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Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Kozlov A. D. Mathematical theory of dynamical chaos and its applications: review part 1. Pseudohyperbolic attractors. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, iss. 2, pp. 4-36. DOI: https://doi.org/10.18500/0869-6632-2017-25-2-4-36

# Mathematical theory of dynamical chaos and its applications: review part 1. Pseudohyperbolic attractors

We consider important problems of modern theory of dynamical chaos and its applications. At present, it is customary to assume that in the finite-dimensional smooth dynamical systems three fundamentally different forms of chaos can be observed. This is the dissipative chaos, whose mathematical image is a strange attractor; the conservative chaos, for which the whole phase space is a large «chaotic sea» with elliptical islands randomly disposed within it; and the mixed dynamics which is characterized by the principle inseparability, in the phase space, of attractors, repellers and orbits with conservative behavior. In the first part of this series of our works, we present some elements of the theory of pseudohyperbolic attractors of multidimensional maps. Such attractors, the same as hyperbolic ones, are genuine strange attractors, however, they allow homoclinic tangencies. We also give a description of phenological scenarios of the appearance of pseudohyperbolic attractors of various types for one parameter families of three-dimensional diffeomorphisms, and, moreover, consider some examples of such attractors in three-dimensional orientable and nonorientable Henon maps. ? In the second part, we will give a review of the theory of spiral attractors. Such type of strange attractors are very important and are often observed type in dynamical systems. The third part will be dedicated to mixed dynamics – a new type of chaos which is typical, in particulary, for (time) reversible systems i.e. systems which are invariant with respect to some changes of coordinates and time reversing. It is well known that such systems occur in many problems of mechanics, electrodynamics, and other areas of natural sciences. DOI: 10.18500/0869-6632-2017-25-2-4-36 Paper reference: Gonchenko A.S., Gonchenko S.V., Kazakov A.O., Кozlov A.D. Mathematical theory of dynamical chaos and its applications: Review. Part 1. Pseudohyperbolic attractors. Izvestiya VUZ. Applied Nonlinear Dynamics. 2017. Vol. 25. Issue 2. P. 4–36. Download full version

1. Conley C.C. Isolated invariant sets and the Morse index. American Mathematical Soc. 1978. No.38. 2. Gonchenko S.V., Shilnikov L.P., Turaev D.V. On Newhouse regions of two-dimensional diffeomorphisms close to a diffeomorphism with a nontransversal heteroclinic cycle. Proc. Steklov Inst. Math. Vol. 216. P. 70–118. 3. Gonchenko S. Reversible mixed dynamics: A concept and examples. Discontinuity, Nonlinearity, and Complexity. 2016. Vol. 5, No. 4. Pp.345–354. 4. Gonchenko S., Turaev D. On three types of dynamics, and the notion of attractor. To appear. 5. Aframovich V.S., Shilnikov L.P. Strange attractors and quasiattractors. Nonlinear Dynamics and Turbulence./ Eds G.I. Barenblatt, G. Iooss, D.D. Joseph. Boston, Pitmen, 1983. 6. Shilnikov V.S. Mathematical problems of nonlinear dynamics: A tutorial. Int. J. Bif. and Chaos. 1997. Vol. 9, No. 7. Pp. 1953–2001. 7. Gonchenko S.V., Shilnikov L.P., Turaev D.V. Quasiattractors and homoclinic tangencies. Computers Math. Applic. 1997. Vol. 34, No. 2-4. Pp. 195–227. 8. Gavrilov N.K., Shilnikov L.P. On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. Part 1. Math. USSR Sb. 1972. Vol. 17. Pp. 467–485; Part 2. Math. USSR Sb. 1973. Vol. 19. Pp. 139–156. 9. Gonchenko S.V. On stable periodic motions in systems close to a system with a nontransversal homoclinic curve. Mat. Zametki. 1983. Vol. 33, No. 5. Pp. 745–755. 10. Gonchenko S.V., Turaev D.V., Shilnikov L.P. Dynamical phenomena in multidimensional systems with non-rough Poincare homoclinic curve. ? Doklady Mathematics. 1993. Vol. 330, No. 2. Pp. 144–147. 11. Gonchenko S.V., Shilnikov L.P., Turaev D.V. Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits. Chaos. 1996. Vol. 6, No. 1. Pp. 15–31. 12. Gonchenko S.V., Shilnikov L.P., Turaev D.V. On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. Nonlinearity. 