= Softening Rayleigh-Duffing phase portrait
With $a=-A>0$, a stable cycle grows from the origin. At $a_h\simeq0.33144619$, symmetric saddle connections form a repelling <heteroclinic cycle>, while the stable inner cycle persists. Just above it an unstable outer cycle coexists with the stable one; they merge at $a_f\simeq0.33147066$ in a <saddle-node bifurcation of periodic orbits>. These approximate values follow from the half-return equations $H_a(p)=p$, with $H_a'(p)=1$ at the fold, and saddle-manifold shooting at the connection. A coarse parameter scan can miss the narrow coexistence interval. The saddle <eigenvalue> ratio is below one at the connection, proving that the loop is repelling and preventing identification of the stable cycle's disappearance with that connection.
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