RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mat. Sb., 1998, Volume 189, Number 2, Pages 137–160 (Mi msb300)  

This article is cited in 73 scientific papers (total in 73 papers)

An example of a wild strange attractor

D. V. Turaeva, L. P. Shilnikovb

a Weierstrass Institute for Applied Analysis and Stochastics
b Research Institute for Applied Mathematics and Cybernetics, N. I. Lobachevski State University of Nizhnii Novgorod

Abstract: It is proved that in the space of $C^r$-smooth ($r\geqslant 4$) flows in $\mathbb R^n$ ($n\geqslant 4$) there exist regions filled by systems that each have an attractor (here: a completely stable chain-transitive closed invariant set) containing a non-trivial basic hyperbolic set together with its unstable manifold, which has points of non-transversal intersection with the stable manifold. A construction is given for such a wild attractor containing an equilibrium state of saddle-focus type.

DOI: https://doi.org/10.4213/sm300

Full text: PDF file (384 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 1998, 189:2, 291–314

Bibliographic databases:

UDC: 517.95
MSC: Primary 58F14; Secondary 58F13, 58F15, 58F25, 58F14, 58F12, 58F10, 34C37, 3
Received: 20.01.1997

Citation: D. V. Turaev, L. P. Shilnikov, “An example of a wild strange attractor”, Mat. Sb., 189:2 (1998), 137–160; Sb. Math., 189:2 (1998), 291–314

Citation in format AMSBIB
\Bibitem{TurShi98}
\by D.~V.~Turaev, L.~P.~Shilnikov
\paper An example of a wild strange attractor
\jour Mat. Sb.
\yr 1998
\vol 189
\issue 2
\pages 137--160
\mathnet{http://mi.mathnet.ru/msb300}
\crossref{https://doi.org/10.4213/sm300}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1622321}
\zmath{https://zbmath.org/?q=an:0927.37017}
\elib{https://elibrary.ru/item.asp?id=13278608}
\transl
\jour Sb. Math.
\yr 1998
\vol 189
\issue 2
\pages 291--314
\crossref{https://doi.org/10.1070/sm1998v189n02ABEH000300}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000073979600013}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0032220873}


Linking options:
  • http://mi.mathnet.ru/eng/msb300
  • https://doi.org/10.4213/sm300
  • http://mi.mathnet.ru/eng/msb/v189/i2/p137

