ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Kuznetsov S. P. Hyperbolic strange attractors of physically realizable systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 4, pp. 5-34. DOI: 10.18500/0869-6632-2009-17-4-5-34

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 175)
Article type: 

Hyperbolic strange attractors of physically realizable systems

Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences

A review of studies aimed on revealing or constructing physical systems with hyperbolic strange attractors, like Plykin attractor and Smale–Williams solenoid, is presented. Examples of iterated maps, differential equations, and simple electronic devices with chaotic dynamics associated with such attractors are presented and discussed. A general principle is considered and illustrated basing on manipulation of phases in alternately excited oscillators and time-delay systems. Alternative approaches are reviewed outlined in literature, as well as the prospects of further researches.

  1. Smale S. Differentiable dynamical systems. Bull. Amer. Math. Soc. (NS). 1967;73(6):747–817. DOI: 10.1090/S0002-9904-1967-11798-1.
  2. Williams RF. Expanding attractors. Publications mathematiques de l’I.H.E.S. 1974;43(1):169–203.
  3. Sinai YG. Stochasticity of dynamic systems. Nonlinear waves. Ed. Gapanov-Grehov AV. Moscow: Nauka; 1979. 361 p. (In Russian).
  4. Modern problems of mathematics. Fundamental directions. Results of Science and Technology, V. 2 Ed. Gamkrelidze RV. Moscow: VINITI; 1985. 310 p. (In Russian).
  5. Eckmann J-P, Ruelle D. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 1985;57(3):617–656. DOI: 10.1103/RevModPhys.57.617.
  6. Shilnikov L. Mathematical problems of nonlinear dynamics: a tutorial. Int. J. of Bifurcation and Chaos. 1997;7(9):1353–2001. DOI: 10.1016/S0016-0032(97)00039-2.
  7. Katok AB, Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems. Moscow: Faktorial; 1999. 768 p. (In Russian).
  8. Hookenheimer J, Holmes F. Nonlinear oscillations, dynamic systems, and vector field bifurcations. Moscow-Izhevsk: Institute of Computer Sciences; 2002. 560 p. (In Russian).
  9. Afraimovich V, Hsu S-B. Lectures on chaotic dynamical systems, AMS/IP Studies in Advanced Mathematics, 28, American Mathematical Society, Providence, RI; International Press, Somerville, MA; 2003. 353 p.
  10. Katok AB, Hasselblatt B. Introduction to the theory of dynamic systems with an overview of recent achievements. Moscow: MCCME; 2005. 464 p. (In Russian).
  11. Anishchenko VS, Astakhov VV, Vadivasova TE, Neiman AB, Strelkova GI, Schimansky-Geier L. Nonlinear Effects in Chaotic and Stochastic Systems. Izhevsk-Moscow: Institute of Computer Sciences; 2003. (In Russian).
  12. Barreira L, Pesin Y. Lectures on Lyapunov exponents and smooth ergodic theory. In book: «Smooth Ergodic Theory and Its Applications», AMS, Proceedings of Symposia in Pure Mathematics, 2001. 147 p.
  13. Bonatti C, Diaz LJ, Viana M. Dynamics Beyond Uniform Hyperbolicity. A global geometric and probobalistic perspective. Encyclopedia of Mathematical Sciences. Vol. 102. Berlin, Heidelberg, New-York: Springer; 2005. 384 p.
  14. Benedicks M, Carleson L. The dynamics of the Henon map. Ann. of Math. 1991;133(2):73–169. DOI: 10.2307/2944326.
  15. Halbert J.T., Yorke J.A. Modeling a chaotic machine’s dynamics as a linear map on a «square sphere». Topology Proceedings. 2014;44:257–284.
  16. Sinai JG, Vul EB. Hyperbolicity conditions for the Lorenz model. Physica D2. 1981;2(1):3–7.
  17. Kuznetsov AP, Sataev IR. Verification of hyperbolicity conditions for a chaotic attractor in a system of coupled nonautonomous van der Pol oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(5):3–29. DOI: 10.18500/0869-6632-2006-14-5-3-29.
  18. Kuznetsov SP. Plykin-type attractor in nonautonomous coupled oscillators. CHAOS. 2009;19(1):013114. DOI: 10.1063/1.3072777.
  19. Kuznetsov SP, Sataev IR. Hyperbolic attractor in a system of coupled non-auto-nomous van der Pol oscillators: Numerical test for expanding and contracting cones. Physics Letters. 2007;365(1-2):97–104. DOI: 10.1016/j.physleta.2006.12.071.
  20. Kuznetsov SP. An example of a non-autonomous continuous-time system with attractor of Plykin type in the Poincaré map. Nelin. Dinam. 2009;5(3):403–424.
  21. Newhouse SE. Lectures on dynamical systems. In Dynamical Systems C.I.M.E. Lectures Bressanone. Progress in Mathematics, No 8. Boston: Birkhauser–Boston; 1978. 1–114. DOI: 10.1007/978-3-642-13929-1_5.
  22. Hunt TJ. Low Dimensional Dynamics: Bifurcations of Cantori and Realisations of Uniform Hyperbolicity. PhD Thesis. Cambridge: Univercity of Cambridge; 2000. 121 p.
  23. Ahiezer NI. Elements of the theory of elliptic functions. Moscow: Nauka; 1970. 304 p. (In Russian).
