In this paper, we study effects of the secondary bifurcations in the neighborhood of the primary codimension-two bifurcation point. The twin-well potential Duffing oscillator is considered and the investigations are focused on the new scenario of destruction of the cross-well chaotic attractor. The phenomenon belongs to the category of the subduction scenario and relies on the replacement of the cross-well chaotic attractor by a pair of unsymmetric 2T-periodic attractors. The exploration of a sequence of accompanying bifurcations throws more light on the complex phenomena that may occur in the neighborhood of the primary codimension-two bifurcation point. It shows that in the close vicinity of the point there appears a transition zone in the system parameter plane, the zone which separates the two so-far investigated scenarios of annihilation of the cross-well chaotic attractor.