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Gonchenko A.S., Samylina E.A., “On the Region of Existence of a Discrete Lorenz Attractor in the Nonholonomic Model of a Celtic Stone”, Radiophys. Quantum Electron.  crossref  isi
    2. Carballo, CM, “Maximal transitive sets with singularities for generic C-1 vector fields”, Boletim da Sociedade Brasileira de Matematica, 31:3 (2000), 287  crossref  mathscinet  zmath  isi
    3. Turaev D., Zelik S., “Homoclinic bifurcations and dimension of attractors for damped nonlinear hyperbolic equations”, Nonlinearity, 16:6 (2003), 2163–2198  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    4. S. V. Gonchenko, V. S. Gonchenko, “On Bifurcations of Birth of Closed Invariant Curves in the Case of Two-Dimensional Diffeomorphisms with Homoclinic Tangencies”, Proc. Steklov Inst. Math., 244 (2004), 80–105  mathnet  mathscinet  zmath
    5. N E Klinshpont, E A Sataev, R V Plykin, “Geometrical and dynamical properties of Lorenz type system”, J. Phys. Conf. Ser., 23 (2005), 96–104  crossref  adsnasa  isi  elib  scopus  scopus  scopus
    6. Gonchenko, SV, “Three-dimensional Henon-like maps and wild Lorenz-like attractors”, International Journal of Bifurcation and Chaos, 15:11 (2005), 3493  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    7. Morales, CA, “Expanding Lorenz attractors through resonant double homoclinic loops”, SIAM Journal on Mathematical Analysis, 36:6 (2005), 1836  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    8. E. A. Sataev, “Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type”, Sb. Math., 196:4 (2005), 561–594  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. Gonchenko, SV, “Chaotic dynamics of three-dimensional Henon maps that originate from a homoclinic bifurcation”, Regular & Chaotic Dynamics, 11:2 (2006), 191  mathnet  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    10. Sergey Gonchenko, Dmitry Turaev, Leonid Shilnikov, “Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps”, Nonlinearity, 20:2 (2007), 241  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    11. Gonchenko, SV, “Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized henon maps”, Regular & Chaotic Dynamics, 12:3 (2007), 233  mathnet  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    12. S V Gonchenko, L P Shilnikov, D V Turaev, “On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I”, Nonlinearity, 21:5 (2008), 923  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    13. Metzger, R, “Sectional-hyperbolic systems”, Ergodic Theory and Dynamical Systems, 28 (2008), 1587  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    14. Morales, CA, “A singular-hyperbolic closing Lemma”, Michigan Mathematical Journal, 56:1 (2008), 29  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    15. Turaev, DV, “Pseudohyperbolicity and the problem on periodic perturbations of Lorenz-type attractors”, Dokl. Math/, 77:1 (2008), 17  crossref  mathscinet  zmath  isi  elib
    16. S. V. Gonchenko, L. P. Shilnikov, D. V. Turaev, “On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors”, Reg Chaot Dyn, 14:1 (2009), 137  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    17. E. A. Sataev, “Some properties of singular hyperbolic attractors”, Sb. Math., 200:1 (2009), 35–76  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. Elhadj Z., Sprott J.C., “Some explicit formulas of Lyapunov exponents for three-dimensional quadratic mappings”, Frontiers of Physics in China, 4:4 (2009), 549–555  crossref  mathscinet  adsnasa  isi  elib  scopus  scopus  scopus
    19. Elhadj Z., Sprott J.C., “Classification of three-dimensional quadratic diffeomorphisms with constant Jacobian”, Frontiers of Physics in China, 4:1 (2009), 111–121  crossref  adsnasa  isi  elib  scopus  scopus  scopus
    20. E. A. Sataev, “Invariant measures for singular hyperbolic attractors”, Sb. Math., 201:3 (2010), 419–470  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. Gonchenko S., Li M.-Ch., “Shilnikov's Cross-map method and hyperbolic dynamics of three-dimensional H,non-like maps”, Regular & Chaotic Dynamics, 15:2–3 (2010), 165–184  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    22. Bautista S., Morales C., “A sectional-Anosov connecting lemma”, Ergodic Theory and Dynamical Systems, 30:2 (2010), 339–359  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    23. S. V. Gonchenko, I. I. Ovsyannikov, “O bifurkatsiyakh trekhmernykh diffeomorfizmov s negrubym geteroklinicheskim konturom, soderzhaschim sedlo-fokusy”, Nelineinaya dinam., 6:1 (2010), 61–77  mathnet  elib
    24. E. A. Sataev, “Stokhasticheskie svoistva singulyarno giperbolicheskikh attraktorov”, Nelineinaya dinam., 6:1 (2010), 187–206  mathnet  elib
