Geodesic
On the classification of homoclinic attractors of three-dimensional flows
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 4, pp. 443-459.

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

For three-dimensional dynamical systems with continuous time (flows), a classification of strange homoclinic attractors containing an unique saddle equilibrium state is constructed. The structure and properties of such attractors are determined by the triple of eigenvalues of the equilibrium state. The method of a saddle charts is used for the classification of homoclinic attractors. The essence of this method is in the construction of an extended bifurcation diagram for a wide class of three-dimensional flows (whose linearization matrix is written in the Frobenius form). Regions corresponding to different configurations of eigenvalues are marked in this extended bifurcation diagram. In the space of parameters defining the linear part of the considered class of three-dimensional flows bifurcation surfaces bounding 7 regions are constructed. One region corresponds to the stability of the equilibrium states while other 6 regions correspond to various homoclinic attractors of the following types: Shilnikov attractor, 2 types of spiral figure-eight attractors, Lorenz-like attractor, saddle Shilnikov attractor and attractor of Lyubimov-Zaks-Rovella. The paper also discusses questions related to the pseudohyperbolicity of homoclinic attractors of three-dimensional flows. It is proved that only homoclinic attractors of two types can be pseudohyperbolic: Lorenz-like attractors containing a saddle equilibrium with a two-dimensional stable manifold whose saddle value is positive and saddle Shilnikov attractors containing a saddle equilibrium state with a two-dimensional unstable manifold.
Keywords: strange attractor, homoclinic trajectory, pseudohyperbolicity, Lorenz attractor
Mots-clés : spiral chaos. 
@article{SVMO_2019_21_4_a3,
     author = {A. O. Kazakov and E. Yu. Karatetskaya and A. D. Kozlov and K. A. Saphonov},
     title = {On the classification of homoclinic attractors of three-dimensional flows},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {443--459},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a3/}
}
TY  - JOUR
AU  - A. O. Kazakov
AU  - E. Yu. Karatetskaya
AU  - A. D. Kozlov
AU  - K. A. Saphonov
TI  - On the classification of homoclinic attractors of three-dimensional flows
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2019
SP  - 443
EP  - 459
VL  - 21
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a3/
LA  - ru
ID  - SVMO_2019_21_4_a3
ER  - 
%0 Journal Article
%A A. O. Kazakov
%A E. Yu. Karatetskaya
%A A. D. Kozlov
%A K. A. Saphonov
%T On the classification of homoclinic attractors of three-dimensional flows
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2019
%P 443-459
%V 21
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a3/
%G ru
%F SVMO_2019_21_4_a3
A. O. Kazakov; E. Yu. Karatetskaya; A. D. Kozlov; K. A. Saphonov. On the classification of homoclinic attractors of three-dimensional flows. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 4, pp. 443-459. http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a3/

[1] L.P. Shilnikov, “On a case of existence of a countable set of periodic motions”, Dokl. Akad. Nauk SSSR, 160:3 (1965), 558-561 | Zbl

[2] L.O. Chua, M. Komuro, T. Matsumoto, “The double scroll family”, Circuits and Systems, IEEE Transactions on., 33:11 (1986), 1072-1118 | DOI | MR | Zbl

[3] L.P. Shilnikov, “The theory of bifurcations and turbulence”, Methods of qualitative theory of differential equations, Gorkiy, 1986, 150-163

[4] L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, L.O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, v. 2, World Scientific Publishing Co Pte Ltd, Singapore, Singapore, 2001, 592 pp. | MR | Zbl

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

[6] A.D. Kozlov, “Examples of strange attractors in three-dimentional nonoriented maps”, J. SVMO, 19:2 (2017), 62–75 | Zbl

[7] D.V. Turaev, L.P. Shilnikov, “An example of a wild strange attractor”, Sb. Math., 189 (1998), 291-314 | DOI | DOI | MR | Zbl

[8] S.V. Gonchenko, A.O. Kazakov, D. Turaev, Wild pseudohyperbolic attractors in a four-dimensional Lorenz system, 2018, arXiv: 1809.07250

[9] A.S. Gonchenko, S.V. Gonchenko, A.O. Kazakov, A.D. Kozlov, “Elements of Contemporary Theory of Dynamical Chaos: A Tutorial. Part I Pseudohyperbolic Attractors”, International Journal of Bifurcation and Chaos, 28:11 (2018), 291–314 | DOI | MR

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

[11] I.M. Ovsyannikov, L.P. Shilnikov, “On systems with homoclinic curves of saddle-focus type”, Mat. Sb., 130(172):4(8) (1986), 552-570

[12] P. Coullet, C. Tresser, A. Arneodo, “Transition to stochasticity for a class of forced oscillators”, Physics letters A., 72:4-5 (1979), 268-270 | DOI | MR

[13] F.R. Gantmacher, The theory of matrices, Chelsea Pub. Co., New York, 1959 | MR

[14] L.P. Shilnikov, “Some instances of generation of periodic motions in n-space”, Dokl. Akad. Nauk SSSR, 143:2 (1962), 289–292 | Zbl

[15] D.V. Lyubimov, M.A. Zaks, “Two mechanisms of the transition to chaos in finite-dimensional models of convection”, Physica D: Nonlinear Phenomena, 9:1-2 (1983), 52-64 | DOI | MR | Zbl

[16] A. Rovella, “The dynamics of perturbations of the contracting Lorenz attractor”, Boletim da Sociedade Brasileira de Matematica-Bulletin/Brazilian Mathematical Society, 24:2 (1993), 233-259 | DOI | MR | Zbl

[17] A.O. Kazakov , A.D. Kozlov, “The asymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems”, Zhurnal SVMO, 20:2 (2018), 187–198

[18] P. Coullet, C. Tresser, A. Arneodo, “Possible new strange attractors with spiral structure”, Communications in Mathematical Physics, 79:4 (1981), 573-579 | DOI | MR | Zbl

[19] A.L. Shilnikov, L.P. Shilnikov, “On the nonsymmetrical Lorenz model”, International Journal of Bifurcation and Chaos, 1:4 (1991), 773-776 | DOI | MR | Zbl

[20] S.E. Newhouse, “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