On some simple examples of mechanical systems with hyperbolic chaos

  • S. P. KuznetsovEmail author
  • V. P. Kruglov


Examples of mechanical systems with hyperbolic chaos are discussed, including the Thurston–Weeks–Hunt–MacKay hinge mechanism, in which conservative Anosov dynamics is realized, and dissipative systems with Smale–Williams type attractors (a particle on a plane under periodic kicks, interacting particles sliding on two alternately rotating disks, and a string with parametric excitation by modulated pump). The examples considered in the paper are interesting from the viewpoint of filling hyperbolic theory, as a well-developed field of the mathematical theory of dynamical systems, with physical content. The results of computer tests for hyperbolicity of the systems are presented that are based on the analysis of the statistics of intersection angles of stable and unstable manifolds.


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  1. 1.
    V. S. Afraimovich, S. V. Gonchenko, L. M. Lerman, A. L. Shilnikov, and D. V. Turaev, “Scientific heritage of L. P. Shilnikov,” Regul. Chaotic Dyn. 19 (4), 435–460 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    V. Afraimovich and S.-B. Hsu, Lectures on Chaotic Dynamical Systems (International Press, Somerville, MA, 2003).CrossRefzbMATHGoogle Scholar
  3. 3.
    A. D. Aleksandrov and N. Yu. Netsvetaev, Geometry (Nauka, Moscow, 1990) [in Russian].zbMATHGoogle Scholar
  4. 4.
    A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators (Fizmatgiz, Moscow, 1959; Pergamon, Oxford, 1966).zbMATHGoogle Scholar
  5. 5.
    V. S. Anishchenko, A. S. Kopeikin, J. Kurths, T. E. Vadivasova, and G. I. Strelkova, “Studying hyperbolicity in chaotic systems,” Phys. Lett. A 270 (6), 301–307 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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 (1969)].zbMATHGoogle Scholar
  7. 7.
    D. V. Anosov, “Dynamical systems in the 1960s: The hyperbolic revolution,” in Mathematical Events of the Twentieth Century (Springer, Berlin, 2006), pp. 1–17.Google Scholar
  8. 8.
    D. V. Anosov, S. Kh. Aranson, V. Z. Grines, R. V. Plykin, E. A. Sataev, A. V. Safonov, V. V. Solodov, A. N. Starkov, A. M. Stepin, and S. V. Shlyachkov, Dynamical Systems with Hyperbolic Behaviour (VINITI, Moscow, 1991), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 66: Dynamical Systems–9; Engl. transl. in Dynamical Systems IX (Springer, Berlin,1995), Encycl. Math. Sci. 66.zbMATHGoogle Scholar
  9. 9.
    N. L. Balazs and A. Voros, “Chaos on the pseudosphere,” Phys. Rep. 143 (3), 109–240 (1986).MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. S. Baptista, “Cryptography with chaos,” Phys. Lett. A 240 (1–2), 50–54 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Parts 1, 2,” Meccanica 15, 9–20, 21–30 (1980).CrossRefzbMATHGoogle Scholar
  12. 12.
    G. M. Bernstein and M. A. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans. Circuits Syst. 37 (9), 1157–1164 (1990).MathSciNetCrossRefGoogle Scholar
  13. 13.
    I. A. Bizyaev, A. V. Borisov, and A. O. Kazakov, “Dynamics of the Suslov problem in a gravitational field: Reversal and strange attractors,” Regul. Chaotic Dyn. 20 (5), 605–626 (2015) [Nelinein. Din. 12 (2), 263–287 (2016)].MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    E. M. Bollt and J. D. Meiss, “Targeting chaotic orbits to the Moon through recurrence,” Phys. Lett. A 204 (5–6), 373–378 (1995).CrossRefGoogle Scholar
  15. 15.
    C. Bonatti, L. J. Díaz, and M. Viana, Dynamics beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective (Springer, Berlin, 2005).zbMATHGoogle Scholar
  16. 16.