    25. Gonchenko S.V., Gonchenko V.S., Shilnikov L.P., “On a Homoclinic Origin of Henon-like Maps”, Regular & Chaotic Dynamics, 15:4–5 (2010), 462–481  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    26. Araujo V., Castro A., Pacifico M.J., Pinheiro V., “Multidimensional Rovella-like attractors”, J Differential Equations, 251:11 (2011), 3163–3201  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    27. Drubi F., Ibanez S., Angel Rodriguez J., “Hopf-pitchfork singularities in coupled systems”, Physica D-Nonlinear Phenomena, 240:9–10 (2011), 825–840  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    28. Arbieto A., Morales C., Senos L., “On the sensitivity of sectional-Anosov flows”, Math Z, 270:1–2 (2012), 545–557  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    29. A. S. Gonchenko, S. V. Gonchenko, L. P. Shilnikov, “K voprosu o stsenariyakh vozniknoveniya khaosa u trekhmernykh otobrazhenii”, Nelineinaya dinam., 8:1 (2012), 3–28  mathnet
    30. ROBERTO BARRIO, ANDREY SHILNIKOV, LEONID SHILNIKOV, “KNEADINGS, SYMBOLIC DYNAMICS AND PAINTING Lorenz CHAOS”, Int. J. Bifurcation Chaos, 22:04 (2012), 1230016  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    31. Sergey Kryzhevich, “Dynamics near nonhyperbolic fixed points or nontransverse homoclinic points”, Mathematics and Computers in Simulation, 2012  crossref  mathscinet  isi  scopus  scopus  scopus
    32. A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov, “O nekotorykh novykh aspektakh khaoticheskoi dinamiki «keltskogo kamnya»”, Nelineinaya dinam., 8:3 (2012), 507–518  mathnet
    33. S V Gonchenko, C Simó, A Vieiro, “Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight”, Nonlinearity, 26:3 (2013), 621  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    34. A. S. Gonchenko, S. V. Gonchenko, “O suschestvovanii attraktorov lorentsevskogo tipa v negolonomnoi modeli «keltskogo kamnya»”, Nelineinaya dinam., 9:1 (2013), 77–89  mathnet
    35. Stefanie Hittmeyer, Bernd Krauskopf, H.M.. Osinga, “Interacting Global Invariant Sets in a Planar Map Model of Wild Chaos”, SIAM J. Appl. Dyn. Syst, 12:3 (2013), 1280  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    36. S.V. Gonchenko, I.I. Ovsyannikov, L. Lerman, D. Turaev, V. Vougalter, M. Zaks, “On Global Bifurcations of Three-dimensional Diffeomorphisms Leading to Lorenz-like Attractors”, Math. Model. Nat. Phenom, 8:5 (2013), 71  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    37. S.V. Gonchenko, A.S. Gonchenko, I.I. Ovsyannikov, D.V. Turaev, L. Lerman, “Examples of Lorenz-like Attractors in Hénon-like Maps”, Math. Model. Nat. Phenom, 8:5 (2013), 48  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    38. Alexander S. Gonchenko, Sergey V. Gonchenko, Alexey O. Kazakov, “Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone”, Regul. Chaotic Dyn., 18:5 (2013), 521–538  mathnet  crossref  mathscinet  zmath
    39. Arbieto A., Morales C.A., “A Dichotomy for Higher-Dimensional Flows”, Proc. Amer. Math. Soc., 141:8 (2013), 2817–2827  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    40. Valentin S. Afraimovich, Sergey V. Gonchenko, Lev M. Lerman, Andrey L. Shilnikov, Dmitry V. Turaev, “Scientific Heritage of L.P. Shilnikov”, Regul. Chaotic Dyn., 19:4 (2014), 435–460  mathnet  crossref  mathscinet  zmath
    41. Sergey V. Gonchenko, Ivan I. Ovsyannikov, Joan C. Tatjer, “Birth of Discrete Lorenz Attractors at the Bifurcations of 3D Maps with Homoclinic Tangencies to Saddle Points”, Regul. Chaotic Dyn., 19:4 (2014), 495–505  mathnet  crossref  mathscinet  zmath
    42. Tingli Xing, Roberto Barrio, Andrey Shilnikov, “Symbolic Quest into Homoclinic Chaos”, Int. J. Bifurcation Chaos, 24:08 (2014), 1440004  crossref  mathscinet  zmath  scopus  scopus  scopus
    43. Alexander Gonchenko, Sergey Gonchenko, Alexey Kazakov, Dmitry Turaev, “Simple Scenarios of Onset of Chaos in Three-Dimensional Maps”, Int. J. Bifurcation Chaos, 24:08 (2014), 1440005  crossref  mathscinet  zmath  scopus  scopus  scopus
    44. Vitor Araujo, Stefano Galatolo, M.J.osé Pacifico, “Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives”, Int. J. Bifurcation Chaos, 24:10 (2014), 1430028  crossref  mathscinet  zmath  scopus  scopus  scopus
    45. E. A. Sataev, “Mixing and eigenfunctions of singular hyperbolic attractors”, Sb. Math., 206:4 (2015), 572–599  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    46. Turaev D., “Maps Close To Identity and Universal Maps in the Newhouse Domain”, Commun. Math. Phys., 335:3 (2015), 1235–1277  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    47. D.V.. Savin, A.P.. Kuznetsov, A.V.. Savin, Ulrike Feudel, “Different types of critical behavior in conservatively coupled Hénon maps”, Phys. Rev. E, 91:6 (2015)  crossref  mathscinet  scopus  scopus  scopus
    48. Hittmeyer S., Krauskopf B., Osinga H.M., “From Wild Lorenz-Like To Wild Rovella-Like Dynamics”, Dynam. Syst., 30:4, SI (2015), 525–542  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    49. Bautista S., Morales C.A., “Recent Progress on Sectional-Hyperbolic Systems”, Dynam. Syst., 30:4, SI (2015), 369–382  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    50. Barrientos P.G. Ibanez S. Angel Rodriguez J., “Robust cycles unfolding from conservative bifocal homoclinic orbits”, Dynam. Syst., 31:4 (2016), 546–579  crossref  mathscinet  zmath  isi  scopus
