Skip to main content
SpringerLink
Log in

On three types of dynamics and the notion of attractor

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We propose a theoretical framework for explaining the numerically discovered phenomenon of the attractor–repeller merger. We identify regimes observed in dynamical systems with attractors as defined in a paper by Ruelle and show that these attractors can be of three different types. The first two types correspond to the well-known types of chaotic behavior, conservative and dissipative, while the attractors of the third type, reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regimes. We prove that every elliptic orbit of a generic non-conservative time-reversible system is a reversible core. We also prove that a generic reversible system with an elliptic orbit is universal; i.e., it displays dynamics of maximum possible richness and complexity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. V. Anosov, Geodesic Flows on Closed Riemann Manifolds with Negative Curvature (Nauka, Moscow, 1967), Tr. Mat. Inst. im. V. A. Steklova, Akad. Nauk SSSR 90 [Proc. Steklov Inst. Math. 90 (1967)].

    MATH  Google Scholar 

  2. D. V. Anosov and I. U. Bronshtein, “Smooth dynamical systems,” Ch. 3: “Topological dynamics,” in Dynamical Systems–1 (VINITI, Moscow, 1985), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 1, pp. 204–229; Engl. transl. in Dynamical Systems I (Springer, Berlin,1988), Encycl. Math. Sci. 1.

    Google Scholar 

  3. D. V. Anosov and V. V. Solodov, “Dynamical systems with hyperbolic behavior,” Ch. 1: “Hyperbolic sets,” in Dynamical Systems–9 (VINITI, Moscow, 1991), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 66, pp. 12–99; Engl. transl. in Dynamical Systems IX (Springer, Berlin,1995), Encycl. Math. Sci. 66.

    Google Scholar 

  4. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (VINITI, Moscow, 1985), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 3; Engl. transl. in Dynamical Systems III (Springer, Berlin,1993), Encycl. Math. Sci. 3.

    MATH  Google Scholar 

  5. S. Bochner, “Compact groups of differentiable transformations,” Ann. Math., Ser. 2, 46 (3), 372–381 (1945).

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Bonatti and K. Shinohara, “Volume hyperbolicity and wildness,” arXiv: 1505.07901v1 [math.DS].

  7. C. Conley, Isolated Invariant Sets and the Morse Index (Amer. Math. Soc., Providence, RI, 1978), Reg. Conf. Ser. Math., No. 38.

    Book  MATH  Google Scholar 

  8. A. Delshams, S. V. Gonchenko, M. S. Gonchenko, and J. T. Lázaro, “Mixed dynamics of two-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies,” arXiv: 1412.1128 [math.DS].

  9. A. Delshams, S. V. Gonchenko, V. S. Gonchenko, J. T. Lázaro, and O. Sten’kin, “Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps,” Nonlinearity 26 (1), 1–33 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Fiedler and D. Turaev, “Coalescence of reversible homoclinic orbits causes elliptic resonance,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6 (6), 1007–1027 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  11. N. K. Gavrilov and L. P. Shil’nikov, “On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. I, II,” Mat. Sb. 88 (4), 475–492 (1972) [Math. USSR, Sb. 17 (4), 467–485 (1972)]; Mat. Sb. 90 (1), 139–156 (1973) [Math. USSR, Sb. 19 (1), 139–156 (1973)].

    MathSciNet  Google Scholar 

  12. A. S. Gonchenko, S. V. Gonchenko, and A. O. Kazakov, “Richness of chaotic dynamics in nonholonomic models of a Celtic stone,” Regul. Chaotic Dyn. 18 (5), 521–538 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov, and D. V. Turaev, “On the phenomenon of mixed dynamics in Pikovsky–Topaj system of coupled rotators,” Physica D 350, 45–57 (2017).

    Article  MathSciNet  Google Scholar 

  14. S. Gonchenko, “On mixed dynamics in reversible systems,” in Abstr. Int. Conf. Diff. Eqns. Dyn. Syst., Suzdal, July 2–7, 2010, pp. 211–212.

    Google Scholar 

  15. S. Gonchenko, “Mixed dynamics as a new form of dynamical chaos,” in Abstr. 10th AIMS Int. Conf. Dyn. Syst. Diff. Eqns. Appl., Madrid, July 10–14, 2014, p. 14.

    Google Scholar 

  16. S. V. Gonchenko, “Reversible mixed dynamics: A concept and examples,” Discontin. Nonlinearity Complex. 5 (4), 365–374 (2016).

    Article  MATH  Google Scholar 

  17. S. V. Gonchenko, J. S. W. Lamb, I. Rios, and D. Turaev, “Attractors and repellers near generic elliptic points of reversible maps,” Dokl. Akad. Nauk 454 (4), 375–378 (2014) [Dokl. Math. 89 (1), 65–67 (2014)].

    MATH  Google Scholar 

  18. S. V. Gonchenko and I. I. Ovsyannikov, “On bifurcations of three-dimensional diffeomorphisms with a nontransverse heteroclinic cycle containing saddle–foci,” Nelinein. Din. 6 (1), 61–77 (2010).

