An example of a wild strange attractor

D. V. Turaev   ; L. P. Shilnikov

Sbornik. Mathematics, Tome 189 (1998) no. 2, pp. 291-314

Résumé

It is proved that in the space of $C^r$-smooth ($r\geqslant 4$) flows in $\mathbb R^n$ ($n\geqslant 4$) there exist regions filled by systems that each have an attractor (here: a completely stable chain-transitive closed invariant set) containing a non-trivial basic hyperbolic set together with its unstable manifold, which has points of non-transversal intersection with the stable manifold. A construction is given for such a wild attractor containing an equilibrium state of saddle-focus type.

Bibliographie

[1] Newhouse S. E., “The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms”, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 101–151 | DOI | MR

[2] Newhouse S. E., “Diffeomorphisms with infinitely many sinks”, Topology, 13 (1974), 9–18 | DOI | MR | Zbl

[3] Gonchenko S. V., Shil'nikov L. P., Turaev D. V., “On models with non-rough Poincare homoclinic curves”, Phys. D, 62:1–4 (1993), 1–14 | DOI | MR | Zbl

[4] Gonchenko S. V., Turaev D. V., Shilnikov L. P., “Dinamicheskie yavleniya v mnogomernykh sistemakh s negruboi gomoklinicheskoi krivoi Puankare”, Dokl. AN, 330:2 (1993), 144–147 | MR | Zbl

[5] Gonchenko S. V., Shil'nikov L. P., Turaev D. V., “Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits”, Interdisc. J. Nonlinear Sci. CHAOS, 6:1 (1996), 1–17 | DOI | MR

[6] Afraimovich V. S., Bykov V. V., Shilnikov L. P., “O vozniknovenii i strukture attraktorov Lorentsa”, Dokl. AN SSSR, 234:2 (1977), 336–339 | MR | Zbl

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

[8] Ovsyannikov I. M., Shilnikov L. P., “O sistemakh s gomoklinicheskoi krivoi sedlo–fokusa”, Matem. sb., 130 (172):4 (1986), 552–570 | MR

[9] Hirsh M. W., Pugh C. C., Shub M., Invariant manifolds, Lect. Notes in Math., 583, 1977

[10] Guckenheimer J., Williams R.-F., “Structural stability of Lorenz attractors”, Astérisque, 50 (1979), 307–320 | MR

[11] Anosov D. V., Geodezicheskie potoki na zamknutykh rimanovykh mnogoobraziyakh otritsatelnoi krivizny, Tr. MIAN, 90, Nauka, M., 1967 | MR

[12] Pesin Ya. B., “Kharakteristicheskie pokazateli Lyapunova i gladkaya ergodicheskaya teoriya”, UMN, 32:4 (1977), 55–111 | MR

[13] Anosov D. V., Aranson S. Kh., Bronshtein I. U., Grines V. Z., “Gladkie dinamicheskie sistemy, II”, Itogi nauki i tekhn. Sovr. probl. matem. Fundam. napr., 1, VINITI, M., 1985, 151–242 | MR

[14] Auslender J., “Generalized recurrence in dynamical systems”, Contributions to differential equations, V. 3, John Wiley and Sons, Inc., New York, 1964, 65–74 | MR

[15] Shilnikov L. P., “K voprosu o strukture rasshirennoi okrestnosti grubogo sostoyaniya ravnovesiya tipa sedlo–fokus”, Matem. sb., 81 (123):1 (1970), 92–113

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

[17] Palis J., Viana M., “High-dimension diffeomorphisms displaying infinitely many periodic attractors”, Ann. of Math., 140 (1994), 207–250 | DOI | MR | Zbl

[18] Hayashi S., On the solution of $C^1$-stability conjecture for flow, Preprint

[19] Feroe J. A., “Homoclinic orbits in a parametrized saddle-focus system”, Phys. D, 62:1–4 (1993), 254–262 | DOI | MR | Zbl

[20] Shilnikov L. P., “Ob odnoi zadache Puankare–Birkgofa”, Matem. sb., 74(116):3 (1967), 378–397 | MR | Zbl

[21] Turaev D. V., “On dimension of non-local bifurcational problems”, Int J. Bifur. Chaos. Appl. Sci. Engrg., 6:5 (1996), 919–948 | DOI | MR | Zbl

[22] Sandstede B., Center manifolds for homoclinic solutions, Preprint, 1994 | MR