ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Gonchenko S. V., Kainov M. N., Kazakov A. O., Turaev D. V. On methods for verification of the pseudohyperbolicity of strange attractors. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 1, pp. 160-185. DOI: 10.18500/0869-6632-2021-29-1-160-185

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Language: 
Russian
Heading: 
Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos.
Article type: 
Review
UDC: 
517.925 + 517.93
DOI: 
10.18500/0869-6632-2021-29-1-160-185

On methods for verification of the pseudohyperbolicity of strange attractors

Autors: 
Gonchenko Sergey Vladimirovich, Lobachevsky State University of Nizhny Novgorod
Kainov M. N., National Research University "Higher School of Economics"
Kazakov Aleksej Olegovich, National Research University "Higher School of Economics"
Turaev D. V., Imperial College London
Abstract: 

The topic of the paper is strange attractors of multidimensional maps and flows. Strange attractors can be divided into two groups: genuine attractors, that keep their chaoticity under small perturbations, and quasi-attractors (according to Afraimovich–Shilnikov), inside which stable periodic orbits can arise under small perturbations. Main goal of this work is to construct effective criteria that make it possible to distinguish such attractors, as well as to verify these criteria by means of numerical experiments. Under «genuine» attractors, we mean the so-called pseudohyperbolic attractors. We give their definition and describe characteristic properties, on the basis of which two numerical methods are constructed, which allow to check the principally important property of pseudohyperbolic attractors: the continuity of strong contracting subspaces and subspaces where volumes are expanded. As examples on which numerical methods for checking pseudohyperbolicity have been tested, we consider the classical Henon map, the singularly hyperbolic Lozi map, the Anosov diffeomorphism of two-dimensional torus, the classical Lorenz and Shimizu–Morioka systems, as well as a three-dimensional Henon-like maps.

Key words: 
chaotic attractor
pseudohyperbolicity
quasiattractor
Lyapunov exponents
Lorenz attractor
Henon map
Acknowledgments: 
This work was supported by the RSF grants No. 19-11-00280 (Introduction, Sections 1 and 3.3) and 19-71-10048 (Sections 2 and 3). Numerical results with models in Section 3 were supported by the Laboratory of Dynamical Systems and Applications NRU HSE, of the Russian Ministry of Science and Higher Education (Grant No. 075-15-2019-1931). Authors also thank RFBR (grants 18-29-10081 and 19-01-00607) and the Theoretical Physics and Mathematics Advancement Foundation «BASIS». The authors also thank P.V. Kuptsov for fruitful discussion and useful comments.
Reference: 
  1. Anosov DV, Aranson SH, Grines VZ et al. Dynamical systems with hyperbolic behavior. Results of Science and Technology. Ser. «Modern problems of mathematics. Fundamental directions». All-Russian Institute of Scientific and Technical Information, Moscow. 1991;66:5–242 (in Russian).
  2. Lorenz EN. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences. 1963;20(2): 130–141. DOI: 10.1175/1520-0469(1963)020<_x0030_130:DNF>2.0.CO;2.
  3. Afraimovich VS, Bykov VV, Shilnikov LP. On the origin and structure of the Lorenz attractor. Proceedings of the Academy of Sciences of the USSR. 1977;234(2):336–339 (in Russian).
  4. Afraimovich V.S., Bykov V.V., Shilnikov L.P. Attractive nonrough limit sets of Lorenz-attractor type. Proceedings of the Moscow Mathematical Society. 1982;44:150–212 (in Russian).
  5. Plykin RV. Sources and sinks of A-diffeomorphisms of surfaces. Mat. sb. 2007;23(2):233–253. DOI: 10.1070/SM1974v023n02ABEH001719.
  6. Anosov DV. Dynamical systems in the 1960s: the hyperbolic revolution. In the book: Mathematical Events of the Twentieth Century. Springer, Berlin, Heidelberg. 2006:1–17. DOI: 10.1007/3-540-29462-7_1.
  7. Turaev DV, Shilnikov LP. An example of a wild strange attractor Mat. sb. 1998;189(2): 291–314. DOI: DOI: 10.4213/sm300.
  8. Gonchenko SV, Kazakov AO, Turaev DV. Wild pseudohyperbolic attractor in a four-dimensional Lorenz system. Nonlinearity. 2021;34(2):1–30.
