Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type
Sbornik. Mathematics, Tome 196 (2005) no. 4, pp. 561-594 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Systems generalizing Lorenz's are considered in a bounded subdomain of $\mathbb R^3$. It is shown that under certain conditions of uniform hyperbolicity small non-autonomous perturbations do not lead to the formation of stable trajectories.
@article{SM_2005_196_4_a4,
     author = {E. A. Sataev},
     title = {Non-existence of stable trajectories in non-autonomous perturbations of systems of {Lorenz} type},
     journal = {Sbornik. Mathematics},
     pages = {561--594},
     year = {2005},
     volume = {196},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2005_196_4_a4/}
}
TY  - JOUR
AU  - E. A. Sataev
TI  - Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type
JO  - Sbornik. Mathematics
PY  - 2005
SP  - 561
EP  - 594
VL  - 196
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_2005_196_4_a4/
LA  - en
ID  - SM_2005_196_4_a4
ER  - 
%0 Journal Article
%A E. A. Sataev
%T Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type
%J Sbornik. Mathematics
%D 2005
%P 561-594
%V 196
%N 4
%U http://geodesic.mathdoc.fr/item/SM_2005_196_4_a4/
%G en
%F SM_2005_196_4_a4
E. A. Sataev. Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type. Sbornik. Mathematics, Tome 196 (2005) no. 4, pp. 561-594. http://geodesic.mathdoc.fr/item/SM_2005_196_4_a4/

[1] Saltzman B., “Finite amplitude free convectiions as an initial value problem. I”, J. Atmospheric Sci., 19:2 (1962), 329–341 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[2] Lorenz E. N., “Deterministic nonperiodic flow”, J. Atmospheric Sci., 20:2 (1963), 130–141 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[3] McLaughlin B., Martin P. C., “Transition to turbulence of statically stressed fluids”, Phys. Rev. Lett., 33:1 (1974), 1189–1192 | DOI

[4] Yorke J. A., Yorke E. D., “Metastable chaos: The transition to sustained chaotic oscillations in a model of Lorenz”, J. Statist. Phys., 21 (1979), 263–277 | DOI | MR

[5] Guckenheimer J., Williams R. F.,, “Structural stability of Lorenz attractors”, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59–72 | DOI | MR | Zbl

[6] Rand D., “The topological classification of Lorenz attractors”, Math. Proc. Cambridge Philos. Soc., 83:3 (1978), 451–460 | DOI | MR | Zbl

[7] Afraimovich V. S., Bykov V. V., Shilnikov L. P., “O prityagivayuschikh negrubykh predelnykh mnozhestvakh tipa attraktora Lorentsa”, Trudy MMO, 44, 1982, 150–212 | MR | Zbl

[8] Šlačkov S. V., “Pseudo-orbit tracing property and structural stability of expanding maps of the interval”, Ergodic Theory Dynam. Systems, 12 (1992), 573–587 | MR | Zbl

[9] Sparrow C., The Lorenz attractor: Bifurcations, chaos and strange attractors, Springer-Verlag, New York, 1982 | MR | Zbl

[10] Bunimovich L. A., Sinai Ya. G., “Stokhastichnost attraktora v modeli Lorentsa”, Nelineinye volny, ed. A. V. Gaponov-Grekhov, Nauka, M., 1979, 212–226

[11] Sataev E. A., “Invariantnye mery dlya giperbolicheskikh otobrazhenii s osobennostyami”, UMN, 47:1 (1992), 147–202 | MR | Zbl

[12] Sataev E. A., “Gibbsovskie mery dlya odnomernykh attraktorov giperbolicheskikh otobrazhenii s osobennostyami”, Izv. RAN. Ser. matem., 56:6 (1992), 1328–1344 | Zbl

[13] Pesin Ya. B., “Dynamical systems with generalised hyperbolic attractors: hyperbolic, ergodic and topological properties”, Ergodic Theory Dynam. Systems, 12 (1992), 123–151 | DOI | MR | Zbl

[14] Tucker W., “A rigorous ODE solver and Smale's 14th problem”, Found. Comput. Math., 2 (2002), 53–117 | MR | Zbl

[15] Rychlik M., “Lorenz attractor through Sil'nicov-type bifurcation. I”, Ergodic Theory Dynam. Systems, 10:4 (1990), 793–821 | DOI | MR | Zbl

[16] Robinson C., “Homoclinic bifurcation to a transitive attractor of Lorenz type”, Nonlinearity, 2:4 (1989), 495–518 | DOI | MR | Zbl

[17] Plykin R. V., Sataev E. A., Shlyachkov S. V., “Strannye attraktory”, Dinamicheskie sistemy, Itogi nauki i tekhniki. Sovr. problemy matem. Fundament. napravleniya, 9, VINITI, M., 1991, 72–144 | MR | Zbl

[18] Sternberg Sh., “On the structure of local homeomorphisms of Euclidean $n$-space. II”, Amer. J. Math., 80 (1958), 623–631 | DOI | MR | Zbl

[19] Chen K. T., “Equivalence and decomposition of vector fields about an elementary critical point”, Amer. J. Math., 85 (1963), 693–722 | DOI | MR | Zbl

[20] Kelly A., “The stable, center-stable, center, center-unstable, unstable manifolds”, J. Differential Equations, 3:4 (1967), 546–570 | DOI | MR

[21] Anosov D. V., “Geodezicheskie potoki na zamknutykh rimanovykh mnogoobraziyakh otritsatelnoi krivizny”, Trudy MIAN, 90, 1967, 3–210 | MR

[22] Gonchenko S. V., Turaev D. V., Shilnikov L. P., “O suschestvovanii oblastei Nyukhausa vblizi sistem s negruboi gomoklinicheskoi krivoi Puankare (mnogomernyi sluchai)”, Dokl. RAN, 329:4 (1993), 404–407 | MR | Zbl

[23] Turaev D. V., Shilnikov L. P., “Primer dikogo strannogo attraktora”, Matem. sb., 189:2 (1998), 137–160 | MR | Zbl