Radiophysics and Quantum Electronics

, Volume 61, Issue 8–9, pp 650–658 | Cite as

On the Appearance of Mixed Dynamics as a Result of Collision of Strange Attractors and Repellers in Reversible Systems

  • A.O. KazakovEmail author

In this work, we propose a scenario of appearance of mixed dynamics in reversible two-dimensional diffeomorphisms. A jump-like increase in the sizes of the strange attractor and strange repeller, which is due to the heteroclinic intersections of the invariant manifolds of the saddle points belonging to the attractor and the repeller, is the key point of the scenario. Such heteroclinic intersections appear immediately after the collisions of the strange attractor and the strange repeller with the boundaries of their attraction and repulsion basins, respectively, after which the attractor and the repeller intersect. Then the dissipative chaotic dynamics related to the existence of the mutually separable strange attractor and strange repeller immediately becomes mixed when the attractor and the repeller are essentially inseparable. The possibility of realizing the proposed scenario is demonstrated using a well-known problem of the rigid-body dynamics, namely, the nonholonomic model of the Suslov top.


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  1. 1.
    C. Conley, in: CBMS Regional Conf. Series in Mathematics, Vol. 38, American Mathematical Society, Providence, RI (1978), p. 89.Google Scholar
  2. 2.
    D. Ruelle, Commun. Math. Phys., 82, No. 1, 137 (1981).ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    M. Hurley, Trans. Am. Math. Soc., 269, No. 1, 247 (1982).CrossRefGoogle Scholar
  4. 4.
    S. V. Gonchenko and D. V. Turaev, in: Proc. V. A. Steklov Math. Inst. Rus. Acad. Sci., 297, 133 (2017).Google Scholar
  5. 5.
    S. V. Gonchenko, D. V. Turaev, and L. P. Shil’nikov, in: Proc. V. A. Steklov Math. Inst. Rus. Acad. Sci., 216, 76 (1997).Google Scholar
  6. 6.
    J. S. W. Lamb and O. V. Stenkin, Nonlinearity, 17, No. 4, 1217 (2004).ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    L. M. Lerman and D. Turaev, Regul. Chaot. Dyn., 17, Nos. 3–4, 318 (2012).ADSCrossRefGoogle Scholar
  8. 8.
    A. Politi, G. L. Oppo, and R. Badii, Phys. Rev. A, 33, No. 6, 4055 (1986).ADSCrossRefGoogle Scholar
  9. 9.
    J. A. G. Roberts and G. R. W. Quispel, Phys. Rep., 216, Nos. 2–3, 63 (1992).ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    J. S. W. Lamb and J. A. G. Roberts, Physica D, 112, No. 1, 1 (1998).ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    D. V. Anosov and I. U. Bronshtein, “Smooth dynamical systems, Ch. 3, Topologic dynamics,” in: Itogi Nauki Tekhn., Ser. Probl. Mat. Fund. Napr., 1, 204 (1985).Google Scholar
  12. 12.
    D. Topaj and A. Pikovsky, Physica D, 170, No. 2, 118 (2002).ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov, and D. V. Turaev, Physica D, 350, 45 (2017).ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    A. S. Gonchenko, S. V. Gonchenko, and A. O. Kazakov, Regul. Chaot. Dyn., 18, No. 5, 521 (2013).ADSCrossRefGoogle Scholar
  15. 15.
    A. O. Kazakov, Regul. Chaot. Dyn., 18, No. 5, 508 (2013).ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Kazakov, in: Dynamics, Bifurcations and Chaos 2015 (DBC II): Extended Abstract of Int. Conf. and School, Nizhny Novgorod, July 20–24, 2015, p. 21.Google Scholar
  17. 17.
    S. P. Kuznetsov, Europhys. Lett., 118, No. 1, 10007 (2017).ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    S. P. Kuznetsov, Regul. Chaot. Dyn., 23, No. 2, 178 (2018).ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    A. O. Kazakov, arXiv:1801.00150 [math.DS] (2017).Google Scholar
  20. 20.
    A. Delshams, S. V. Gonchenko, V. S. Gonchenko, et al., Nonlinearity, 26, No. 1, 1 (2012).ADSCrossRefGoogle Scholar
  21. 21.
    I. A. Bizyaev, A. V. Borisov, and A. O. Kazakov, Regul. Chaot. Dyn., 20, No. 5, 605 (2015).ADSCrossRefGoogle Scholar
  22. 22.
    M. Hénon, The Theory of Chaotic Attractors, Springer, New York (1976).zbMATHGoogle Scholar
  23. 23.
    M. J. Feigenbaum, Physica D, 7, Nos. 1–3, 16 (1983).ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    G. K. Suslov, Theoretical Mechanics [in Russian], Gostekhizdat, Moscow–Leningrad (1946).Google Scholar
  25. 25.
    V. V. Vagner, in: Proc. Workshop on Vector and Tensor Analysis, No. 5, 301 (1941).Google Scholar
  26. 26.
    V. V. Kozlov, Usp. Mekh., 8, No. 3, 85 (1985).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Research University “Higher School of Economics”Nizhny NovgorodRussia

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