    A. V. Borisov, A. O. Kazakov, and S. P. Kuznetsov, “Nonlinear dynamics of the rattleback: A nonholonomic model,” Usp. Fiz. Nauk 184 (5), 493–500 (2014) [Phys. Usp. 57, 453–460 (2014)].CrossRefGoogle Scholar
  17. 17.
    A. V. Borisov and I. S. Mamaev, “Strange attractors in rattleback dynamics,” Usp. Fiz. Nauk 173 (4), 407–418 (2003) [Phys. Usp. 46, 393–403 (2003)].CrossRefGoogle Scholar
  18. 18.
    A. S. Dmitriev, E. V. Efremova, N. A. Maksimov, and A. I. Panas, Generation of Chaos (Tekhnosfera, Moscow, 2012) [in Russian].Google Scholar
  19. 19.
    A. S. Dmitriev, E. V. Efremova, A. Yu. Nikishov, and A. I. Panas, “Chaotic generators: From vacuum devices to nanoschems,” Radioelektron., Nanosist., Inf. Tekhnol. 1 (1–2), 6–22 (2009).Google Scholar
  20. 20.
    A. S. Dmitriev and A. I. Panas, Dynamical Chaos: New Information Carriers for Communication Systems (Fizmatlit, Moscow, 2002) [in Russian].Google Scholar
  21. 21.
    M. Drutarovsk´y and P. Galajda, “A robust chaos-based true random number generator embedded in reconfigurable switched-capacitor hardware,” Radioengineering 16 (3), 120–127 (2007).Google Scholar
  22. 22.
    J. D. Farmer, “Chaotic attractors of an infinite-dimensional dynamical system,” Physica D 4 (3), 366–393 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    F. R. Gantmakher, Lectures in Analytical Mechanics, 2nd ed. (Nauka, Moscow, 1966; Mir, Moscow, 1970); 3rd ed. (Fizmatlit, Moscow,2005) [in Russian].zbMATHGoogle Scholar
  24. 24.
    A. V. Gaponov-Grekhov and M. I. Rabinovich, “Problems of present-day nonlinear dynamics,” Vestn. Ross. Akad. Nauk 67 (7), 608–614 (1997) [Herald Russ. Acad. Sci. 67 (4), 257–262 (1997)].Google Scholar
  25. 25.
    F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, and A. Politi, “Characterizing dynamics with covariant Lyapunov vectors,” Phys. Rev. Lett. 99 (13), 130601 (2007).CrossRefGoogle Scholar
  26. 26.
    H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd ed. (Addison-Wesley, San Francisco, 2001).zbMATHGoogle Scholar
  27. 27.
    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).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    H. Gritli, N. Khraief, and S. Belghith, “Chaos control in passive walking dynamics of a compass-gait model,” Commun. Nonlinear Sci. Numer. Simul. 18 (8), 2048–2065 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Handbook of Chaos Control, Ed. by E. Schöll and H. G. Schuster (Wiley-VCH, Weinheim, 2008).Google Scholar
  30. 30.
    T. J. Hunt and R. S. MacKay, “Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor,” Nonlinearity 16 (4), 1499–1510 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    O. B. Isaeva, A. Yu. Jalnine, and S. P. Kuznetsov, “Arnold’s cat map dynamics in a system of coupled nonautonomous van der Pol oscillators,” Phys. Rev. E 74 (4), 046207 (2006).CrossRefGoogle Scholar
  32. 32.
    O. B. Isaeva, A. S. Kuznetsov, and S. P. Kuznetsov, “Hyperbolic chaos of standing wave patterns generated parametrically by a modulated pump source,” Phys. Rev. E 87 (4), 040901 (2013).CrossRefGoogle Scholar
  33. 33.
    O. B. Isaeva, A. S. Kuznetsov, and S. P. Kuznetsov, “Hyperbolic chaos in parametric oscillations of a string,” Nelinein. Din. 9 (1), 3–10 (2013).CrossRefGoogle Scholar
  34. 34.
    A. Yu. Jalnine, “A new information transfer scheme based on phase modulation of a carrier chaotic signal,” Izv. Vyssh. Uchebn. Zaved., Prikl. Nelinein. Din. 22 (5), 3–12 (2014).Google Scholar
  35. 35.
