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

For citation:

Ajdarova J. S., Kuznetsov S. P. Chaotic dynamics of Hunt model – artificially constructed flow system with a hyperbolic attractor. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 3, pp. 176-196. DOI: 10.18500/0869-6632-2008-16-3-176-196

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 260)
Article type: 

Chaotic dynamics of Hunt model – artificially constructed flow system with a hyperbolic attractor

Ajdarova Julija Serikovna, Saratov State University
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences

We study numerically chaotic behavior associated with the presence of a hyperbolic strange attractor of Plykin type in the model of Hunt, that is an artificially constructed dynamical system with continuous time. There are presented portraits of the attractor, plots of realizations for chaotic signal generated by the system, illustrations of the sensitive dependence on initial conditions for the trajectories on the attractor. Quantitative characteristics of the attractor are estimated, including the Lyapunov exponents and the attractor dimension. We discuss the symbolic dynamics on the attractor, find out and analyze some unstable periodic orbits belonging to the attractor.

Key words: 
  1. Sinai YG. Stochasticity of dynamical systems. In: Gaponov-Grekhov AV, editor. Nonlinear waves. Moscow: Nauka; 1979. P. 192–211 (in Russian).
  2. Gamkrelidze RV, editor. Modern Problems of Mathematics. Fundamental Directions. Results of Science and Technology. Vol. 2. Moscow: All-Russian Institute of Scientific and Technical Information of the USSR Academy of Sciences; 1985. 307 p. (in Russian).
  3. Eckmann JP and Ruelle D. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 1985;57(3):617–656. DOI: 10.1103/RevModPhys.57.617.
  4. Devaney RL. An Introduction to Chaotic Dynamical Systems. Addison–Wesley, New York; 1989. 360 p.
  5. Shilnikov L. Mathematical problems of nonlinear dynamics: A tutorial. Int. J. Bifurcat. Chaos. 1997;7(9):1953–2001. DOI: 10.1142/S0218127497001527.
  6. Katok AB, Hasselblat B. Introduction to the Modern Theory of Dynamical Systems. NY: Cambridge University Press; 1995. 802 p.
  7. Afraimovich V and Hsu SB. Lectures on Chaotic Dynamical Systems. AMS/IP Studies in Advanced Mathematics, 28, American Mathematical Society, Providence, RI; International Press, Somerville, MA; 2003. 353 p.
  8. Rabinovich MI, Trubetskov DI. Oscillations and Waves in Linear and Nonlinear Systems. Berlin: Springer; 1989. 578 p. DOI: 10.1007/978-94-009-1033-1.
  9. Aischenko VS, Astakhov V, Vadivasova T, Neiman A, Shimansky-Geyer L. Nonlinear Dynamics of Chaotic and Stochastic Systems. Berlin: Springer; 2007. 446 p. DOI: 10.1007/978-3-540-38168-6.
  10. Andronov AA, Vitt AA, Khaikin SE. Theory of Oscillators. Pergamon Press; 1966. 848 p.
  11. Neimark YI. The Method of Point Mappings in the Theory of Nonlinear Oscillations. Moscow: Nauka; 1972. 472 p. (in Russian).
  12. Kuznetsov SP. Dynamic Chaos. 2 edition. Moscow: Fizmalit; 2006. 296 p. (in Russian).
  13. 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.
  14. Kuznetsov SP, Seleznev EP. A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system. J. Exp. Theor. Phys. 2006;102(2):355–364. DOI: 10.1134/S1063776106020166.
  15. 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 (in Russian). DOI: 10.18500/0869-6632-2006-14-5-3-29.
  16. Kuznetsov SP and Sataev IR. Hyperbolic attractor in a system of coupled non-autonomous van der Pol oscillators: Numerical test for expanding and contracting cones. Physics Letters A. 2007;365(1–2):97–104. DOI: 10.1016/j.physleta.2006.12.071.
  17. Isaeva OB, Jalnine AY, and Kuznetsov SP. Arnold’s cat map dynamics in a system of coupled nonautonomous van der Pol oscillators. Phys. Rev. E. 2006;74(4):046207. DOI: 10.1103/PhysRevE.74.046207.
  18. Kuznetsov SP and Pikovsky A. Autonomous coupled oscillators with hyperbolic strange attractors. Physica D. 2007;232(2):87–102. DOI: 10.1016/j.physd.2007.05.008.
  19. Grants of DFG and RFBR 04-02-04011 and 06-02-16619.
  20. 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 (in Russian). DOI: 10.18500/0869-6632-2007-15-6-75-85.
  21. Belykh V, Belykh I, and Mosekilde E. Hyperbolic Plykin attractor can exist in neuron models. Int. J. of Bifurcat. Chaos. 2005;15(11):3567–3578. DOI: 10.1142/S0218127405014222.
  22. Hunt TJ. Low Dimensional Dynamics: Bifurcations of Cantori and Realisations of Uniform Hyperbolicity. PhD Thesis. Univ. of Cambridge; 2000.
  23. Plykin RV. Sources and sinks ofa-diffeomorphisms of surfaces. Mathematics of the USSR-Sbornik. 1974;23(2):233–253. DOI: 10.1070/SM1974v023n02ABEH001719.
  24. Sveshnikov AA. Applied Methods of the Theory of Random Functions. Pergamon Press; 1966. 332 p.
  25. Benettin G, Galgani L, Giorgilli A, Strelcyn JM. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part I: Theory. Part II: Numerical application. Meccanica. 1980;15(1):9–30.
Short text (in English):
(downloads: 70)