Regular and Chaotic Dynamics

, Volume 21, Issue 2, pp 160–174 | Cite as

Verification of hyperbolicity for attractors of some mechanical systems with chaotic dynamics

  • Sergey P. KuznetsovEmail author
  • Vyacheslav P. Kruglov


Computer verification of hyperbolicity is provided based on statistical analysis of the angles of intersection of stable and unstable manifolds for mechanical systems with hyperbolic attractors of Smale–Williams type: (i) a particle sliding on a plane under periodic kicks, (ii) interacting particles moving on two alternately rotating disks, and (iii) a string with parametric excitation of standing-wave patterns by a modulated pump. The examples are of interest as contributing to filling the hyperbolic theory of dynamical systems with physical content.


dynamical system chaos attractor hyperbolic dynamics Lyapunov exponent Smale–Williams solenoid parametric oscillations 

MSC2010 numbers

37D20 37D45 70G60 70Q05 


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  1. 1.
    Smale, S., Differentiable Dynamical Systems, Bull. Amer. Math. Soc., 1967, vol. 73, pp. 747–817.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Williams, R. F., Expanding Attractors, Publ. Math. Inst. Hautes Études Sci., 1974, vol. 43, pp. 169–203.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dynamical Systems 9: Dynamical Systems with Hyperbolic Behaviour, D.V.Anosov (Ed.), Encyclopaedia Math. Sci., vol. 66, Berlin: Springer, 1995.Google Scholar
  4. 4.
    Katok, A. and Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., vol. 54, Cambridge: Cambridge Univ. Press, 1995.CrossRefzbMATHGoogle Scholar
  5. 5.
    Afraimovich, V. and Hsu, S.-B., Lectures on Chaotic Dynamical Systems, AMS/IP Stud. Adv. Math., vol. 28, Providence,R.I.: AMS, 2003.zbMATHGoogle Scholar
  6. 6.
    Bonatti, Ch., Díaz, L. J., and Viana, M., Dynamics beyond Uniform Hyperbolicity: A Global Geometric and Probobalistic Perspective, Encyclopaedia Math. Sci., vol. 102, Berlin: Springer, 2005.zbMATHGoogle Scholar
  7. 7.
    Andronov, A. and Pontryagin, L., Systèmes grossiers, Dokl. Akad. Nauk. SSSR, 1937, vol. 14, no. 5, pp. 247–250 (Russian).zbMATHGoogle Scholar
  8. 8.
    Andronov, A.A., Vitt, A. A., and Khaĭkin, S.E., Theory of Oscillators, New York: Pergamon, 1966.zbMATHGoogle Scholar
  9. 9.
    Rabinovich M. I. and Gaponov-Grekhov, A.V., Problems of Present-Day Nonlinear Dynamics, Her. Russ. Acad. Sci., 1997, vol. 67, no. 4, pp. 257–262; see also: Vestn. Ross. Akad. Nauk, 1997, vol. 67, no. 7, pp. 608–614.MathSciNetGoogle Scholar
  10. 10.
    Shilnikov, L.P., Shilnikov, A. L., Turaev, D., and Chua, L.O., Methods of Qualitative Theory in Nonlinear Dynamics: Part 1, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol. 4, River Edge, N.J.: World Sci., 1998.CrossRefzbMATHGoogle Scholar
  11. 11.
    Shilnikov, L.P., Shilnikov, A. L., Turaev, D., and Chua, L.O., Methods of Qualitative Theory in Nonlinear Dynamics: Part 2, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol. 5, River Edge,N.J.: World Sci., 2001.zbMATHGoogle Scholar
  12. 12.
    Afraimovich, V. S., Gonchenko, S.V., Lerman, L.M., Shilnikov, A. L., and Turaev, D. V., Scientific Heritage of L.P. Shilnikov, Regul. Chaotic Dyn., 2014, vol. 19, no. 4, pp. 435–460.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kuznetsov, A.P., Kuznetsov, S.P., and Ryskin, N. M., Nonlinear Oscillations, Moscow: Fizmatlit, 2002 (Russian).zbMATHGoogle Scholar
  14. 14.
    Borisov, A.V., Kazakov, A.O., and Kuznetsov, S.P., Nonlinear Dynamics of the Rattleback: A Nonholonomic Model, Physics–Uspekhi, 2014, vol. 57, no. 5, pp. 453–460; see also: Uspekhi Fiz. Nauk, 2014, vol. 184, no. 5, pp. 493–500.Google Scholar
  15. 15.
    Gonchenko, A. S., Gonchenko, S.V., and Kazakov, A.O., Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 521–538.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kuznetsov, S.P., Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finite-Dimensional Models, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 345–382.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Monin, A. S., On the Nature of Turbulence, Sov. Phys. Usp., 1978, vol. 21, no. 5, pp. 429–442; see also: Uspekhi Fiz. Nauk, 1978, vol. 125, no. 5, pp. 97–122.CrossRefGoogle Scholar
