We found a novel connecting orbit in the averaged Duffing-Rayleigh equation. The orbit starts from an unstable manifold of a saddle type equilibrium point and reaches to a stable manifold of a node type equilibrium. Although the connecting orbit is structurally stable in terms of the conventional definition of structural stability, it is structually unstable since a one-deimensional manifold into which the connecting orbit flows is unstable. We can consider the orbit is one of global bifurcations governing the differentiability of the closed orbit.