    Article  Google Scholar 

  19. S. V. Gonchenko and I. I. Ovsyannikov, “On global bifurcations of three-dimensional diffeomorphisms leading to Lorenz-like attractors,” Math. Model. Nat. Phenom. 8 (5), 71–83 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  20. S. V. Gonchenko, L. P. Shilnikov, and O. V. Stenkin, “On Newhouse regions with infinitely many stable and unstable invariant tori,” in Progress in Nonlinear Science: Proc. Int. Conf., Nizhni Novgorod, 2001, Vol. 1: Mathematical Problems of Nonlinear Dynamics (Nizhni Novgorod, 2002), pp. 80–102.

    Google Scholar 

  21. S. V. Gonchenko, L. P. Shilnikov, and D. V. Turaev, “On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors,” Regul. Chaotic Dyn. 14 (1), 137–147 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  22. S. V. Gonchenko, O. V. Stenkin, and L. P. Shilnikov, “On the existence of infinitely many stable and unstable invariant tori for systems from Newhouse regions with heteroclinic tangencies,” Nelinein. Din. 2 (1), 3–25 (2006).

    Article  Google Scholar 

  23. S. V. Gonchenko, D. V. Turaev, and L. P. Shil’nikov, “On Newhouse domains of two-dimensional diffeomorphisms which are close to a diffeomorphism with a structurally unstable heteroclinic cycle,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 216, 76–125 (1997) [Proc. Steklov Inst. Math. 216, 70–118 (1997)].

    MATH  Google Scholar 

  24. S. V. Gonchenko, D. V. Turaev, and L. P. Shil’nikov, “Homoclinic tangencies of arbitrarily high order in conservative two-dimensional maps,” Dokl. Akad. Nauk 407 (3), 299–303 (2006) [Dokl. Math. 73 (2), 210–213 (2006)].

    MathSciNet  MATH  Google Scholar 

  25. S. Gonchenko, D. Turaev, and L. Shilnikov, “Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps,” Nonlinearity 20 (2), 241–275 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Hurley, “Attractors: Persistence, and density of their basins,” Trans. Am. Math. Soc. 269 (1), 247–271 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  27. A. O. Kazakov, “Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane,” Regul. Chaotic Dyn. 18 (5), 508–520 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Kazakov, “On chaotic dynamics in the Suslov problem,” in Dynamics, Bifurcations and Chaos 2015 (DBC II): Ext. Abstr. Int. Conf.–Sch., Nizhni Novgorod, July 20–24, 2015 (Lobachevsky State Univ., Nizhni Novgorod, 2015), pp. 21–30.

    Google Scholar 

  29. J. S. W. Lamb and G. R. W. Quispel, “Reversing k-symmetries in dynamical systems,” Physica D 73 (4), 277–304 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  30. J. S. W. Lamb and O. V. Stenkin, “Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits,” Nonlinearity 17 (4), 1217–1244 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  31. S. E. Newhouse, “Diffeomorphisms with infinitely many sinks,” Topology 13, 9–18 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  32. S. E. Newhouse, “Quasi-elliptic periodic points in conservative dynamical systems,” Am. J. Math. 99, 1061–1087 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  33. S. E. Newhouse, “The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms,” Publ. Math., Inst. Hautes études Sci. 50, 101–151 (1979).

    Article  MATH  Google Scholar 

  34. D. Ruelle, “Small random perturbations of dynamical systems and the definition of attractors,” Commun. Math. Phys. 82, 137–151 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  35. M. B. Sevryuk, Reversible Systems (Springer, Berlin, 1986), Lect. Notes Math. 1211.

    Book  MATH  Google Scholar 

  36. D. Topaj and A. Pikovsky, “Reversibility vs. synchronization in oscillator lattices,” Physica D 170 (2), 118–130 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  37. D. Treschev, “Closures of asymptotic curves in a two-dimensional symplectic map,” J. Dyn. Control Syst. 4 (3), 305–314 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  38. D. V. Treschev, Introduction to the Perturbation Theory of Hamiltonian Systems (Phasis, Moscow, 1998), Bibliot. Stud.-Mat. 6; Ext. Engl. transl.: D. Treschev and O. Zubelevich, Introduction to the Perturbation Theory of Hamiltonian Systems (Springer, Berlin,2010), Springer Monogr. Math.

    MATH  Google Scholar 

  39. D. Turaev, “On dimension of nonlocal bifurcational problems,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6 (5), 919–948 (1996).

    Article  MATH  Google Scholar 

  40. D. Turaev, “Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic areapreserving maps,” Nonlinearity 16 (1), 123–135 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  41. D. Turaev, “Richness of chaos in the absolute Newhouse domain,” in Proc. Int. Congr. Math., Hyderabad (India), 2010, Vol. 3: Invited Lectures (World Scientific, Hackensack, NJ, 2011), pp. 1804–1815.

    Google Scholar 

  42. D. Turaev, “Maps close to identity and universal maps in the Newhouse domain,” Commun. Math. Phys. 335 (3), 1235–1277 (2015).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lobachevsky State University of Nizhni Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950, Russia

    S. V. Gonchenko & D. V. Turaev

  2. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

    D. V. Turaev

Authors
  1. S. V. Gonchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. V. Turaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. V. Gonchenko.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 297, pp. 133–157.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gonchenko, S.V., Turaev, D.V. On three types of dynamics and the notion of attractor. Proc. Steklov Inst. Math. 297, 116–137 (2017). https://doi.org/10.1134/S0081543817040071

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543817040071

Search

Navigation