Certain diffeomorphisms of two-dimensional manifolds are considered. These diffeomorphisms have a finite hyperbolic limit set which contains a limit set cycle. The only nontransverse cycle connection in these cycles is a complete coincidence of one component of the unstable manifold of one periodic point with one component of the stable manifold of some other periodic point. A one-parameter family of diffeomorphisms containing the original diffeomorphism is described. It is shown that for parameter values arbitrarily near the parameter value corresponding to the original map these diffeomorphisms have a much enlarged limit set and are Ω \Omega -stable.