    A. Yu. Jalnine, “Hyperbolic and non-hyperbolic chaos in a pair of coupled alternately excited FitzHugh–Nagumo systems,” Commun. Nonlinear Sci. Numer. Simul. 23 (1–3), 202–208 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1978) [in Russian].Google Scholar
  37. 37.
    J. L. Kaplan and J. A. Yorke, “Chaotic behavior of multidimensional difference equations,” in Functional Differential Equations and Approximation of Fixed Points, Ed. by H.-O. Peitgen and H.-O. Walther (Springer, Berlin, 1979), Lect. Notes Math. 730, pp. 204–227.Google Scholar
  38. 38.
    A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge Univ. Press, Cambridge, 1995).CrossRefzbMATHGoogle Scholar
  39. 39.
    M. E. Kazaryan, A Course of Differential Geometry (2001–2002) (MTsNMO, Moscow, 2002) [in Russian].Google Scholar
  40. 40.
    A. A. Koronovskii, O. I. Moskalenko, and A. E. Hramov, “On the use of chaotic synchronization for secure communication,” Usp. Fiz. Nauk 179 (12), 1281–1310 (2009) [Phys. Usp. 52, 1213–1238 (2009)].CrossRefGoogle Scholar
  41. 41.
    M. Kourganoff, “Anosov geodesic flows, billiards and linkages,” arXiv: 1503.04305 [math.DS].Google Scholar
  42. 42.
    V. V. Kozlov, “Topological obstructions to the integrability of natural mechanical systems,” Dokl. Akad. Nauk SSSR 249 (6), 1299–1302 (1979) [Sov. Math., Dokl. 20, 1413–1415 (1979)].MathSciNetzbMATHGoogle Scholar
  43. 43.
    V. P. Kruglov, A. S. Kuznetsov, and S. P. Kuznetsov, “Hyperbolic chaos in systems with parametrically excited patterns of standing waves,” Nelinein. Din. 10 (3), 265–277 (2014).zbMATHGoogle Scholar
  44. 44.
    V. P. Kruglov, S. P. Kuznetsov, and A. Pikovsky, “Attractor of Smale–Williams type in an autonomous distributed system,” Regul. Chaotic Dyn. 19 (4), 483–494 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    P. V. Kuptsov, “Computation of Lyapunov exponents for spatially extended systems: Advantages and limitations of various numerical methods,” Izv. Vyssh. Uchebn. Zaved., Prikl. Nelinein. Din. 18 (5), 93–112 (2010).zbMATHGoogle Scholar
  46. 46.
    P. V. Kuptsov, “Fast numerical test of hyperbolic chaos,” Phys. Rev. E 85 (1), 015203 (2012).CrossRefGoogle Scholar
  47. 47.
    P. V. Kuptsov, S. P. Kuznetsov, and A. Pikovsky, “Hyperbolic chaos of Turing patterns,” Phys. Rev. Lett. 108 (19), 194101 (2012).CrossRefGoogle Scholar
  48. 48.
    A. P. Kuznetsov, S. P. Kuznetsov, and N. M. Ryskin, Nonlinear Oscillations (Fizmatlit, Moscow, 2002) [in Russian].zbMATHGoogle Scholar
  49. 49.
    A. P. Kuznetsov, N. A. Migunova, I. R. Sataev, Yu. V. Sedova, and L. V. Turukina, “From chaos to quasiperiodicity,” Regul. Chaotic Dyn. 20 (2), 189–204 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    S. P. Kuznetsov, Dynamic Chaos (Fizmatlit, Moscow, 2001) [in Russian].Google Scholar
  51. 51.
    S. P. Kuznetsov, “Example of a physical system with a hyperbolic attractor of the Smale–Williams type,” Phys. Rev. Lett. 95 (14), 144101 (2005).CrossRefGoogle Scholar
  52. 52.
    S. P. Kuznetsov, “Dynamical chaos and uniformly hyperbolic attractors: From mathematics to physics,” Usp. Fiz. Nauk 181 (2), 121–149 (2011) [Phys. Usp. 54, 119–144 (2011)].CrossRefGoogle Scholar
  53. 53.
