On discrete Lorenz attractors of various types

A. S. Gonchenko

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 32 (2024) no. 6, pp. 832-857.

Résumé

The purpose of this work is to develop the theory of discrete attractors of Lorenz type in the case of three-dimensional maps. In this case, special attention will be paid to standard discrete Lorenz attractors, as well as discrete Lorenz attractors with axial symmetry (i.e. with symmetry $x \to -x$, $y\to -y$, $z \to -z$ characteristic of flows with the Lorenz attractors). The main results of the work are related to the construction of elements of classification of such attractors. For various types of discrete Lorenz attractors, we will describe their basic geometric and dynamical properties, and also present the main phenomenological bifurcation scenarios in which they arise. In the work we also consider specific examples of discrete Lorenz attractors of various types in three-dimensional quadratic maps such as three-dimensional Henon maps and quadratic maps with axial symmetry and constant Jacobian. For the latter, their normal forms will be constructed — universal maps, to which any map from a given class can be reduced by means of linear coordinate transformations.

Détail

Keywords: Lorenz attractor, bifurcation, three-dimensional Henon map, global symmetry, bifurcation scenario

Bibliographie

[1] Lorenz E. N., “Deterministic nonperiodic flow”, Journal of the Atmospheric Sciences, 20:2 (1963), 130–141 | 2.0.co;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[2] Guckenheimer J., “A strange, strange attractor”, The Hopf Bifurcation Theorem and its Applications, Applied Mathematical Sciences, 19, Springer, New York, 1976, 368–381 | DOI | MR

[3] Williams R. F., “The structure of Lorenz attractor”, Lecture Notes in Math., 615 (1977), 94–112 | DOI | MR

[4] Afraimovich V. S., Bykov V. V., Shilnikov L. P., “O vozniknovenii i strukture attraktora 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”, Trudy MMO, 44 (1982), 150–212 | Zbl

[6] Kaplan J. L., Yorke J. A., “Predturbulence: A regime observed in a fluid flow model of Lorenz”, Comm. Math. Phys., 67:2 (1979), 93–108 | DOI | MR | Zbl

[7] Shilnikov L. P., “Teoriya bifurkatsii i kvazigiperbolicheskie attraktory”, Uspekhi Mat. Nauk, 36:4 (1981), 240

[8] Shilnikov L. P., “Teoriya bifurkatsii i model Lorentsa”, Bifurkatsiya rozhdeniya tsikla i ee prilozheniya, eds. Marsden Zh., MakKraken M., Mir, M., 1980, 317–336

[9] Bunimovich L. A., Sinai Ya. G., “Stokhastichnost attraktora v modeli Lorentsa”, Nelineinye volny, Nauka, M., 1979, 212–260

[10] Bunimovich L., “Statistical properties of Lorenz attractors”, Nonlinear Dynamics and Turbulence, ed. G.I. Barenblatt, Pitman, Boston, 1983, 1-22 | MR

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

[12] Malkin M. I., “O topologicheskoi sopryazhennosti razryvnykh otobrazhenii intervala”, UMZh, 32:5 (1980), 610–616 | MR

[13] Morales C. A., Pacifico M. J., Pujals E. R., “On C1 robust singular transitive sets for three-dimensional flows”, C. R. Acad. Sci. Ser. I Math., 326 (1998), 81–86 | MR | Zbl

[14] Sataev E. A., “Nekotorye svoistva singulyarnykh giperbolicheskikh attraktorov”, Mat. sb, 200:1 (2009), 35–76 | DOI | MR | Zbl

[15] Turaev D. V., Shilnikov L. P., “Primer dikogo strannogo attraktora”, Mat. sb, 189 (1998), 291–314 | DOI | MR | Zbl

[16] Turaev D. V., Shilnikov L. P., “Psevdogiperbolichnost i zadacha o periodicheskom vozmuschenii attraktorov lorentsevskogo tipa”, Doklady RAN, 77b:1b (2008), 17–21 | MR | Zbl

[17] Gonchenko S., Ovsyannikov I., Simo C., Turaev D., “Three-dimensional Henon-like maps and wild Lorenz-like attractors”, Int. J. of Bifurcation and Chaos, 15:11 (2005), 3493–3508 | DOI | MR | Zbl

[18] Gonchenko A. S., Gonchenko S. V., Shilnikov L. P., “K voprosu o stsenariyakh vozniknoveniya khaosa v trekhmernykh otobrazheniyakh”, Nelineinaya Dinamika, 8 (2012), 3-28

[19] Gonchenko S. V., Gonchenko A. S., Ovsyannikov I. I., Turaev D. V., “Examples of Lorenz-like Attractors in Henon-like Maps”, Math. Model. Nat. Phenom, 8:5 (2013), 48–70 | DOI | MR | Zbl