  9. Guckenheimer J, Williams RF. Structural stability of Lorenz attractors. Inst. Hautes Etudes Sci. ´ Publ. Math. 1979;50(5):59–72. DOI: 10.1007/BF02684769.
  10. Shilnikov LP. A case of the existence of a denumerable set of periodic motions. Proceedings of the Academy of Sciences of the USSR. 1965;160(3):558–561 (in Russian).
  11. Shilnikov LP. A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of the saddle-focus type. Mat. sb. 1970;10(1):91–102. DOI: 10.1070/SM1970v010n01ABEH001588.
  12. Kuznetsov SP. Example of a physical system with a hyperbolic attractor of the Smale–Williams type. Phys. Rev. Lett. 2005;95(14):144101. DOI: 10.1103/PhysRevLett.95.144101.
  13. Kuznetsov SP. Dynamical Chaos and Hyperbolic Attractors. From Mathematics to Physics. Izhevsk Institute of Computer Science, Moscow, Izhevsk. 2013:488. (in Russian)
  14. Kuznetsov SP. Some lattice models with hyperbolic chaotic attractors. Russian Journal of Nonlinear Dynamics. 2020;16(1):13–21. DOI: 10.20537/nd200102.
  15. Aframovich VS, Shilnikov LP. Strange attractors and quasiattractors. Nonlinear Dynamics and Turbulence (eds. G.I. Barenblatt, G. Iooss, D.D. Joseph), Boston, Pitmen. 1983.
  16. Gonchenko SV, Shilnikov LP, Turaev DV. Quasiattractors and homoclinic tangencies. Computers and Mathematics with Applications. 1997;34(2–4):195–227. DOI: 10.1016/S0898-1221(97)00124-7.
  17. Galias Z, Tucker W. Is the Henon attractor chaotic? International Journal of Bifurcation and ´ Chaos. 2015;25(3):033102. DOI: 10.1063/1.4913945.
  18. Gonchenko SV, Ovsyannikov II, Simo C, Turaev DV. Three-dimensional H ´ enon-like maps ´ and wild Lorenz-like attractors. International Journal of Bifurcation and Chaos. 2005;15(11): 3493–3508. DOI: 10.1142/S0218127405014180.
  19. Gonchenko AS, Gonchenko SV, Shilnikov LP. Towards scenarios of chaos appearance in threedimensional maps. Russian Journal of Nonlinear Dynamics. 2012;8(1):3–28. DOI: 10.20537/nd1201001. (in Russian)
  20. Gonchenko AS, Gonchenko SV. Variety of strange pseudohyperbolic attractors in three-dimensional generalized Henon maps. Physica D. 2016;337:43–57. ´ DOI: 10.1016/j.physd.2016.07.006.
  21. Kuznetsov SP, Sataev IR. Hyperbolic attractor in a system of coupled non-autonomous van der Pol oscillators: Numerical test for expanding and contracting cones. Phys. Lett. A. 2007; 365(1–2):97–104. DOI: 10.1016/j.physleta.2006.12.071.
  22. Kuptsov PV. Fast numerical test of hyperbolic chaos. Phys. Rev. E. 2012;85(1):015203. DOI: 10.1103/PhysRevE.85.015203.
  23. Kruglov VP. Technique and results of numerical test for hyperbolic nature of attractors for reduced models of distributed systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2014;22(6): 79–93. DOI: 10.18500/0869-6632-2014-22-6-79-93. (In Russian)
  24. Kuznetsov SP, Kruglov VP. Verification of hyperbolicity for attractors of some mechanical systems with chaotic dynamics. Regular and Chaotic Dynamics. 2016;21(2):160–174. DOI: 10.1134/S1560354716020027.
  25. Kuptsov PV, Kuznetsov SP. Numerical test for hyperbolicity in chaotic systems with multiple time delays. Communications in Nonlinear Science and Numerical Simulation. 2018;56(3): 227–239. DOI: 10.1016/j.cnsns.2017.08.016.
