Towards scenarios of chaos appearance in three-dimensional maps

A. S. Gonchenko ; S. V. Gonchenko ; L. P. Shilnikov

Russian journal of nonlinear dynamics, Tome 8 (2012) no. 1, pp. 3-28.

Détail

Keywords: strange attractor, chaotic dynamics, spiral attractor, invariant curve, three-dimensional Hénon map.
Mots-clés : torus-chaos, homoclinic orbit

Bibliographie

[1] Shilnikov L. P., “Ob odnom sluchae suschestvovaniya schetnogo mnozhestva periodicheskikh dvizhenii”, Dokl. AN SSSR, 160:3 (1965), 558–561 | MR | Zbl

[2] Shilnikov L. P., “K voprosu o strukture rasshirennoi okrestnosti grubogo sostoyaniya ravnovesiya tipa sedlo–fokus”, Matem. sb., 81:1 (1970), 92–103 | MR | Zbl

[3] Shilnikov L. P., “Teoriya bifurkatsii i turbulentnost”, Metody kachestvennoi teorii differentsialnykh uravnenii, Mezhvuz. sb., ed. E. A. Leontovich-Andronova, GGU, Gorkii, 1986, 150–163

[4] 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

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

[6] Shilnikov L. P., “Teoriya bifurkatsii i model Lorentsa”, Bifurkatsiya rozhdeniya tsikla i ee prilozheniya, eds. Dzh. Marsden, M. Mak-Kraken, Mir, M., 1980, 317–335

[7] Shilnikov L. P., “Mathematical problems of nonlinear dynamics: A Tutorial”, J. Franklin Inst. B, 334:5–6 (1997), 793–864 | DOI | MR

[8] Afraimovich V. S., Shilnikov L. P., “Invariantnye tory, ikh razrushenie i stokhastichnost”, Metody kachestvennoi teorii differentsialnykh uravnenii, Mezhvuz. tematich. sb. nauch. tr., ed. E. A. Leontovich-Andronova, GGU, Gorkii, 1983, 3–26

[9] Benedicks M., Carleson L., “The dynamics of the Hénon map”, Ann. of Math. (2), 133 (1991), 73–169 | DOI | MR | Zbl

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

[11] Turaev D. V., Shilnikov L. P., “Psevdogiperbolichnost i zadacha o periodicheskikh vozmuscheniyakh attraktorov lorentsevskogo tipa”, Dokl. RAN, 418:1 (2008), 23–27 | Zbl

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

[13] Gonchenko S. V., Shilnikov L. P., Turaev D. V., “Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits”, Chaos, 6:1 (1996), 15–31 | DOI | MR | Zbl

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

[15] Gonchenko S. V., Turaev D. V., Shilnikov L. P., “O dinamicheskikh svoistvakh diffeomorfizmov s gomoklinicheskimi kasaniyami”, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 7 (2003), 92–118

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

[17] Aframovich V. S., Shilnikov L. P., “Strange attractors and quasiattractors”, Nonlinear Dynamics and Turbulence, eds. G. I. Barenblatt, G. Iooss, D. D. Joseph, Pitmen, Boston, 1983, 1–34 | MR

[18] Gonchenko S. V., Ovsyannikov I. I., Simó C., Turaev D. V., “Three-dimensional Hénon-like maps and wild Lorenz-like attractors”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3493–3508 | DOI | MR | Zbl

[19] Morozov A. D., Dragunov T. N., Vizualizatsiya i analiz invariantnykh mnozhestv dinamicheskikh sistem, NITs «Regulyarnaya i khaoticheskaya dinamika», Inst. kompyutern. issled., M.–Izhevsk, 2003, 304 pp.

[20] Morozov A. D., Dragunov T. N., Boykova S. A., Malysheva O. V., Invariant sets for Windows, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 37, World Sci., Singapore, 1999, 259 pp. | MR | Zbl

[21] Lomeli H. E., Meiss J. D., “Quadratic volume preserving maps”, Nonlinearity, 11 (1998), 557–574 | DOI | MR | Zbl

[22] Gonchenko S. V., Meiss J. D., Ovsyannikov I. I., “Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation”, Regul. Chaotic Dyn., 11:2 (2006), 191–212 | DOI | MR | Zbl

[23] Gonchenko S. V., Shilnikov L. P., Turaev D. V., “On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: 1”, Nonlinearity, 21 (2008), 923–972 | DOI | MR | Zbl

[24] Gonchenko S. V., Gonchenko V. S., Shilnikov L. P., “On homoclinic origin of Hénon-like maps”, Regul. Chaotic Dyn., 15:4–5 (2010), 462–481 | DOI | MR | Zbl

[25] Gonchenko S. V., Shilnikov L. P., Turaev D. V., “On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: 2”, Nonlinearity (to appear)

[26] Borisov A. V., Mamaev I. S., “Strannye attraktory v dinamike keltskikh kamnei”, UFN, 173:4 (2003), 407–418

[27] Borisov A. V., Mamaev I. S., “Strannye attraktory v dinamike keltskikh kamnei”, Negolonomnye dinamicheskie sistemy: Integriruemost, khaos, strannye attraktory, Sb. st., eds. A. V. Borisov, I. S. Mamaev, Inst. kompyutern. issled., M.–Izhevsk, 2002, 293–316

[28] Gonchenko S. V., Turaev D. V., Shilnikov L. P., “Ob oblastyakh Nyukhausa dvumernykh diffeomorfizmov, blizkikh k diffeomorfizmu s negrubym geteroklinicheskim konturom”, Tr. MIAN, 216 (1997), 76–125 | Zbl

[29] Lamb J. S. W., Stenkin O. V., “Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits”, Nonlinearity, 17:4 (2004), 1217–1244 | DOI | MR | Zbl

[30] Delshams A., Gonchenko S. V., Gonchenko V. S., Lazaro J. T., Stenkin O. V., “Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps”, Nonlinearity (to appear)

[31] Lukyanov V. I., Shilnikov L. P., “O nekotorykh bifurkatsiyakh dinamicheskikh sistem s gomoklinicheskimi strukturami”, Dokl. AN SSSR, 243:1 (1978), 26–29 | MR | Zbl

[32] Arnold V. I., Afraimovich V. S., Ilyashenko Yu. S., Shilnikov L. P., “Teoriya bifurkatsii”, Dinamicheskie sistemy — 5, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 5, ed. V. I. Arnold, VINITI, M., 1986, 9–218

[33] Bykov V. V., “O bifurkatsiyakh dinamicheskikh sistem, blizkikh k sistemam s separatrisnym konturom, soderzhaschim sedlo–fokus”, Metody kachestvennoi teorii differentsialnykh uravnenii, Mezhvuz. temat. sb. nauchn. tr., GGU, Gorkii, 1980, 44–72

[34] Bykov V. V., “The bifurcations of separatrix contours and chaos”, Phys. D, 62 (1993), 290–299 | DOI | MR | Zbl

[35] Afraimovich V. S., Shilnikov L. P., “O nekotorykh globalnykh bifurkatsiyakh, svyazannykh s ischeznoveniem nepodvizhnoi tochki tipa sedlo–uzel”, Dokl. AN SSSR, 219:6 (1974), 1281–1285

[36] Turaev D. V., Shilnikov L. P., “Bifurkatsii kvaziattraktorov tor–khaos”, Matem. mekhanizmy turbulentnosti, Kiev, 1986, 113–121

[37] Anischenko V. S., Slozhnye kolebaniya v prostykh sistemakh, Nauka, M., 1990, 312 pp.