the returning orbit; unlike the positive branch, it is not a <global bifurcation> through a saddle <homoclinic orbit>.
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