  24. Ajdarova JS, Kuznetsov SP. Chaotic dynamics of Hunt model – artificially constructed flow system with a hyperbolic attractor. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(3):176–196. DOI: 10.18500/0869-6632-2008-16-3-176-196.
  25. Kuznetsov SP. A non-autonomous flow system with Plykin type attractor. Communications in Nonlinear Science and Numerical Simulation. 2009;14(9-10):3487–3491. DOI: 10.1016/J.CNSNS.2009.02.002.
  26. Kuznetsov SP. Example of a physical system with a hyperbolic attractor of the Smale–Williams type. Phys. Rev. Lett. 2005;95(14):144101. DOI: 10.1103/PhysRevLett.95.144101.
  27. Kuznetsov SP, Isaeva OB, Osbaldestin A. Complex analytic dynamics phenomena in a system of coupled nonautonomous oscillators with alternative excitation. Technical Physics Letters. 2007;33(9):748–751. DOI: 10.1134/S1063785007090106.
  28. Jalnine AYu, Kuznetsov SP. On the realization of the Hunt-Ott strange nonchaotic attractor in a physical system. Technical Physics. 2007;52(4):401–408. DOI: 10.1134/S1063784207040020.
  29. Kuznetsov SP, Pikovsky A. Autonomous coupled oscillators with hyperbolic strange attractors. Physica. 2007;232(2):87–102. DOI: 10.1016/j.physd.2007.05.008.
  30. Kuznetsov SP. On the feasibility of a parametric generator of hyperbolic chaos. Journal of Experimental and Theoretical Physics. 2008;106(2):380–387. DOI: 10.1007/s11447-008-2016-x.
  31. Kuznetsov AP, Kuznetsov SP, Pikovsky AS, Turukina LV. Chaotic dynamics in the systems of coupling nonautonomous oscillators with resonance and nonresonance communicator of the signal. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007;15(6):75–85. DOI: 10.18500/0869-6632-2007-15-6-75-85.
  32. Kuznetsov SP, Pikovsky AS, Sataev IR. Hyperbolic Smale–Williams attractor in Poincare map of a four-dimensional autonomous system. Proc. of the III Int. Conf. «Frontiers of Nonlinear Physics». Nizhny Novgorod–Saratov–Nizhny Novgorod, July 3–9; 2007. 66–67.
  33. Kuznetsov SP, Ponomarenko VI. Realization of a strange attractor of the Smale-Williams type in a radiotechnical delay-feedback oscillator. Technical Physics Letters. 2008;34(9):771–773. DOI: 10.1134/S1063785008090162.
  34. Kuznetsov SP, Pikovsky AS. Hyperbolic chaos in the phase dynamics of a Q-switched oscillator with delayed nonlinear feedbacks. Europhysics Letters. 2008;84(1):10013. DOI: 10.1209/0295-5075/84/10013.
  35. Baranov SV, Kuznetsov SP, Ponomarenko VI. Chaos in the phase dynamics of q-switched van der Pol oscillator with additional delayed feedback loop. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(1):12–23. DOI: 10.18500/0869-6632-2010-18-1-12-23.
  36. Kuznetsov SP, Seleznev EP. A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system. Journal of Experimental and Theoretical Physics. 2006;102(2):355–364. DOI: 10.1134/S1063776106020166.
  37. Ruel D, Takens F. On the nature of turbulence. In book: Strange attractors. Ed. Sinai YaG, Shilnikov PL. Moscow: Mir; 1981. P. 117. (In Russian).
  38. Newhouse S, Ruelle D, Takens F. Occurrence of strange Axiom-A attractors near quasi periodic flows on Tm, m≥3. Comm. Math. Phys. 1978-1979;64(1):35-40. DOI: 10.1007/BF01940759.
  39. Hunt TJ, MacKay RS. Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor. Nonlinearity. 2003;16(4):1499–1510. DOI:10.1088/0951-7715/16/4/318.
  40. Belykh V, Belykh I, Mosekilde E. The hyperbolic Plykin attractor can exist in neuron models. Int. J. of Bifurcation and Chaos. 2005;15(11):3567–3578. DOI: 10.1142/S0218127405014222.
  41. Shil’nikov LP, Turaev DV. Blue sky catastrophes. Dokl. Akad. Nauk. 1995;342(5):596–599.
  42. Shil’nikov LP, Turaev DV. Simple bifurcations leading to hyperbolic attractors. Computers Math. Appl. 1997;34(2–4):173–193. DOI: 10.1016/S0898-1221(97)00123-5.
  43. Gavrilov NK, Shilnikov AL. An example of blue sky catastrophe. In: Methods of qualitative theory of differential equations and related topics. Amer. Math. Soc. Transl., II Ser. Vol. 200, AMS, Providence, RI; 1999. 165–188.
  44. Shilnikov A, Cymbalyuk G. Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe. Phys. Rev. Lett. 2005;94(4):048101. DOI: 10.1103/PhysRevLett.94.048101.
  45. Lorentz E. Deterministic non-periodic current. In book: Strange attractors. Ed. Sinai YaG, Shilnikov PL. Moscow: Mir; 1981. P. 88.
  46. Tucker W. A rigorous ODE solver and Smale’s 14th problem. Comp. Math. 2002;2(1):53–117. DOI: 10.1007/s002080010018.
  47. Morales CA. Lorenz attractor through saddle-node bifurcations. Ann. de l’Inst. Henri Poincare. 1996;13(5):589–617.
  48. Dmitriev AS, Panas AI. Dynamic chaos: New media for communication systems. Moscow: Fizmatlit. 2002. 252 p. (In Russian).
Short text (in English):
(downloads: 60)