[20] Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Turaev D., “Simple scenarios of onset of chaos in three-dimensional maps”, Int. J. Bif. and Chaos, 24:8 (2014), 25 | DOI | MR | Zbl

[21] Gonchenko A. S., Gonchenko S. V., “Variety of strange pseudohyperbolic attractors in three-dimensional generalized Henon maps”, Physica D, 337(4) (2016), 43–57 | DOI | MR | Zbl

[22] Hénon M., “A two-dimensional mapping with a strange attractor”, The Theory of Chaotic Attractors, eds. Hunt B.R, Li T.Y, Kennedy J.A, Nusse H.E., Springer, New York, 1976, 94–102 | DOI | MR

[23] Gonchenko S., Gonchenko A., “On discrete Lorenz-like attractors in three-dimensional maps with axialsymmetry”, Chaos, 33:12 (2023), 123104 | DOI | MR | Zbl

[24] Gonchenko S., Gonchenko A., Kazakov A., and Samylina E., “On discrete Lorenz-like attractors”, Chaos, 31:2 (2021), 023117 | DOI | MR | Zbl

[25] Gonchenko S., Karatetskaia E., Kazakov A., Kruglov V., “Conjoined Lorenz twins — a new pseudohyperbolicattractor in three-dimensional maps and flows”, Chaos, 32 (2022), 121107 | DOI | MR | Zbl

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

[27] Gonchenko S. V., Turaev D. V., Shilnikov L. P., “Dinamicheskie yavleniya v mnogomernykh sistemakh s negruboi gomoklinicheskoi krivoi Puankare”, Dokl. Ros. Akad. Nauk, 330:2 (1993), 144-147 | Zbl

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

[29] Gonchenko S. V., Shilnikov L. P., Turaev D. V., “On dynamical properties ofmultidimensional diffeomorphisms from Newhouse regions”, Nonlinearity, 21:5 (2008), 923–972 | DOI | MR | Zbl

[30] Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Kozlov A. D., “Elements of contemporary theory of dynamical chaos: A tutorial. Part I. Pseudohyperbolic attractors”, Int. J. Bifurcation and Chaos, 28:11 (2018), 1830036 | DOI | MR | Zbl

[31] Gonchenko S. V., Kazakov A. O., Turaev D., “Wild pseudohyperbolic attractors in a four-dimensional Lorenz system”, Nonlinearity, 34:4 (2021), 2018–2047 | DOI | MR | Zbl

[32] Shilnikov L. P., “Teoriya bifurkatsii i turbulentnost”, Metody kachestvennoi teorii differentsialnykh uravnenii, Gorkii, 1986, 150-163

[33] Gonchenko S. V., Gonchenko A. S., Kazakov A. O., “Richness of chaotic dynamics in nonholonomic models of a Celtic stone”, Regular and Chaotic Dynamics, 15:5 (2013), 521-538 | DOI | MR

[34] Gonchenko A. S., Gonchenko S. V., Turaev D., “Doubling of invariant curves and chaos inthree-dimensional diffeomorphisms”, Chaos, 31 (2021), 113130 | DOI | MR | Zbl

[35] Gonchenko A. S., Samylina E. A., “Ob oblasti suschestvovaniya diskretnogo attraktora Lorentsa v negolonomnoi modeli keltskogo kamnya”, Izv. vuzov. Radiofizika, 62:5 (2019)

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

[37] Gonchenko S. V., Turaev D. V., Shilnikov L. P., “O suschestvovanii oblastei Nyukhausa v okrestnosti sistem s negruboi gomoklinicheskoi krivoi Puankare (mnogomernyi sluchai)”, Dokl. Ros. Akad. Nauk, 329 (1993) | Zbl

[38] Gonchenko S. V., Shilnikov L. P., “Ob invariantakh $\Omega$-sopryazhennosti diffeomorfizmov s negruboi gomoklinicheskoi traektoriei”, Ukr. mat. zhurnal, 42:2 (1990), 153-159 | MR | Zbl

[39] Gonchenko S. V., Turaev D. V., Shilnikov L. P., “Gomoklinicheskie kasaniya proizvolnogo poryadka v oblastyakh Nyukhausa”, Itogi nauki i tekhniki, ser.Sovremennaya matematika i ee prilozheniya; tematicheskie obzory, 67 (1999), 69-128 | Zbl

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

[41] Shimizu T., Morioka N., “On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model”, Phys. Lett. A, 76 (1980), 201–204 | DOI | MR

[42] Shilnikov A. L., Shilnikov L. P., Turaev D. V., “Normal forms andLorenz attractors”, Int. J. of Bifurcation and chaos, 3 (1993), 1123–1139 | DOI | MR | Zbl

[43] Shilnikov A. L., “On bifurcations of the Lorenz attractor in the Shimuizu-Morioka model”, Physica D, 62 (1993), 338–346 | DOI | MR | Zbl