2008. Vol. 21, No. 5. Pp. 923–972. 13. Mora L., Viana M. Abundance of strange attractors. Acta Math. 1993. Vol. 171. Pp. 1–71. 14. Palis J., Viana M. High dimension diffeomorphisms displaying infinitely many sinks. Ann. Math. 1994. Vol. 140. Pp. 91–136. 15. Gonchenko S.V., Sten’kin O.V., Turaev D.V. Complexity of homoclinic bifurcations and ?-moduli. Int.Journal of Bifurcation and Chaos. 1996. Vol. 6, No. 6. Pp. 969– 989. 16. Colli E. Infinitely many coexisting strange attractors. Ann. IHP. Anal. Non Lineaire. 1998. Vol. 15. Pp. 539–579. 17. Homburg A.J. Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria. Nonlinearity. 2002. Vol. 15. Pp. 1029– 1050. 18. Gonchenko S.V., Meiss J.D. and Ovsyannikov I.I. Chaotic dynamics of three-dimensional Henon maps that originate from a homoclinic bifurcation. ? Regul. Chaotic Dyn. 2006. Vol. 11. Pp. 191–212. 19. Gonchenko S.V., Shilnikov L.P. and Turaev D.V. On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors. Regul. Chaotic Dyn. 2009. Vol. 14. P. 137–147. 20. Gonchenko S.V. and Ovsyannikov I.I. On global bifurcations of three-dimensional diffeomorphisms leading to Lorenz-like attractors. Mat. Model. of Nat. Phenom. 2013. Vol. 8. Pp. 71–83. 21. Gonchenko S.V., Ovsyannikov I.I. and Tatjer J.C. Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points. Regul. Chaotic Dyn. 2014. Vol. 19. Pp. 495–505. 22. Gonchenko S.V. and Ovsyannikov I.I. Homoclinic tangencies to resonant saddles and discrete Lorenz attractors. Discr. and Cont. Dyn. Sys. Series S. 2017. Vol. 10, No. 2. Pp. 365–374. 23. Lozi R. Un attracteur de Henon. J. Phys. 1978. Vol. 39. Coll-C5. Pp. 9–10. 24. Belykh V.N. Chaotic and strange attractors of a two-dimensional map. Sb. Mathematics. 1995. Vol. 186, No. 3. Pp. 3–18. 25. Feudel U., Kuznetsov S., Pikovsky A. Strange nonchaotic attractors: Dynamics between order and chaos in quasiperiodically forced systems. Nonlinear Science. Vol.56. World scientific series on nonlinear science: Monographs and treatises, 2006. 26. Afraimovich V.S., Shilnikov L.P. Invariant two-dimensional tori, their breakdown and stochasticity. Amer. Math. Soc. Transl. 1991. Vol. 149, No. 2. Pp. 201–212. 27. Shilnikov L.P. Chua’s circuit: Rigorous result and future problems. Int. J. Bifurcation and Chaos. 1994. Vol. 4, No. 3. Pp. 489–519. 28. Henon M. A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 1976. Vol. 50. Pp. 69–77. 29. Benedicks M., Carleson L. The dynamics of the Henon map. Ann. Math. 1991. Vol. 133. Pp. 73–169. 30. Gonchenko S.V., Simo C. and Vieiro A. Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight. Nonlinearity. 2013. Vol. 26, No. 3. Pp. 621–678. 31. Rossler O. E. An equation for continuous chaos. ? Physics Letters A. 1976. Vol. 57, No. 5. Pp. 397–398. 32. Rossler O.E. Different types of chaos in two simple differential equations. ? Zeitschrift fur Naturforschung A ? . 1976. Vol. 31, No. 12. Pp. 1664–1670. 33. Arneodo A., Coullet P., Tresser C. Occurence of strange attractors in three-dimensional Volterra equations. Physics Letters A. 1980. Vol. 79, No. 4. Pp. 259–263. 34. Arneodo A., Coullet P., C. Tresser. Possible new strange attractors with spiral structure. Commun. Math. Phys. 1981. Vol. 79. Pp. 573–579. 35. Arneodo A., Coullet P., Tresser C. Oscillators with chaotic behavior: An illustration of a theorem by Shilnikov. ? Journal of Statistical Physics. 1982. Vol. 27, No. 1. Pp. 171–182. 36. Shilnikov L.P. The theory of bifurcations and turbulence. Selecta Math. Sovietica. 1991. Vol. 10. Pp. 43–53. 37. Gonchenko A.S., Gonchenko S.V., Shilnikov L.P. Towards scenarios of chaos appearance in three-dimensional maps. Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics]. 2012. Vol. 8, No. 1. Pp. 3–28. 38. Gonchenko A.S., Gonchenko S.V., Kazakov A.O. and Turaev D. Simple scenarios of onset of chaos in three-dimensional maps. Int. J. Bif. and Chaos. 2014. Vol. 24, No. 8. 25 pages. 39. Gonchenko S.V., Gonchenko A.S., Ovsyannikov I.I., Turaev D.V. Examples of Lorenz-like attractors in Henon-like maps. Math. Model. Nat. Phen. 2013. Vol. 8, No. 5. Pp. 48–70. 40. Gonchenko A., Gonchenko S. Variety of strange pseudohyperbolic attractors in three-dimensional generalized Henon maps. Physica D. 2016. Vol. 337. Pp. 43-57. 