  18. 18.
    Letellier, Ch., Chaos in Nature, Singapore: World Sci., 2013.CrossRefzbMATHGoogle Scholar
  19. 19.
    Scott, S. K., Chemical Chaos, Oxford: Oxford Univ. Press, 1993.Google Scholar
  20. 20.
    Thompson, J.M.T. and Stewart, H. B., Nonlinear Dynamics and Chaos, New York: Wiley, 1986.zbMATHGoogle Scholar
  21. 21.
    Landa, P. S., Nonlinear Oscillations and Waves in Dynamical Systems, Dordrecht: Springer, 1996.CrossRefzbMATHGoogle Scholar
  22. 22.
    Shannon, C.E., A Mathematical Theory of Communication, Bell Syst. Tech. J., 1948, vol. 27, pp. 379–423, 623–656.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kuznetsov, S.P., Example of a Physical System with a Hyperbolic Attractor of the Smale–Williams Type, Phys. Rev. Lett., 2005, vol. 95, no. 14, 144101, 4 pp.CrossRefGoogle Scholar
  24. 24.
    Kuznetsov, S.P. and Seleznev, E.P., Strange Attractor of Smale–Williams Type in the Chaotic Dynamics of a Physical System, J. Exp. Theor. Phys., 2006, vol. 102, no. 2, pp. 355–364; see also: Zh. Èksper. Teoret. Fiz., 2006, vol. 129, no. 2, pp. 400–412.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Isaeva, O.B., Jalnine, A.Yu., and Kuznetsov, S.P., Arnold’s Cat Map Dynamics in a System of Coupled Nonautonomous van der Pol Oscillators, Phys. Rev. E, 2006, vol. 74, no. 4, 046207, 5 pp.CrossRefGoogle Scholar
  26. 26.
    Kuznetsov, S.P. and Pikovsky, A., Autonomous Coupled Oscillators with Hyperbolic Strange Attractors, Phys. D, 2007, vol. 232, no. 2, pp. 87–102.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kuznetsov, S.P., Example of Blue Sky Catastrophe Accompanied by a Birth of Smale–Williams Attractor, Regul. Chaotic Dyn., 2010, vol. 15, nos. 2-3, pp. 348–353.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kuznetsov, S.P., Hyperbolic Chaos: A Physicist’s View, Berlin: Springer, 2012.CrossRefzbMATHGoogle Scholar
  29. 29.
    Kuznetsov, S.P., Dynamical Chaos and Uniformly Hyperbolic Attractors: From Mathematics to Physics, Phys. Uspekhi, 2011, vol. 54, no. 2, pp. 119–144; see also: Uspekhi Fiz. Nauk, 2011, vol. 181, no. 2, pp. 121–149.CrossRefGoogle Scholar
  30. 30.
    Kuznetsov, S.P., Plykin Type Attractor in Electronic Device Simulated in MULTISIM, Chaos, 2011, vol. 21, no. 4, 043105, 10 pp.CrossRefzbMATHGoogle Scholar
  31. 31.
    Isaeva, O.B., Kuznetsov, S.P., and Mosekilde, E., Hyperbolic Chaotic Attractor in Amplitude Dynamics of Coupled Self-Oscillators with Periodic Parameter Modulation, Phys. Rev. E., 2011, vol. 84, no. 1, 016228, 10 pp.CrossRefGoogle Scholar
  32. 32.
    Isaeva, O.B., Kuznetsov, S.P., Sataev, I.R., Savin, D.V., and Seleznev, E.P., Hyperbolic Chaos and Other Phenomena of Complex Dynamics Depending on Parameters in a Nonautonomous System of Two Alternately Activated Oscillators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2015, vol. 25, no. 12, 1530033, 15 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kuznetsov, S.P., Ponomarenko, V. I., and Seleznev, E.P., Autonomous System Generating Hyperbolic Chaos: Circuit Simulation and Experiment, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 2013, vol. 21, no. 5, pp. 17–30 (Russian).Google Scholar
  34. 34.
    Jalnine, A.Yu., Hyperbolic and Non-Hyperbolic Chaos in a pair of Coupled Alternately Excited FitzHugh–Nagumo Systems, Commun. Nonlinear Sci. Numer. Simul., 2015, vol. 23, nos. 1–3, pp. 202–208.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kuznetsov, S.P., Some Mechanical Systems Manifesting Robust Chaos, Nonlinear Dynamics & Mobile Robotics, 2013, vol. 1, no. 1, pp. 3–22.Google Scholar
  36. 36.
    Kuznetsov, S.P., Hyperbolic Strange Attractors of Physically Realizable Systems, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 2009, vol. 17, no. 4, pp. 5–34 (Russian).Google Scholar