    S. P. Kuznetsov, “Plykin type attractor in electronic device simulated in MULTISIM,” Chaos 21 (4), 043105 (2011).CrossRefzbMATHGoogle Scholar
  54. 54.
    S. P. Kuznetsov, Hyperbolic Chaos: A Physicist’s View (Springer, Berlin, 2012).CrossRefzbMATHGoogle Scholar
  55. 55.
    S. P. Kuznetsov, Dynamic Chaos and Hyperbolic Attractors: From Mathematics to Physics (Inst. Komp’yut. Issled., Moscow, 2013) [in Russian].Google Scholar
  56. 56.
    S. P. Kuznetsov, “Some mechanical systems manifesting robust chaos,” Nonlinear Dyn. Mob. Rob. 1 (1), 3–22 (2013).MathSciNetGoogle Scholar
  57. 57.
    S. P. Kuznetsov, “Plate falling in a fluid: Regular and chaotic dynamics of finite-dimensional models,” Nelinein. Din. 11 (1), 3–49 (2015) [Regul. Chaotic Dyn. 20 (3), 345–382 (2015)].MathSciNetCrossRefGoogle Scholar
  58. 58.
    S. P. Kuznetsov, “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. 20 (6), 649–666 (2015) [Nelinein. Din. 12 (1), 121–143 (2016)].MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    S. P. Kuznetsov, “Chaos in a system of three coupled rotators: From Anosov’s dynamics to a hyperbolic attractor,” Izv. Saratov Univ., Nov. Ser., Fiz. 15 (2), 5–17 (2015).Google Scholar
  60. 60.
    S. P. Kuznetsov, “From geodesic flow on a surface of negative curvature to electronic generator of robust chaos,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 26 (14), 1650232 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    S. P. Kuznetsov and A. Pikovsky, “Autonomous coupled oscillators with hyperbolic strange attractors,” Physica D 232 (2), 87–102 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    S. P. Kuznetsov and V. I. Ponomarenko, “Realization of a strange attractor of the Smale–Williams type in a radiotechnical delay-feedback oscillator,” Pisma Zh. Tekh. Fiz. 34 (18), 1–8 (2008) [Tech. Phys. Lett. 34 (9), 771–773 (2008)].Google Scholar
  63. 63.
    S. P. Kuznetsov and I. R. Sataev, “Verification of hyperbolicity conditions for a chaotic attractor in a system of coupled nonautonomous van der Pol oscillators,” Izv. Vyssh. Uchebn. Zaved., Prikl. Nelinein. Din. 14 (5), 3–29 (2006).zbMATHGoogle Scholar
  64. 64.
    S. P. Kuznetsov and I. R. Sataev, “Hyperbolic attractor in a system of coupled non-autonomous van der Pol oscillators: Numerical test for expanding and contracting cones,” Phys. Lett. A 365 (1–2), 97–104 (2007).CrossRefzbMATHGoogle Scholar
  65. 65.
    S. P. Kuznetsov and E. P. Seleznev, “A strange attractor of the Smale–Williams type in the chaotic dynamics of a physical system,” Zh. Eksp. Teor. Fiz. 129 (2), 400–412 (2006) [J. Exp. Theor. Phys. 102 (2), 355–364 (2006)].MathSciNetGoogle Scholar
  66. 66.
    S. P. Kuznetsov and L. V. Turukina, “Attractors of Smale–Williams type in periodically kicked model systems,” Izv. Vyssh. Uchebn. Zaved., Prikl. Nelinein. Din. 18 (5), 80–92 (2010).zbMATHGoogle Scholar
  67. 67.
    Y.-C. Lai, C. Grebogi, J. A. Yorke, and I. Kan, “How often are chaotic saddles nonhyperbolic?,” Nonlinearity 6 (5), 779–798 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    C. Letellier, Chaos in Nature (World Sci., Hackensack, NJ, 2013).CrossRefzbMATHGoogle Scholar
  69. 69.
