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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Reversible Mixed Dynamics: A Concept and Examples

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 365--374 | DOI:10.5890/DNC.2016.12.003

S.V. Gonchenko

Nizhny Novgorod State University, Nizhny Novgorod, Russia

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We observe some recent results related to the new type of dynamical chaos, the so-called, “mixed dynamics” which can be considered as an intermediate link between “strange attractor“ and “conservative chaos”. We propose a mathematical concept of mixed dynamics for two-dimensional reversible maps and consider several examples.


The author thanks D. Turaev for very useful remarks. This work is particularly supported by RSciF-grant 14-41-00044 and RFBR-grants 16-01-00364 and 14-01-00344. Section “Examples” is carried out by RSciF-grant 14-12-00811.


  1. [1]  Newhouse, S. (1974,) Diffeomorphisms with infinitely many sinks, Topology, 13, 9-18.
  2. [2]  Newhouse, S.E. (1979), The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. IHES, 50, 101-151.
  3. [3]  Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V. (1993), On the existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case), Rus. Acad. Sci.Dokl.Math., 47, No.2, 268-283.
  4. [4]  Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V. (1993), Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits, Russian Acad. Sci. Dokl. Math., 47, No.3, 410-415.
  5. [5]  Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V. (1996), Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Chaos, 6, 15-31.
  6. [6]  Gonchenko, S.V., Shilnikov, L.P., and Turaev, D. (2008), On dynamical properties of multidimensional diffeomorphisms from Newhouse regions, Nonlinearity, 21(5), 923-972.
  7. [7]  Gonchenko, S.V., Shilnikov, L.P., and Turaev, D. (2009), On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors, Regular and Chaotic Dynamics, 14(1), 137-147.
  8. [8]  Gonchenko, S.V. and Ovsyannikov, I.I. (2013), On Global Bifurcations of Three-dimensional Diffeomorphisms Leading to Lorenz-like Attractors, Math. Model. Nat. Phenom., 8(5), 80-92.
  9. [9]  Afraimovich, V.S. and Shilnikov, L.P. (1983), in Nonlinear Dynamics and Turbulence (eds G.I.Barenblatt, G.Iooss, D.D.Joseph (Boston,Pitmen).
  10. [10]  Afraimovich, V.S. (1984), Strange attractors and quaiattractors, Nonlinear and Turbulent Processes in Physics, ed. by R.Z.Sagdeev, Gordon and Breach, Harwood Academic Publishers, 3, 1133-1138.
  11. [11]  Afraimovich, V.S., Bykov, V.V., and Shilnikov, L.P. (1980), On the existence of stable periodic motions in the Lorenz model, Sov. Math. Survey, 35, No. 4(214), 164-165.
  12. [12]  Bykov, V.V. and Shilnikov, A.L. (1989), On boundaries of existence of the Lorenz attractor, Methods of the Qualitative Theory anf Bifurcaion Theory: L.P. Shilnikov Ed., Gorky State Univ., 151-159.
  13. [13]  Shilnikov, L.P. (1994), Chua's Circuit: Rigorous result and future problems, Int.J. Bifurcation and Chaos, 4(3), 489- 519.
  14. [14]  Turaev, D.V. and Shilnikov, L.P. (1998), An example of a wild strange attractor, Sb. Math., 189(2), 137-160.
  15. [15]  Sataev, E.A. (2005), Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type, Sb. Math., 196, 561-594.
  16. [16]  Turaev, D.V. and Shilnikov, L.P. (2008), Pseudo-hyperbolisity and the problem on periodic perturbations of Lorenz-like attractors, Rus. Dokl. Math., 467, 23-27.
  17. [17]  Gonchenko, S., Ovsyannikov I., Simo, C., and Turaev, D. (2005), Three-dimensional Henon-like maps and wild Lorenz-like attractors, Int. J. of Bifurcation and chaos, 15(11), 3493-3508.
  18. [18]  Gonchenko, S.V., Gonchenko, A.S., Ovsyannikov, I.I., and Turaev, D. (2013), Examples of Lorenz-like Attractors in Henon-likeMaps, Math. Model. Nat. Phenom., 8(5), 32-54.
