 Softening Rayleigh-Duffing phase portrait

With $a=-A>0$, a stable cycle grows from the origin. At $a_h\simeq 0.33144619$, 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 $a_f\simeq 0.33147066$ in a saddle-node bifurcation of periodic orbits. These approximate values follow from the half-return equations $H_a(p)=p$, with $H_a^{\prime }(p)=1$ 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.