  26. Gonchenko AS, Gonchenko SV, Kazakov AO, Kozlov AD. Mathematical theory of dynamical chaos and its applications: review part 1. Pseudohyperbolic attractors. Izvestiya VUZ. Applied Nonlinear Dynamics. 2017;25(2):4–36. DOI: 10.18500/0869-6632-2017-25-2-4-36. (in Russian)
  27. Gonchenko AS, Gonchenko SV, Kazakov AO, Kozlov AD. Elements of contemporary theory of dynamical chaos: a tutorial. Part I. Pseudohyperbolic attractors. International Journal of Bifurcation and Chaos. 2018;28(11):1830036. DOI: 10.1142/S0218127418300367.
  28. Kuptsov PV, Politi A. Large-deviation approach to space-time chaos. Phys. Rev. Lett. 2011; 107(11):114101. DOI: 10.1103/PhysRevLett.107.114101.
  29. Turaev DV, Shilnikov LP. Pseudohyperbolicity and the problem on periodic perturbations of Lorenz-type attractors. Doklady Mathematics. 2008;77(1):17–21. DOI: 10.1007/s11472-008-1005-4.
  30. Tucker W. The Lorenz attractor exists. Comptes Rendus de l’Academie des Sciences – Series I – ´ Mathematics. 1999;328(12):1197–1202. DOI: 10.1016/S0764-4442(99)80439-X.
  31. Gonchenko A.S., Gonchenko S.V., Kazakov A.O., Turaev D. Simple scenarios of onset of chaos in three-dimensional maps. International Journal of Bifurcation and Chaos. 2014;24(8):1440005. DOI: 10.1142/S0218127414400057.
  32. Ginelli F, Poggi P, Turchi A, Chate H, Livi R, Politi A. Characterizing dynamics with covariant ´ Lyapunov vectors. Phys. Rev. Lett. 2007;99(13):130601. DOI: 10.1103/PhysRevLett.99.130601.
  33. Wolfe CL, Samelson RM. An efficient method for recovering Lyapunov vectors from singular vectors. Tellus A: Dynamic Meteorology and Oceanography. 2007;59(3):355–366. DOI: 10.1111/j.1600-0870.2007.00234.x.
  34. Kuptsov PV, Parlitz U. Theory and computation of covariant Lyapunov vectors. Journal of Nonlinear Science. 2012;22(5):727–762. DOI: 10.1007/s00332-012-9126-5.
  35. Shilnikov AL, Shilnikov LP, Turaev DV Normal forms and Lorenz attractors. International Journal of Bifurcation and Chaos. 1993;3(5):1123–1139. DOI: 10.1142/S0218127493000933.
  36. Gonchenko AS, Gonchenko SV, Kazakov AO, Samylina E. On discrete Lorenz-like attractors. International Journal of Bifurcation and Chaos. 2021 (in print).
  37. Benedicks M, Carleson L. The dynamics of the Henon map. Annals of Mathematics. 1991; ´ 133(1):73–169. DOI: 10.2307/2944326.
  38. Mora L, Viana M. Abundance of strange attractors. Acta mathematica. 1993;171(1):1–71. DOI: 10.1007/BF02392766.
  39. Wang Q, Young L-S. Toward a theory of rank one attractors. Annals of Mathematics. 2008;167(2): 349–480. DOI: 10.4007/annals.2008.167.349.
  40. Chigarev V, Kazakov AO, Pikovsky AS. Kantorovich–Rubinstein–Wasserstein distance between overlapping attractor and repeller. International Journal of Bifurcation and Chaos. 2020;30(7): 073114. DOI: 10.1063/5.0007230.
  41. Bykov VV, Shilnikov AL. On the boundaries of the domain of existence of the Lorenz attractor. Selecta Mathematica. 1992;11(4):375–382.
  42. Creaser JL, Krauskopf B, Osinga HM. Finding first foliation tangencies in the Lorenz system. SIAM Journal on Applied Dynamical Systems. 2017;16(4):2127–2164. DOI: 10.1137/17M1112716.
  43. Roshchin NV. Unsafe stability boundaries of the Lorentz model. Journal of Applied Mathematics and Mechanics. 1978;42(5):1038–1041. DOI: 10.1016/0021-8928(78)90049-7.
  44. Shilnikov LP. Bifurcation theory and the Lorenz model. Appendix to Russian edition of «The Hopf Bifurcation and Its Applications». Eds. J. Marsden and M. McCraken. 1980:317–335.