    51. Ovsyannikov I.I. Turaev V D., “Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model”, Nonlinearity, 30:1 (2016), 115–137  crossref  mathscinet  isi  scopus
    52. Gonchenko A.S. Gonchenko S.V., “Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps”, Physica D, 337 (2016), 43–57  crossref  mathscinet  zmath  isi  scopus
    53. Li D., “Homoclinic bifurcations that give rise to heterodimensional cycles near a saddle-focus equilibrium”, Nonlinearity, 30:1 (2017), 173–206  crossref  mathscinet  zmath  isi  scopus
    54. Gonchenko S. Ovsyannikov I., “Homoclinic tangencies to resonant saddles and discrete Lorenz attractors”, Discret. Contin. Dyn. Syst.-Ser. S, 10:2 (2017), 273–288  crossref  mathscinet  zmath  isi  scopus
    55. Li D. Turaev D.V., “Existence of Heterodimensional Cycles Near Shilnikov Loops in Systems With a Z(2) Symmetry”, Discret. Contin. Dyn. Syst., 37:8 (2017), 4399–4437  crossref  mathscinet  zmath  isi
    56. 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  crossref  mathscinet  zmath  isi  scopus
    57. Mammeri M., “Symmetry and Periodic-Chaos in 3-D Sinusoid Discrete Maps”, Bull. Math. Anal. Appl., 9:1 (2017), 1–8  mathscinet  zmath  isi
    58. Morales C.A., San Martin B., “Contracting Singular Horseshoe”, Nonlinearity, 30:11 (2017), 4208–4219  crossref  mathscinet  zmath  isi  scopus
    59. Creaser J.L., Krauskopf B., Osinga H.M., “Finding First Foliation Tangencies in the Lorenz System”, SIAM J. Appl. Dyn. Syst., 16:4 (2017), 2127–2164  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    60. Hittmeyer S., Krauskopf B., Osinga H.M., Shinohara K., “Existence of Blenders in a Henon-Like Family: Geometric Insights From Invariant Manifold Computations”, Nonlinearity, 31:10 (2018), R239–R267  crossref  mathscinet  zmath  isi
    61. Eilertsen J., Magnan J., “On the Chaotic Dynamics Associated With the Center Manifold Equations of Double-Diffusive Convection Near a Codimension-Four Bifurcation Point At Moderate Thermal Rayleigh Number”, Int. J. Bifurcation Chaos, 28:8 (2018), 1850094  crossref  mathscinet  zmath  isi  scopus
    62. Conchenko A.S. Conchenko V S. Kazakovt V O. Kozlov A.D., “Elements of Contemporary Theory of Dynamical Chaos: a Tutorial. Part i. Pseudohyperbolic Attractors”, Int. J. Bifurcation Chaos, 28:11 (2018), 1830036  crossref  mathscinet  isi  scopus
    63. Pavel V. Kuptsov, Sergey P. Kuznetsov, “Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction”, Regul. Chaotic Dyn., 23:7-8 (2018), 908–932  mathnet  crossref
    64. Capinski M.J., Turaev D., Zgliczynski P., “Computer Assisted Proof of the Existence of the Lorenz Attractor in the Shimizu-Morioka System”, Nonlinearity, 31:12 (2018), 5410–5440  crossref  mathscinet  zmath  isi  scopus
    65. A. O. Kazakov, A. D. Kozlov, “Nesimmetrichnyi attraktor Lorentsa kak primer novogo psevdogiperbolicheskogo attraktora v trekhmernykh sistemakh”, Zhurnal SVMO, 20:2 (2018), 187–198  mathnet  crossref
    66. Diaz L.J., Perez S.A., “Henon-Like Families and Blender-Horseshoes At Nontransverse Heterodimensional Cycles”, Int. J. Bifurcation Chaos, 29:3 (2019), 1930006  crossref  mathscinet  zmath  isi  scopus
    67. San Martin B., Vivas K.J., “Asymptotically Sectional-Hyperbolic Attractors”, Discret. Contin. Dyn. Syst., 39:7 (2019), 4057–4071  crossref  mathscinet  zmath  isi  scopus
    68. A. O. Kazakov, E. Yu. Karatetskaya, A. D. Kozlov, K. A. Safonov, “O klassifikatsii gomoklinicheskikh attraktorov trekhmernykh potokov”, Zhurnal SVMO, 21:4 (2019), 443–459  mathnet  crossref
    69. Garashchuk I.R., Sinelshchikov D.I., Kazakov A.O., Kudryashov N.A., “Hyperchaos and Multistability in the Model of Two Interacting Microbubble Contrast Agents”, Chaos, 29:6 (2019), 063131  crossref  zmath  isi
    70. Belykh V.N. Barabash N.V. Belykh I.V., “A Lorenz-Type Attractor in a Piecewise-Smooth System: Rigorous Results”, Chaos, 29:10 (2019), 103108  crossref  mathscinet  zmath  isi
    71. Borisov V A., Vetchanin E.V., Mamaev I.S., “Motion of a Smooth Foil in a Fluid Under the Action of External Periodic Forces. i”, Russ. J. Math. Phys., 26:4 (2019), 412–427  crossref  mathscinet  zmath  isi
    72. S. V. Gonchenko, M. S. Gonchenko, I. O. Sinitsky, “On mixed dynamics of two-dimensional reversible diffeomorphisms with symmetric non-transversal heteroclinic cycles”, Izv. Math., 84:1 (2020), 23–51  mathnet  crossref  crossref  isi  elib
    73. S. V. Gonchenko, A. S. Gonchenko, A. O. Kazakov, “Three Types of Attractors and Mixed Dynamics of Nonholonomic Models of Rigid Body Motion”, Proc. Steklov Inst. Math., 308 (2020), 125–140  mathnet  crossref  crossref  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:1091
    Full text:353
    References:65
    First page:2

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020