On three types of dynamics and the notion of attractor

S. V. Gonchenko ; D. V. Turaev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Order and chaos in dynamical systems, Tome 297 (2017), pp. 133-157.

Résumé

We propose a theoretical framework for explaining the numerically discovered phenomenon of the attractor-repeller merger. We identify regimes observed in dynamical systems with attractors as defined in a paper by Ruelle and show that these attractors can be of three different types. The first two types correspond to the well-known types of chaotic behavior, conservative and dissipative, while the attractors of the third type, reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regimes. We prove that every elliptic orbit of a generic non-conservative time-reversible system is a reversible core. We also prove that a generic reversible system with an elliptic orbit is universal; i.e., it displays dynamics of maximum possible richness and complexity.

Bibliographie

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

[2] Anosov D. V., Bronshtein I. U., “Gladkie dinamicheskie sistemy. Gl. 3: Topologicheskaya dinamika”, Dinamicheskie sistemy – 1, Itogi nauki i tekhniki. Sovr. probl. matematiki. Fund. napr., 1, VINITI, M., 1985, 204–229

[3] Anosov D. V., Solodov V. V., “Dinamicheskie sistemy s giperbolicheskim povedeniem. Gl. 1: Giperbolicheskie mnozhestva”, Dinamicheskie sistemy – 9, Itogi nauki i tekhniki. Sovr. probl. matematiki. Fund. napr., 66, VINITI, M., 1991, 12–99 | MR | Zbl

[4] Arnold V. I., Kozlov V. V., Neishtadt A. I., “Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki”, Dinamicheskie sistemy – 3, Itogi nauki i tekhniki. Sovr. probl. matematiki. Fund. napr., 3, VINITI, M., 1985, 5–290 | MR | Zbl

[5] Bochner S., “Compact groups of differentiable transformations”, Ann. Math. Ser. 2, 46:3 (1945), 372–381 | DOI | MR | Zbl

[6] Bonatti C., Shinohara K., Volume hyperbolicity and wildness, E-print, 2015, arXiv: 1505.07901v1[math.DS]

[7] Conley C., Isolated invariant sets and the Morse index, Reg. Conf. Ser. Math., 38, Amer. Math. Soc., Providence, RI, 1978 | MR | Zbl

[8] Delshams A., Gonchenko S. V., Gonchenko M. S., Lázaro J. T., Mixed dynamics of two-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies, E-print, 2014, arXiv: 1412.1128[math.DS]

[9] Delshams A., Gonchenko S. V., Gonchenko V. S., Lázaro J. T., Sten'kin O., “Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps”, Nonlinearity, 26:1 (2013), 1–33 | DOI | MR | Zbl

[10] Fiedler B., Turaev D., “Coalescence of reversible homoclinic orbits causes elliptic resonance”, Int. J. Bifurcation Chaos Appl. Sci. Eng., 6:6 (1996), 1007–1027 | DOI | MR | Zbl

[11] Gavrilov N. K., Shilnikov L. P., “O trekhmernykh dinamicheskikh sistemakh, blizkikh k sistemam s negruboi gomoklinicheskoi krivoi. I”, Mat. sb., 88(130):4 (1972), 475–492 ; “II”, Мат. сб., 90(132):1 (1973), 139–156 | MR | Zbl | MR | Zbl

[12] Gonchenko A. S., Gonchenko S. V., Kazakov A. O., “Richness of chaotic dynamics in nonholonomic models of a Celtic stone”, Regul. Chaotic Dyn., 18:5 (2013), 521–538 | DOI | MR | Zbl

[13] Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Turaev D. V., “On the phenomenon of mixed dynamics in Pikovsky–Topaj system of coupled rotators”, Physica D, 350 (2017), 45–57 | DOI | MR

[14] Gonchenko S., “On mixed dynamics in reversible systems”, Abstr. Int. Conf. Diff. Eqns. Dyn. Syst. (Suzdal, July 2–7, 2010), 211–212

[15] Gonchenko S., “Mixed dynamics as a new form of dynamical chaos”, Abstr. 10th AIMS Int. Conf. Dyn. Syst. Diff. Eqns. Appl. (Madrid, July 10–14, 2014), 14

[16] Gonchenko S. V., “Reversible mixed dynamics: A concept and examples”, Discontin. Nonlinearity Complex., 5:4 (2016), 365–374 | DOI | Zbl

[17] Gonchenko C. V., Lemb I. S. V., Rios I., Turaev D., “Attraktory i repellery v okrestnosti ellipticheskikh tochek obratimykh sistem”, DAN, 454:4 (2014), 375–378 | Zbl