41. Gonchenko A.S., Kozlov A.D. On scenaria of chaos appearance in three-dimension nonorientable maps. J. SVMO. 2016. Vol. 18, No. 4. Pp. 17–29. 42. Turaev D.V., Shilnikov L.P. An example of a wild strange attractor. Sb. Math. 1998. Vol. 189. Pp. 291–314. 43. Newhouse S.E. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math. Inst. Hautes Etudes Sci. 1979. Vol. 50. Pp. 101–151. 44. Ruelle D. Small random perturbations of dynamical systems and the definition of attractors. Comm. Math. Phys. 1981. Vol. 82. Pp. 137–151. 45. Gonchenko S., Ovsyannikov I., Simo C., Turaev D. Three-dimensional Henon-like maps and wild Lorenz-like attractors. Int. J. of Bifurcation and Chaos. 2005. Vol. 15, No. 11. Pp. 3493–3508. 46. Shilnikov A.L. Bifurcation and chaos in the Morioka–Shimizu system. Selecta Math. Soviet. 1991. Vol. 10, No. 2. Pp. 105–117. 47. Shilnikov A.L. On bifurcations of the Lorenz attractor in the Shimuizu–Morioka model. Physica D. 1993. Vol. 62. Pp. 338–346. 48. Tucker W. The Lorenz attractor exists. Comptes Rendus de l’Academie des Sciences-Series I-Mathematics. 1999. Vol. 328, No. 12. Pp. 1197–1202. 49. Ovsyannikov I.I., Turaev D. Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model. Nonlinearity. 2017. Vol. 30. Pp. 115–137. 50. Turaev D.V. and Shilnikov L.P. Pseudo-hyperbolisity and the problem on periodic perturbations of Lorenz-like attractors. Doklady Mathematics. 2008. Vol. 77, No. 1. Pp. 17–21. 51. Gonchenko S.V., Turaev D.V., Shilnikov L.P. On an existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case). ?Doklady Akademii Nauk. 1993. Vol. 329, No. 4. Pp. 404–407. 52. Tatjer J.C. Three-dimensional dissipative diffeomorphisms with homoclinic tangencies. Ergodic Theory Dynam. Systems. 2001. Vol. 21. Pp. 249–302. 53. Gonchenko S.V., Gonchenko V.S., Tatjer J.C. Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Henon maps. ? Regular and Chaotic Dynamics. 2007. Vol. 12, No.3. Pp. 233–266. 54. Sataev E.A. Stochastic properties of the singular hyperbolic attractors. Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics]. 2010. Vol. 6, No. 1. Pp. 187–206. 55. Gonchenko S.V., Gonchenko A.S., Kazakov A.O. Richness of chaotic dynamics in nonholonomic models of a Celtic stone. Regular and Chaotic Dynamics. 2013. Vol. 15, No. 5. Pp. 521–538. 56. Gonchenko S.V., Turaev D.V., Shilnikov L.P. Homoclinic tangencies of an arbitrary order in Newhouse domains. Journal of Mathematical Sciences. 2001. Vol. 105, No. 1. Pp. 1738–1778. 57. Gonchenko S., Gonchenko A. and Ming-Chia Li. On topological and hyperbolic properties of systems with homoclinic tangencies. In book «Nonlinear Dynamics New Directions». Springer International Publishing Switzerland, 2015. 27 pages. 58. Gonchenko S., Ovsyannikov I., Tatjer J.C. Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points. Regular and Chaotic Dynamics. 2014. Vol. 19, No. 4. Pp. 495–505. 59. Grines V., Levchenko Yu., Medvedev V. and Pochinka O. The topological classification of structurally stable 3-diffeomorphisms with two-dimensional basic sets. Nonlinearity. 2015. Vol. 28. Pp. 4081–4102. 60. Kuznetsov S.P. Example of a physical system with a hyperbolic attractor of the Smale–Williams type. Phys. Rev. Lett. 2005. Vol. 95. 144101. 61. Kuznetsov S.P. and Seleznev E.P. Strange attractor of Smale–Williams type in the chaotic dynamics of a physical system. Zh. Eksper. Teoret. Fiz. 2006. Vol. 129, No.2. Pp. 400–412 [J. Exp. Theor. Phys. 2006. Vol. 102, No. 2. Pp. 355–364]. 62. Kuznetsov S.P. and Pikovsky A. Autonomous coupled oscillators with hyperbolic strange attractors. Physica D. 2007. Vol. 232. Pp. 87–102. 63. Kuznetsov S.P. Example of blue sky catastrophe accompanied by a birth of the Smale–Williams attractor. Regular and Chaotic Dynamics. 2007. Vol. 12, No. 3. Pp. 233–266. 64. Shilnikov A.L., Shilnikov L.P., Turaev D.V. Normal forms and Lorenz attractors. Int. J. of Bifurcation and Chaos. 1993. Vol. 3. Pp. 1123–1139. 65. Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L.O. Methods of qualitative theory in nonlinear dynamics. World Scientific. Part I, 1998; Part 2, 2002 . 66. Anosov D.V. (ed.). Geodesic flows on closed Riemann manifolds with negative curvature. American Mathematical Society. 1969. No. 90.

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