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

For citation:

Arzhanukhina D. S., Kuznetsov S. P. Robust chaos in autonomous time-delay system. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 2, pp. 36-49. DOI: 10.18500/0869-6632-2014-22-2-36-49

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Language: 
English
Heading: 
Deterministic Chaos
Article type: 
Article
UDC: 
517.929:621.373
DOI: 
10.18500/0869-6632-2014-22-2-36-49

Robust chaos in autonomous time-delay system

Autors: 
Arzhanukhina Darja Sergeevna, Saratov State University
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

We consider an autonomous system constructed as modification of the logistic differential equation with delay that generates successive trains of oscillations with phases evolving according to chaotic maps. The system contains two feedback loops characterized by two generally distinct  retarding time parameters. In the case of their equality, chaotic dynamics is associated with the  Smale–Williams attractor that corresponds to the double-expanding circle map for the phases of the carrier of the oscillatory trains. Alternatively, at appropriately chosen two different delays attractor is close to torus with Anosov dynamics on it as the phases are governed by the Fibonacci map. In both cases the attractors manifest robustness (absence of regularity windows under variation of parameters) and presumably relate to the class of structurally stable hyperbolic attractors.

Key words: 
Attractor
hyperbolic chaos
maps
Anosov dynamics
Arnold cat
Fibonacci map
Smale–Williams attractor
Reference: 
  1. Anosov DV, Gould GG, Aranson SK, Grines VZ, Plykin RV, Safonov AV, Sataev EA, Shlyachkov SV, Solodov VV, Starkov AN, Stepin AM. Dynamical Systems with Hyperbolic Behaviour. Encyclopaedia of Mathematical Sciences. Dynamical Systems IX. Berlin:Springer. 1995. 235 p. DOI: 10.1007/978-3-662-03172-8
  2. Smale S. Differentiable dynamical systems. Bull. Amer. Math. Soc. (NS). 1967;73(6):747–817. DOI: 10.1090/S0002-9904-1967-11798-1
  3. Williams RF. Expanding attractors. Publications mathematiques de l’I.H. E.S. 1974; 43:169–203.
  4. Afraimovich V, Hsu S-B. Lectures on chaotic dynamical systems. Vol.28. American Mathematical Society, Providence RI, International Press, Somerville, MA; 2003. 353 p.
  5. Devaney RL. An Introduction to Chaotic Dynamical Systems. NY: Addison – Wesley; 1989. 336 p. DOI:10.1155/S1048953390000077
  6. Shilnikov L. Mathematical problems of nonlinear dynamics: A tutorial. Int. J. of Bif. & Chaos. 1997;7(9):1953–2001.
  7. Elhadj Z, Sprott JC. Robust Chaos and Its Applications. Singapore: World Scientific Publishing Company; 2011. 471 p.
  8. Banerjee S, Yorke JA, Grebogi C. Robust Chaos. Phys. Rev. Lett. 1998;80(14):3049–3052.
  9. 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.
  10. Hunt TJ. Low Dimensional Dynamics: Bifurcations of Cantori and Realisations of Uniform Hyperbolicity. PhD Thesis. University of Cambridge; 2000. 121 p.
  11. 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
  12. Morales CA. Lorenz attractor through saddle-node bifurcations. Ann. de l’Inst. Henri Poincare. 1996;13(5):589–617. DOI:10.1016/S0294-1449(16)30116-0
  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. JETP. 2006;102(2):355–364. DOI:10.1134/S1063776106020166
  15. Isaeva OB, Jalnine Ayu, Kuznetsov SP. Arnold’s cat map dynamics in a system of coupled nonautonomous van der Pol oscillators. Phys. Rev. E. 2006;74(2):046207. DOI: 10.1103/PhysRevE.74.046207
  16. Kuznetsov SP, Pikovsky A. Autonomous coupled oscillators with hyperbolic strange attractors. Physica. 2007;D232(2):87–102. DOI:10.1016/j.physd.2007.05.008
  17. Kuznetsov SP. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics. Physics-Uspekhi. 2011; 54(2):119–144. DOI: 10.3367/UFNe.0181.201102a.0121
  18. Kuznetsov SP. Hyperbolic Chaos: A Physicist’s View. Higher Education Press: Beijing and Springer-Verlag: Berlin, Heidelberg, 2012. 320 p.
  19. Arzhanukhina DS, Kuznetsov SP. A system of three non-autonomous oscillators with hyperbolic chaos. I. The model with dynamics on attractor governed by Arnold’s cat map on torus. Izvestiya VUZ. Applied Nonlinear Dynamics. 2012; 20(6):56–66. (in Russian). DOI: 10.18500/0869-6632-2012-20-6-56-66
  20. Arzhanukhina DS, Kuznetsov SP. A system of three non-autonomous oscillators with hyperbolic chaos. II. The model with DA-attractor. Izvestiya VUZ. Applied Nonlinear Dynamics. 2013;21(2):163–172. (in Russian). DOI: 10.18500/0869-6632-2013-21-2-163-172
  21. Kuznetsov SP, Ponomarenko VI. Realization of a strange attractor of the Smale–Williams type in a radiotechnical delay-fedback oscillator. Tech. Phys. Lett. 2008;34(9):771–773. DOI:10.1134/S1063785008090162
  22. 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
  23. Kuznetsov AS, Kuznetsov SP. Parametric generation of robust chaos with time-delayed feedback and modulated pump source. Communications in Nonlinear Science and Numerical Simulation. 2013;18(3):728–734. DOI:10.1016/j.cnsns.2012.08.006
  24. Kuznetsov SP, Pikovsky A. Attractor of Smale–Williams type in an autonomous time-delay system. Preprint nlin. ArXiv: 1011.5972; 2010.
  25. Fowler AC. An asymptotic analysis of the delayed logistic equation when the delay is large. IMA Journal of Applied Mathematics. 1982;28:41–49. DOI:10.1093/IMAMAT/28.1.41
  26. Benettin G, Galgani L, Giorgilli A, Strelcyn J-M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Meccanica. 1980;15(1):9–20. DOI: 10.1007/BF02128236
  27. Schuster HG, Just W. Deterministic chaos: an introduction. Wiley-VCH; 2005. 299 p.
  28. Farmer JD. Chaotic attractors of an infinite-dimensional dynamical system. Physica D. 1982;4(3):366–393. DOI:10.1016/0167-2789(82)90042-2
  29. Baljakin AA, Ryskin NM. Peculiarities of calculation of the Lyapunov exponents set in distributed self-oscillated systems with delayed feedback. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007;15(6):3-21. DOI: 10.18500/0869-6632-2007-15-6-3-21
  30. Arnold VI, Avez A. Ergodic Problems in Classical Mechanics. New York: Benjamin; 1968. 284 p.
  31. Farmer J, Ott E, Yorke J. The dimension of chaotic attractors. Physica D. 1983;7:153–170.
  32. Grassberger P,Procaccia I. Measuring the strangeness of strange attractors. Physica D. 1983;9(1-2):189–208. DOI: 10.1016/0167-2789(83)90298-1
  33. Anishchenko VS, Astakhov VV, Neiman AB, Vadivasova TE, Schimansky-Geier L. Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Development. Berlin:Springer. 2002. 446 p.
Received: 
17.03.2014
Accepted: 
17.03.2014
Published: 
31.07.2014
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