[18] Gonchenko S. V., Ovsyannikov I. I., “O bifurkatsiyakh trekhmernykh diffeomorfizmov s negrubym geteroklinicheskim konturom, soderzhaschim sedlo-fokusy”, Nelineinaya dinamika, 6:1 (2010), 61–77

[19] Gonchenko S. V., Ovsyannikov I. I., “On global bifurcations of three-dimensional diffeomorphisms leading to Lorenz-like attractors”, Math. Model. Nat. Phenom., 8:5 (2013), 71–83 | DOI | MR | Zbl

[20] Gonchenko S. V., Shilnikov L. P., Stenkin O. V., “On Newhouse regions with infinitely many stable and unstable invariant tori”, Progress in nonlinear science, Proc. Int. Conf. (Nizhni Novgorod, 2001), v. 1, Mathematical problems of nonlinear dynamics, Nizhni Novgorod, 2002, 80–102 | MR

[21] Gonchenko S. V., Shilnikov L. P., Turaev D. V., “On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors”, Regul. Chaotic Dyn., 14:1 (2009), 137–147 | DOI | MR | Zbl

[22] Gonchenko S. V., Stenkin O. V., Shilnikov L. P., “O suschestvovanii schetnogo mnozhestva ustoichivykh i neustoichivykh invariantnykh torov u sistem iz oblastei Nyukhausa s geteroklinicheskimi kasaniyami”, Nelineinaya dinamika, 2:1 (2006), 3–25

[23] 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 | MR | Zbl

[24] Gonchenko S. V., Turaev D. V., Shilnikov L. P., “Gomoklinicheskie kasaniya proizvolnogo poryadka v konservativnykh dvumernykh otobrazheniyakh”, DAN, 407:3 (2006), 299–303 | MR | Zbl

[25] Gonchenko S., Turaev D., Shilnikov L., “Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps”, Nonlinearity, 20:2 (2007), 241–275 | DOI | MR | Zbl

[26] Hurley M., “Attractors: Persistence, and density of their basins”, Trans. Amer. Math. Soc., 269:1 (1982), 247–271 | DOI | MR | Zbl

[27] Kazakov A. O., “Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane”, Regul. Chaotic Dyn., 18:5 (2013), 508–520 | DOI | MR | Zbl

[28] Kazakov A., “On chaotic dynamics in the Suslov problem”, Dynamics, bifurcations and chaos 2015, DBC II: Ext. Abstr. Int. Conf.–Sch. (Nizhni Novgorod, July 20–24, 2015), Lobachevsky State Univ., Nizhni Novgorod, 2015, 21–30

[29] Lamb J. S. W., Quispel G. R. W., “Reversing $k$-symmetries in dynamical systems”, Physica D, 73:4 (1994), 277–304 | DOI | MR | Zbl

[30] 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

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

[32] Newhouse S. E., “Quasi-elliptic periodic points in conservative dynamical systems”, Amer. J. Math., 99 (1977), 1061–1087 | DOI | MR | Zbl

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

[34] Ruelle D., “Small random perturbations of dynamical systems and the definition of attractors”, Commun. Math. Phys., 82 (1981), 137–151 | DOI | MR | Zbl

[35] Sevryuk M. B., Reversible systems, Lect. Notes Math., 1211, Springer, Berlin, 1986 | DOI | MR | Zbl

[36] Topaj D., Pikovsky A., “Reversibility vs. synchronization in oscillator lattices”, Physica D, 170:2 (2002), 118–130 | DOI | MR | Zbl

[37] Treschev D., “Closures of asymptotic curves in a two-dimensional symplectic map”, J. Dyn. Control Syst., 4:3 (1998), 305–314 | DOI | MR | Zbl

[38] Treschev D. V., Vvedenie v teoriyu vozmuschenii gamiltonovykh sistem, B-ka studenta-matematika, 6, Fazis, M., 1998 | MR

[39] Turaev D., “On dimension of nonlocal bifurcational problems”, Int. J. Bifurcation Chaos Appl. Sci. Eng., 6:5 (1996), 919–948 | DOI | MR | Zbl

[40] Turaev D., “Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps”, Nonlinearity, 16:1 (2003), 123–135 | DOI | MR | Zbl

[41] Turaev D., “Richness of chaos in the absolute Newhouse domain”, Proc. Int. Congr. Math. (Hyderabad, India, 2010), v. 3, Invited lectures, World Scientific, Hackensack, NJ, 2011, 1804–1815 | MR | Zbl

[42] Turaev D., “Maps close to identity and universal maps in the Newhouse domain”, Commun. Math. Phys., 335:3 (2015), 1235–1277 | DOI | MR | Zbl