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The relevant saddle equilibrium is the origin, with eigenvaluesFor $\lambda <0$, they have opposite signs. Its saddle index isnear the homoclinic curve, because that curve has $\kappa >0$. Along its leading asymptote, $\delta =1-\frac{4}{5}\sqrt{-\lambda }+o(\sqrt{-\lambda })$.

A local passage near the saddle equilibrium contributes a power $s^{\delta }$ to the return map. Because $\delta <1$, its derivative becomes large as the orbit approaches the separatrix. The periodic orbits meeting the loops are therefore unstable, and their periods diverge logarithmically. On the side between the homoclinic and subcritical Hopf curves there are two unstable single-well periodic orbits, surrounding the two sink equilibria. On crossing the double-loop curve downward, they join into one unstable outer periodic orbit surrounding both wells. The attracting outer periodic orbit is not the one that collides with the saddle equilibrium: it persists across this global event.

The unstable outer periodic orbit must subsequently meet that attracting outer periodic orbit in a fold of periodic orbits. This extra curve is required to obtain the six-region diagram. One can see its position from the same energy-balance calculation. After normalizing $a_0=1$, an outer Hamiltonian contour with $H>0$ selectsAt $H=0$, $B=4/5$. The derivative of the denominator diverges positively there, while that of the numerator stays finite, so $B$ initially decreases. Rescaling at large $H$ gives $B(H)\propto \sqrt{H}$, so it eventually increases. At its minimum the stable and unstable selected periodic orbits meet. Thus the outer-cycle fold in a weakly perturbed double-well oscillator has ${\kappa }_{\mathrm{fold}}=c_f(-\lambda )+o(|\lambda |)$ with $0<c_f<4/5$; quadrature gives $c_f\simeq 0.75226$. The decreasing portion of $B(H)$ selects the unstable periodic orbit, and the increasing portion selects the stable one, since the energy drift has sign $\beta -B(H)$.