Geodesic
On three-dimensional dynamical systems close to systems with a struc­tu­rally unstable homoclinic curve. I
Sbornik. Mathematics, Tome 17 (1972) no. 4, pp. 467-485 
N. K. Gavrilov; L. P. Shilnikov. On three-dimensional dynamical systems close to systems with a struc­tu­rally unstable homoclinic curve. I. Sbornik. Mathematics, Tome 17 (1972) no. 4, pp. 467-485. http://geodesic.mathdoc.fr/item/SM_1972_17_4_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper three-dimensional dynamical systems are considered that are close to systems with a structurally unstable homoclinic curve, i.e. with a path biasymptotic to a structurally stable periodic motion of saddle type to which the stable and unstable manifolds are tangent. Under the assumption that the tangency is the simplest structurally unstable one, it is established that in the set of paths lying entirely in an extended neighborhood of a periodic motion there is a subset whose paths are in one-to-one correspondence with the paths of a subsystem of a Bernoulli scheme of three symbols. Figures: 5. Bibliography: 6 titles.

[1] A. A. Andronov, E. A. Leontovich, I. I. Gordon, I. G. Maier, Teoriya bifurkatsii dinamicheskikh sistem na ploskosti, Nauka, Moskva, 1967

[2] E. A. Leontovich, “O rozhdenii predelnykh tsiklov iz separatrisy”, DAN SSSR, 28:4 (1951), 641–644

[3] L. P. Shilnikov, “O rozhdenii periodicheskogo dvizheniya iz traektorii, dvoyakoasimptoticheskoi k sostoyaniyu ravnovesiya tipa sedlo”, Matem. sb., 77(119) (1968), 461–472

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

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

[6] L. P. Shilnikov, “K voprosu o strukture okrestnosti gomoklinicheskoi truby invariantnogo tora”, DAN SSSR, 180:2 (1968), 286–289 | Zbl