  19. [19]  Gonchenko, A.S. and Gonchenko, S.V. (2015), Lorenz-like attractors in a nonholonomic model of a rattleback, Nonlinearity, 28, 3403-3417.
  20. [20]  Borisov, A.V., Kazakov, A.O., and Sataev, I.R. (2014), The reversal and chaotic attractor in the nonholonomic model of Chaplygin's top, Regular and Chaotic Dynamics, 19(6), 718-733.
  21. [21]  Gonchenko, A. and Gonchenko, S. (2016), Variety of strange pseudohyperbolic attractors in three-dimensional generalized H'enon maps, Physica D, 337, 43-57.
  22. [22]  Gonchenko, A.S., Gonchenko, S.V., and Shilnikov, L. P. (2012), Towards scenarios of chaos appearance in threedimensional maps, Rus. Nonlin. Dyn. 8(1), 3-28.
  23. [23]  Gonchenko, A.S., Gonchenko, S.V., Kazakov, A.O., and Turaev, D. (2014), Simple scenarios of oncet of chaos in three-dimensional maps, Int. J. Bif. And Chaos, 24(8), 25 pages.
  24. [24]  Duarte, P. (2000), Persistent homoclinic tangencies for conservative maps near the identity, Ergod. Th. Dyn. Sys., 20, 393-438.
  25. [25]  Gonchenko, S.V. (1995), Bifurcations of two-dimensional diffeomorphisms with a nonrough homoclinic cotour, Int. Conf. on Nonlinear Dynamics, Chaotic and Complex Systems, Zakopane, Poland, 7-12 Nov., 1995.
  26. [26]  Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V. (1996), Bifurcations of two-dimensional diffeomorphisms with non-rough homoclinic contours, J.Techn.Phys., 37(3-4), 349-352.
  27. [27]  Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V. (1997), On Newhouse regions of two-dimensional diffeomorphisms close to a diffeomorphism with a nontransversal heteroclinic cycle. Proc. Steklov Inst. Math., 216, 70-118.
  28. [28]  Turaev, D.V. (1996), On dimension of nonlocal bifurcational problems, Int.J. of Bifurcation and Chaos, 6(5), 919-948.
  29. [29]  Lamb, J.S.W. and Stenkin, O.V. (2004), Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits, Nonlinearity, 17(4), 1217-1244.
  30. [30]  Delshams, A., Gonchenko, S.V., Gonchenko V.S., Lazaro, J.T., and Sten'kin, O.V. (2013), Abundance of attracting, repelling and elliptic orbits in two-dimensional reversible maps, Nonlinearity, 26(1), 1-35.
  31. [31]  Sevryuk, M. (1986), Reversible Systems, Lecture Notes in Mathematics, 1211, (Berlin, Heidelberg, New York: Springer-Verlag).
  32. [32]  Gonchenko, S.V., Lamb, J.S.W., Rios, I., and Turaev, D. (2014), Attractors and repellers near generic elliptic points of reversible maps, Doklady Mathematics, 89(1), 65-67.
  33. [33]  Delshams A., Gonchenko S.V., Gonchenko M.S. and Lázaro, J.T. (2014), Mixed dynamics of two-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies, arXiv:1412.1128v1, 18 pages (to appear in Nonlinearity).
  34. [34]  Lerman, L.M. and Turaev, D.V. (2012), Breakdown of Symmetry in Reversible Systems, Regul. Chaotic Dyn., 17(3-4), 318-336.
  35. [35]  Ruelle, D. (1981), Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys., 82, 137-151.
  36. [36]  Politi, A., Oppo, G.L., and Badii, R. (1986), Coexistence of conservative and dissipative behaviour in reversible dynamicla systems, Phys. Rev. A, 33, 4055-4060.
  37. [37]  Pikovsky, A. and Topaj, D. (2002), Reversibility vs. synchronization in oscillator latties, Physica D, 170, 118-130.
  38. [38]  Gonchenko,A.S., Gonchenko, S.V., and Kazakov, A.O. (2013), Richness of chaotic dynamics in nonholonomic models of Celtic stone, Regular and Chaotic Dynamics, 15(5), 521-538.
  39. [39]  Borisov, A.V. and Mamaev, I.S. (2001), Dynamics of rigid body, RCD-press: Moscow-Izhevsk, 384pp. (in Russian)
  40. [40]  Borisov, A.V. andMamaev, I.S. (2002), Strange attractors in celtic stone dynamics, in book "NonholonomicDynamical Systems", Moscow-Izhevsk, RChD-press, 293-316 (in Russian).
  41. [41]  Borisov, A.V. and Mamaev, I.S. (2003), Strange Attractors in Rattleback Dynamics, Physics-Uspekhi, 46(4), 393-403.