Geodesic
The asymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 187-198.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper a new method of constructing of three-dimensional flow systems with different chaotic attractors is presented. Using this method, an example of three-dimensional system possessing an asymmetric Lorenz attractor is obtained. Unlike the classical Lorenz attractor, the observed attractor does not have symmetry. However, the discovered asymmetric attractor, as well as classical one, belongs to a class of «true» chaotic, or, more precise, pseudohyperbolic attractors; the theory of such attractors was developed by D. Turaev and L.P. Shilnikov. Any trajectory of a pseudohyperbolic attractor has a positive Lyapunov exponent and this property holds for attractors of close systems. In this case, in contrast to hyperbolic attractors, pseudohyperbolic ones admit homoclinic tangencies, but bifurcations of such tangencies do not lead to generation of stable periodic orbits. In order to find the non-symmetric Lorenz attractor we applied the method of «saddle chart». Using diagrams of maximal Lyapunov exponent, we show that there are no stability windows in the neighborhood of the observed attractor. In addition, we verify the pseudohyperbolicity for the non-symmetric Lorenz attractor using the LMP-method developed quite recently by Gonchenko, Kazakov and Turaev.
Keywords: strange attractor, pseudohyperbolicity, Lorenz attractor, Lyapunov exponents.
Mots-clés : homoclinic orbit 
@article{SVMO_2018_20_2_a4,
     author = {A. O. Kazakov and A. D. Kozlov},
     title = {The asymmetric {Lorenz} attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {187--198},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a4/}
}
TY  - JOUR
AU  - A. O. Kazakov
AU  - A. D. Kozlov
TI  - The asymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2018
SP  - 187
EP  - 198
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a4/
LA  - ru
ID  - SVMO_2018_20_2_a4
ER  - 
%0 Journal Article
%A A. O. Kazakov
%A A. D. Kozlov
%T The asymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2018
%P 187-198
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a4/
%G ru
%F SVMO_2018_20_2_a4
A. O. Kazakov; A. D. Kozlov. The asymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 187-198. http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a4/

[1] Aframovich V. S., Shilnikov L. P., “Strange attractors and quasiattractors”, Nonlinear Dynamics and Turbulence, Pitmen, Boston, 1983 | MR

[2] D.V. Turaev, L.P. Shilnikov, “An example of a wild strange attractor”, Sb. Math., 189:2 (1998), 291–314 (In Russ.) | DOI | Zbl

[3] D.V. Turaev, L.P. Shilnikov, “Pseudo-hyperbolisity and the problem on periodic perturbations of Lorenz-like attractors”, Doklady Mathematics, 77:1 (2008), 17–21 (In Russ.) | Zbl

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

[5] A.S. Gonchenko, S.V. Gonchenko, L.P. Shilnikov, “Towards scenarios of chaos appearance in three-dimensional maps”, Rus. J. Nonlin. Dyn., 8:1 (2012), 3–28 (In Russ.)

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

[7] S.V. Gonchenko, A.O.Kazakov, D.Turaev, Wild spiral attractors in a four-dimensional Lorenz model, to appear

[8] A. L. Shilnikov, L. P. Shilnikov, “On the nonsymmetrical Lorenz model”, Int. J. Bifurcation Chaos, 1:4 (1991), 773–776 | DOI | MR | Zbl

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

[10] V. Z. Grines , Y. V. Zhuzhoma, O. V.Pochinka, “Rough diffeomorphisms with basic sets of codimension one”, Journal of Mathematical Sciences, 225 (2017), 195–219 | DOI | MR | Zbl

[11] S.P. Kuznetsov, Dynamical Chaos and hyperbolic attractors: from mathematics to physics, Institute of Computer Studying, M.-Izhevsk, 2013, 488 pp. (In Russ.)

[12] E. N. Lorenz, “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

[13] V.S. Afraimovich , V.V. Bykov, L.P. Shilnikov, “The origin and structure of the Lorenz attractor”, Sov. Phys. Dokl., 234:2 (1977), 336–339 (In Russ.) | MR

[14] W.Tucker, “The Lorenz attractor exists”, Comptes Rendus de l'Academie des Sciences-Series I-Mathematics, 328:12 (1999), 1197-1202 | MR | Zbl

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

[16] S.V. Gonchenko, A.S.Gonchenko, A.O.Kazakov, “Richness of chaotic dynamics in nonholonomic models of a Celtic stone”, Regular and Chaotic Dynamics, 8:5 (2013), 521–538 | DOI | MR

[17] L. Chua, M. Komuro, T. Matsumoto, “The double scroll family”, IEEE transactions on circuits and systems, 33:11 (1986), 1072–1118 | DOI | MR | Zbl

[18] L.P. Shilnikov, “The theory of bifurcations and turbulence”, Methods of qualitative theory of differential equations, Mezhvuz. sb., eds. E. A. Leontovich, GGU, Gorkiy, 1986, 150–163 (In Russ.)

[19] L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, L.O. Chua, Methods Of qualitative theory in nonlinear dynamics, World Sci, Singapore–New Jersey–London–Hong Kong, 2001

[20] D. Turaev, L. P. Shilnikov, “On bifurcations of the homoclinic «figure eight» for a saddle with a negative saddle value”, Sov. Math. Dokl., 34 (1987), 397–401 | MR