Geodesic
On attainable transitions from Morse--Smale systems to systems with many periodic motions
Izvestiya. Mathematics , Tome 8 (1974) no. 6, pp. 1235-1270

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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. 
@article{IM2_1974_8_6_a5,
     author = {V. S. Afraimovich and L. P. Shilnikov},
     title = {On attainable transitions from {Morse--Smale} systems to systems with many periodic motions},
     journal = {Izvestiya. Mathematics },
     pages = {1235--1270},
     publisher = {mathdoc},
     volume = {8},
     number = {6},
     year = {1974},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1974_8_6_a5/}
}
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V. S. Afraimovich; L. P. Shilnikov. On attainable transitions from Morse--Smale systems to systems with many periodic motions. Izvestiya. Mathematics , Tome 8 (1974) no. 6, pp. 1235-1270. http://geodesic.mathdoc.fr/item/IM2_1974_8_6_a5/