Geodesic
On three-dimensional dynamical systems close to systems with a struc­tu­rally unstable homoclinic curve. II
Sbornik. Mathematics, Tome 19 (1973) no. 1, pp. 139-156 
N. K. Gavrilov; L. P. Shilnikov. On three-dimensional dynamical systems close to systems with a struc­tu­rally unstable homoclinic curve. II. Sbornik. Mathematics, Tome 19 (1973) no. 1, pp. 139-156. http://geodesic.mathdoc.fr/item/SM_1973_19_1_a9/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Dynamical systems are considered which are close to systems with a structurally unstable homoclinic curve. A definition of accessibility of a bifuration surface is given, and it is established that a bifurcation surface $H^1$ corresponding to systems with a structurally unstable homoclinic curve will be inaccessible from at least one side. Cases are singled out in which $H^1$ can be the boundary separating Morse–Smale systems from systems with a countable number of periodic motions. Bibliography: 14 titles.

[1] N. K. Gavrilov, L. P. Shilnikov, “O trekhmernykh dinamicheskikh sistemakh, blizkikh k sistemam s negruboi gomoklinicheskoi krivoi, I”, Matem. sb., 88 (130) (1972), 475–492 | MR | Zbl

[2] L. P. Shilnikov, “K voprosu o strukture rasshirennoi okrestnosti grubogo sostoyaniya ravnovesiya tipa sedlo-fokus”, Matem. sb., 81 (123) (1970), 92–103

[3] L. P. Shilnikov, “Ob odnom novom tipe bifurkatsii mnogomernykh dinamicheskikh sistem”, DAN SSSR, 189:1 (1969), 59–62 | MR

[4] S. Smale, “A structurally stable diffeomorphism with an infinite number of periodic points”, Trudy mezhdunar. simpoziuma po nelineinym kolebaniyam, 2, Kiev, 1963, 365–366 | MR | Zbl

[5] V. M. Alekseev, “Kvazisluchainye dinamicheskie sistemy. I: Kvazisluchainye diffeomorfizmy”, Matem. sb., 76 (118) (1968), 72–134 | Zbl

[6] Yu. I. Neimark, “Ob odnom klasse dinamicheskikh sistem”, Trudy V mezhdunarodnoi konferentsii po nelineinym kolebaniyam, 2, Kiev, 1970, 363–376 | Zbl

[7] V. I. Arnold, “O matritsakh, zavisyaschikh ot parametrov”, Uspekhi matem. nauk, XXVI:2 (152) (1971), 101–114 | MR

[8] Dzh. Pali, S. Smeil, “Teoremy strukturnoi ustoichivosti”, Matematika, 13:2 (1969), 145–155

[9] V. S. Afraimovich, L. P. Shilnikov, “Ob osobykh traektoriyakh dinamicheskikh sistem”, Uspekhi matem. nauk, XXVII:3 (165) (1972), 189–190

[10] V. S. Afraimovich, L. P. Shilnikov, “Ob osobykh mnozhestvakh sistem Morsa-Smeila”, Trudy Mosk. matem. ob-va, XXVIII (1973) | MR | Zbl

[11] L. P. Shilnikov, “Ob odnoi zadache Puankare-Birkgofa”, Matem. sb., 74 (116) (1967), 378–397

[12] I. W. Robbin, “A structural stability theorem”, Ann. Math., 94:3 (1971), 447–493 | DOI | MR | Zbl

[13] J. Franks, “Necessary conditions for stability of diffeomorphisms”, Trans. Amer. Math. Soc., 158:2 (1971), 301–309 | DOI | MR

[14] J. Franks, “Differentiably $\Omega$-stable diffeomorphisms”, Topology, 11:2 (1972), 107–113 | DOI | MR | Zbl