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Systems with Time-delay Feedback

  • Sergey P. Kuznetsov

Abstract

Besides the alternately excited oscillators, the principle of manipulation by phases in the course of the excitation transfer may be applied to systems with delayed feedback. In this case, it is sufficient to have a single self-oscillator manifesting successive stages of activity and suppression, while the excitation transfer accompanied with appropriate phase transformation is carried out through the delayed feedback loop, from one stage of activity to another. In practical implementation, these systems can be even simpler than the alternately excited oscillators. From a mathematical point of view, they arc more complex, because presence of the delay implies formally infinite dimension of the state space. Careful mathematical analysis of attractors in such systems, including rigorous foundation of hyperbolicity, is a challenging problem that goes far beyond the scope of this book. In the present chapter, we start with some necessary material concerning the mathematical nature of the differential equations with delay. The next two sections consider models of systems with delay, generating sequences of pulses with the phases of the carrier transformed from pulse to pulse in accordance with expanding circle maps. First, the non-autonomous systems are discussed that operate due to the externally imposed parameter modulation with or without an auxiliary reference signal. Afterwards, we consider an autonomous system with time delay. On the physical level of reasoning it seems confident that chaotic attractors in these systems are of the same nature as those in the above discussed alternately excited oscillatory systems being associated with expanding circle maps for some cyclic variable. Particularly, it is believed that the generated chaos is structurally stable. As one can hypothesize, attractors in these systems as they are treated in a framework of discrete-time description (like the Poincaré maps) belong to the class of solenoids of Smale-Williams embedded in the infinite-dimensional phase space.

Keywords

Lyapunov Exponent Parameter Modulation Chaotic Attractor Iteration Diagram Large Lyapunov Exponent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Balyakin, A.A.. Ryskin, N.M.: Features of computation of spectrum of Lyapunov exponents in distributed self-oscillatory systems with delayed feedback. Izvestija VUZov-Applied Nonlinear Dynamics (Saratov) 15, 3–21 (2007).Google Scholar
  2. Baranov, S.V., Kuznetsov, S.P., Ponomarenko, V.I.: Chaos in the phase dynamics of Q-switched van der Pol oscillator with additional delaycd-feedback loop. Izvestija VUZov-Applied Nonlinear Dynamics (Saratov) 18(1), 11–23 (2010).zbMATHGoogle Scholar
  3. Bellman, R., Cooke, C.L.: Differential-difference Equations. Academic Press, New York (1963).zbMATHGoogle Scholar
  4. Blokhina, E.V.. Kuznetsov, S.P., Rozhnev, A.G.: High-dimensional chaos in a gyrotron. IEEE Transactions on Electron Devices 54, 188–193 (2007).ADSCrossRefGoogle Scholar
  5. Bonatti, C., Diaz, L.J., Viana, M.: Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probobalistic Perspective. Encyclopedia of Mathematical Sciences. Vol.102. Springer, Berlin, Heidelberg, New York (2005).Google Scholar
  6. Chiasson, J.N., Loiseaum, J.J. (eds.): Applications of Time Delay Systems (Lecture Notes in Control and Information Sciences, Vol. 352). Springer, Heidelberg (2007).Google Scholar
  7. El’sgol’ts. L.E., Norkin, S.B.: Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Academic Press, New York (1973).Google Scholar
  8. Farmer, J.D.: Chaotic attractors of an infinite-dimensional dynamical system. Physica D 4, 366–393 (1982).MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. Fowler, A.C.: An asymptotic analysis of the delayed logistic equation when the delay is large. IMA Journal of Applied Mathematics 28, 41–49 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  10. Giacomelli, G., Politi, A.: Relationship between delayed and spatially extended dynamical systems. Phys. Rev. Lett. 76, 2686–2689 (1996).ADSCrossRefGoogle Scholar
  11. Glass, I., Mackey, M.C.: From Clocks to Chaos: The Rhythms of Life. Princeton University Press, Princeton (1988).zbMATHGoogle Scholar
  12. Hu, H.Y., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer, Berlin (2002).zbMATHGoogle Scholar
  13. Jenkins, G.M., Watts, D.G.: Spectral Analysis and its Application. Holden-Day, Inc., San Francisco (1968).Google Scholar
  14. Kuznetsov, S.P.: Complex dynamics of oscillators with delayed feedback. Radiophysics & Quantum Electronics 25, 996–1009 (1982).ADSCrossRefGoogle Scholar
  15. Kuznetsov, S.P., Pikovsky, A.: Hyperbolic chaos in the phase dynamics of a Q-switched oscillator with delayed nonlinear feedbacks. Europhysics Letters 84, 10013 (2008).MathSciNetADSCrossRefGoogle Scholar
  16. Kuznetsov, S.P., Pikovsky, A.: Attractor of S male-Williams type in an autonomous time-delay system. Preprint nlin. arXiv: 1011.5972 (2010).Google Scholar
  17. Kuznetsov, S.P., Ponomarenko, V.I.: Realization of a strange attractor of the Smale-Williams type in a radiotechnical delay-fedback oscillator. Tech. Phys. Lett. 34, 771–773 (2008).ADSCrossRefGoogle Scholar
  18. Le Berre, M., Ressayre, E., Tallet, A., Gibbs, H.M., Kaplan, D.L., Rose M.H.: Conjecture on the dimensions of chaotic attractors of delayed-feedback dynamical systems. Phys. Rev. A 35, 4020–4022 (1987).MathSciNetADSCrossRefGoogle Scholar
  19. Lepri, S., Giacomelli, G., Politi, A., Arecchi, F.T.: High-dimensional chaos in delayed dynamical systems. Physica D 70. 235–249 (1994).MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. Myskis, A.D.: Linear differential equations with retarded argument, Nauka, Moscow (1972).Google Scholar
  21. Pazó, D., López, J.M.: Characteristic Lyapunov vectors in chaotic time-delayed systems. Phys. Rev. E 82, 056201 (2010).ADSCrossRefGoogle Scholar
  22. Pesin, Ya.B.: Lectures on Partial Hyperbolicity and Stable Ergodicity. Zurich lectures in advanced mathematics. European Mathematical Society (2004).Google Scholar
  23. Pesin, Ya.: Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents. Regular and Chaotic Dynamics 12, 476–489 (2007).MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. Vallee R., Delisle, C. Chrostowski J.: Noise versus chavs in acousto-optic bistability. Physical Review A 30, 336–342 (1984).ADSCrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sergey P. Kuznetsov
    • 1
  1. 1.Kotel’nikov’s Institute of Radio-Engineering and Electronics of RASSaratov BranchSaratovRussian Federation

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