  45. Shilnikov LP. Selected works of L.P. Shilnikov. Lobachevsky State University of Nizhny Novgorod (ed. Afraimovich VS, Belyakov LA, Gonchenko SV, Lerman LM, Morozov AD, Turaev DV, Shilnikov AL). 2017 (in Russian).
  46. Shilnikov AL, Shilnikov LP, Turaev DV, Chua LO. Methods of Qualitative Theory in Nonlinear Dynamics. Part 2. World Scientific Series on Nonlinear Science Series A: Volume 5. 2001:592. DOI: 10.1142/4221.
  47. Shilnikov AL. Bifurcations and chaos in the Morioka–Shimizu model. Part II. Methods in Qualitative Theory and Bifurcation Theory, Gorky St. Univ. 1989:130–138 (in Russian).
  48. Golmakani A and Homburg AJ. Lorenz attractors in unfoldings of homoclinic-flip bifurcations. Dynamical Systems. 2011;26(1):61–76. DOI: 10.1080/14689367.2010.503186.
  49. Xing T, Barrio R and Shilnikov AL. Symbolic quest into homoclinic chaos. International Journal of Bifurcation and Chaos. 2014;24(8):1440004. DOI: 10.1142/S0218127414400045.
  50. Sataev EA. Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type. Sb. Math. 2005;196(4):561–594. DOI: https://doi.org/10.4213/sm1288.
  51. Shimizu T, Morioka N. On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model. Phys. Lett. A. 1980;76(3–4):201–204. DOI: 10.1016/0375-9601(80)90466-1.
  52. Shilnikov AL. Bifurcation and chaos in the Morioka–Shimizu system. Selecta Mathematica. 1991;10(2):105–117.
  53. Shilnikov AL. On bifurcations of the Lorenz attractor in the Shimizu–Morioka model. Physica D. 1993;62(1–4):338–346. DOI: 10.1016/0167-2789(93)90292-9.
  54. Capinski MJ, Turaev DV, Zgliczy ´ nski P. Computer assisted proof of the existence of the Lorenz ´ attractor in the Shimizu-Morioka system. Nonlinearity. 2018;31(12):5410–5440. DOI: 10.1088/1361-6544/aae032.
  55. Gonchenko SV, Gonchenko AS, Ovsyannikov II, Turaev DV. Examples of Lorenz-like attractors in Henon-like maps. Mathematical Modelling of Natural Phenomena. 2013;8(5):48–70. ´ DOI: 10.1051/mmnp/20138504.
  56. Gonchenko SV, Meiss JD, Ovsyannikov II. Chaotic dynamics of three-dimensional Henon maps ´ that originate from a homoclinic bifurcation. Regular and Chaotic Dynamics. 2006;11(2): 191–212. DOI: 10.1070/RD2006v011n02ABEH000345.
  57. Gonchenko SV, Ovsyannikov II, Tatjer JC. Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points. Regular and Chaotic Dynamics. 2014;19(4):495–505. DOI: 10.1134/S1560354714040054.
  58. Gonchenko SV, Shilnikov LP, Turaev DV. On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors. Regular and Chaotic Dynamics. 2009;14(1): 137–147. DOI: 10.1134/S1560354709010092.
  59. Gonchenko SV, Ovsyannikov II. On global bifurcations of three-dimensional diffeomorphisms leading to Lorenz-like attractors. Mathematical Modelling of Natural Phenomena. 2013;8(5): 71–83. DOI: 10.1051/mmnp/20138505.
  60. Gonchenko SV, Ovsyannikov II. Homoclinic tangencies to resonant saddles and discrete Lorenz attractors. Discrete & Continuous Dynamical Systems Series S. 2017;10(2):273–288. DOI: 10.3934/dcdss.2017013.
  61. Gonchenko AS, Gonchenko SV, Kazakov AO. Richness of chaotic dynamics in nonholonomic models of a Celtic stone. Regular and Chaotic Dynamics. 2013;18(5):521–538. DOI: 10.1134/S1560354713050055.
  62. Gonchenko AS, Samylina EA. On the region of existence of a discrete Lorenz attractor in the nonholonomic model of a Celtic stone. Radiophysics and Quantum Electronics. 2019;62(5):369– 384. DOI: 10.1007/s11141-019-09984-9.
Received: 
18.12.2020
Accepted: 
30.12.2020
Published: 
01.02.2020
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