  37. 37.
    Hénon, M., A Two-Dimensional Mapping with a Strange Attractor, Commun. Math. Phys., 1976, vol. 50, no. 1, pp. 69–77.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Lai, Y.-Ch., Grebogi, C., Yorke, J. A., and Kan, I., How Often Are Chaotic Saddles Nonhyperbolic?, Nonlinearity, 1993, vol. 6, no. 5, pp. 779–798.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Anishchenko, V. S., Kopeikin, A. S., Kurths, J., Vadivasova, T.E., Strelkova, G. I., Studying Hyperbolicity in Chaotic Systems, Phys. Lett. A, 2000, vol. 270, no. 6, pp. 301–307.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ginelli, F., Poggi, P., Turchi, A., Chaté, H., Livi, R., and Politi, A., Characterizing Dynamics with Covariant Lyapunov Vectors, Phys. Rev. Lett., 2007, vol. 99, no. 13, 130601, 4 pp.CrossRefGoogle Scholar
  41. 41.
    Kuptsov, P.V., Fast Numerical Test of Hyperbolic Chaos, Phys. Rev. E, 2012, vol. 85, no. 1, 015203, 4 pp.MathSciNetCrossRefGoogle Scholar
  42. 42.
    Kuznetsov, S.P., Hyperbolic Chaos in Self-Oscillating Systems Based on Mechanical Triple Linkage: Testing Absence of Tangencies of Stable and Unstable Manifolds for Phase Trajectories, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 649–666.MathSciNetCrossRefGoogle Scholar
  43. 43.
    Kuznetsov, S.P. and Turukina, L. V., Attractors of Smale–Williams Type in Periodically Kicked Model Systems, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 2010, vol. 18, no. 5, pp. 80–92 (Russian).zbMATHGoogle Scholar
  44. 44.
    Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M., Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them: P. 1: Theory; P. 2: Numerical Application, Meccanica, 1980, vol. 15, pp. 9–30.CrossRefzbMATHGoogle Scholar
  45. 45.
    Schuster, H.G. and Just, W., Deterministic Chaos: An Introduction, Weinheim: Wiley-VCH, 2005.CrossRefzbMATHGoogle Scholar
  46. 46.
    Kuznetsov, S.P., Dynamical Chaos, 2nd ed., Moscow: Fizmatlit, 2006 (Russian).Google Scholar
  47. 47.
    Rayleigh, J.W. S., The Theory of Sound, Volume One, 2nd ed., New York: Dover, 2013.zbMATHGoogle Scholar
  48. 48.
    Rowland, D.R., Parametric Resonance and Nonlinear String Vibrations, Am. J. Phys., 2004, vol. 72, no. 6, pp. 758–766.CrossRefGoogle Scholar
  49. 49.
    Isaeva, O.B., Kuznetsov, A. S., and Kuznetsov, S.P., Hyperbolic Chaos of Standing Wave Patterns Generated Parametrically by a Modulated Pump Source, Phys. Rev. E, 2013, vol. 87, no. 4, 040901(R), 4 pp.CrossRefGoogle Scholar
  50. 50.
    Isaeva, O.B., Kuznetsov, A. S., and Kuznetsov, S. P., Hyperbolic Chaos in Parametric Oscillations of a String, Nelin. Dinam., 2013, vol. 9, no. 1, pp. 3–10 (Russian).CrossRefGoogle Scholar
  51. 51.
    Kuznetsov, S.P., Kuznetsov, A. S., and Kruglov, V. P., Hyperbolic Chaos in Systems with Parametrically Excited Patterns of Standing Waves, Nelin. Dinam., 2014, vol. 10, no. 3, pp. 265–277 (Russian).CrossRefzbMATHGoogle Scholar
  52. 52.
    Sinaĭ, Ya.G., The Stochasticity of Dynamical Systems: Selected Translations, Selecta Math. Soviet., 1981, vol. 1, no. 1, pp. 100–119.MathSciNetGoogle Scholar
  53. 53.
    Kuznetsov, S.P. and Sataev, I.R., Hyperbolic Attractor in a System of Coupled Non-Autonomous van derPol Oscillators: Numerical Test for Expanding and Contracting Cones, Phys. Lett. A, 2007, vol. 365, nos. 1-2, pp. 97–104.CrossRefzbMATHGoogle Scholar
  54. 54.
    Wilczak, D., Uniformly Hyperbolic Attractor of the Smale–Williams Type for a Poincaré Map in the Kuznetsov System: With Online Multimedia Enhancements, SIAM J. Appl. Dyn. Syst., 2010, vol. 9, no. 4, pp. 1263–1283.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Sergey P. Kuznetsov
    • 1
    • 2
    • 3
    Email author
  • Vyacheslav P. Kruglov
    • 2
    • 3
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Kotelnikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov BranchSaratovRussia
  3. 3.Saratov State UniversitySaratovRussia

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