    K. A. Lukin, “Noise radar technology,” Telecommun. Radio Eng. 55 (12), 8–16 (2001).CrossRefGoogle Scholar
  70. 70.
    M. L. S. Magalhaes and M. Pollicott, “Geometry and dynamics of planar linkages,” Commun. Math. Phys. 317 (3), 615–634 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    L. I. Mandelstam, Complete Collection of Works, Vol. 4: Lectures on Oscillations (1930–1932) (Akad. Nauk SSSR, Moscow, 1955) [in Russian].Google Scholar
  72. 72.
    A. S. Monin, “On the nature of turbulence,” Usp. Fiz. Nauk 125 (5), 97–122 (1978) [Sov. Phys. Usp. 21, 429–442 (1978)].MathSciNetCrossRefGoogle Scholar
  73. 73.
    Yu. I. Neimark and P. S. Landa, Stochastic and Chaotic Oscillations (Nauka, Moscow, 1987; Kluwer, Dordrecht, 1992).CrossRefzbMATHGoogle Scholar
  74. 74.
    Ya. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity (Eur. Math. Soc., Zürich, 2004), Zurich Lect. Adv. Math.CrossRefzbMATHGoogle Scholar
  75. 75.
    N. Ptitsyn, Application of the Theory of Deterministic Chaos to Cryptography (Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Moscow, 2002) [in Russian].Google Scholar
  76. 76.
    D. R. Rowland, “Parametric resonance and nonlinear string vibrations,” Am. J. Phys. 72 (6), 758–766 (2004).CrossRefGoogle Scholar
  77. 77.
    H. G. Schuster and W. Just, Deterministic Chaos: An Introduction (Wiley-VCH, Weinheim, 2005).CrossRefzbMATHGoogle Scholar
  78. 78.
    S. K. Scott, Chemical Chaos (Clarendon, Oxford, 1993).Google Scholar
  79. 79.
    L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics (World Sci., Singapore, 1998, 2001; Inst. Komp’yut. Issled., Moscow, 2004, 2009), Parts 1, 2.zbMATHGoogle Scholar
  80. 80.
    Ya. G. Sinai, “Stochasticity of dynamical systems,” in Nonlinear Waves, Ed. by A. V. Gaponov-Grekhov (Nauka, Moscow, 1979), pp. 192–212 [in Russian].Google Scholar
  81. 81.
    S. Smale, “Differentiable dynamical systems,” Bull. Am. Math. Soc. 73 (6), 747–817 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    S. Steingrube, M. Timme, F. Wörgötter, and P. Manoonpong, “Self-organized adaptation of a simple neural circuit enables complex robot behaviour,” Nature Phys. 6 (3), 224–230 (2010).CrossRefGoogle Scholar
  83. 83.
    D. J. Struik, Lectures on Classical Differential Geometry (Dover Publ., New York, 1988).zbMATHGoogle Scholar
  84. 84.
    J. W. Strutt, The Theory of Sound (Macmillan, London, 1894), Vol. 1.Google Scholar
  85. 85.
    J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos (J. Wiley & Sons, Chichester, 1986).zbMATHGoogle Scholar
  86. 86.
    W. P. Thurston and J. R. Weeks, “The mathematics of three-dimensional manifolds,” Sci. Am. 251 (1), 108–120 (1984).CrossRefGoogle Scholar
  87. 87.
    L. V. Turukina, “Hyperbolic chaos in systems with periodic impulse kicks,” Nelinein. Mir 8 (2), 72–73 (2010).Google Scholar
  88. 88.
    D. Wilczak, “Uniformly hyperbolic attractor of the Smale–Williams type for a Poincaré map in the Kuznetsov system,” SIAM J. Appl. Dyn. Syst. 9 (4), 1263–1283 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  89. 89.
    R. F. Williams, “Expanding attractors,” Publ. Math., Inst. Hautes étud. Sci. 43, 169–203 (1974).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Saratov Branch of the Kotel’nikov Institute of Radio Engineering and ElectronicsRussian Academy of SciencesSaratovRussia
  3. 3.Saratov State UniversitySaratovRussia

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