On attainable transitions from Morse–Smale systems to systems with many periodic motions

In this paper it is proved that with the disappearance of equilibrium states of the type saddle-saddle there appear singular sets homeomorphic to a suspension over a certain topological Markov chain. It is established that the corresponding bifurcation surface can separate Morse–Smale systems from systems with a countable set of periodic motions and is $\Omega$-attainable on both sides. On the basis of the results obtained a description is given of the structure of basic sets connected with the appearance of homoclinic curves. Cases are indicated when the description of the structure of the neighborhood of a homoclinic curve coincides with the description of a basic set.