Critical saddle-node bifurcations and Morse-Smale maps

We study the saddle-node bifurcation in diffeomorphisms with a "critical" homoclinic orbit to the saddle-node point. In a typical family F-gamma of diffeomorphisms that undergoes a saddle-node bifurcation at gamma = 0, the diffeomorphisms (possibly after reparameterization) for gamma < 0 have two periodic points which coalesce for gamma = 0 and then disappear for gamma > 0. If the saddle-node points has a critical homoclinic orbit it is known that complicated dynamics can occur for gamma > 0. We show that there are families such that for gamma > 0, there are parameter values arbitrarily close to gamma = 0 for which the map is Morse-Smale. Such parameter values are shown to have positive frequency at gamma = 0(+). In the process we show that the boundary of the set of Morse-Smale diffeomorphisms possesses comb-like structures. We also show that there are other families unfolding a saddle-node for which there are no Morse-Smale maps for gamma > 0. These results rely heavily on projecting to circle endomorphisms. We conclude with Some numerical results from such maps. (C) 2004 Elsevier B.V. All rights reserved.