Critical saddle-node bifurcations and Morse–Smale maps

Abstract

We study the saddle-node bifurcation in diffeomorphisms with a “critical” homoclinic orbit to the saddle-node point. In a typical family F γ of diffeomorphisms that undergoes a saddle-node bifurcation at γ = 0 , the diffeomorphisms (possibly after reparameterization) for γ < 0 have two periodic points which coalesce for γ = 0 and then disappear for γ > 0 . If the saddle-node points has a critical homoclinic orbit it is known that complicated dynamics can occur for γ > 0 . We show that there are families such that for γ > 0 , there are parameter values arbitrarily close to γ = 0 for which the map is Morse–Smale. Such parameter values are shown to have positive frequency at γ = 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 γ > 0 . These results rely heavily on projecting to circle endomorphisms. We conclude with some numerical results from such maps.