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Discrete Maps

  • Alexander S. Mikhailov
  • Alexander Yu. Loskutov
Part of the Springer Series in Synergetics book series (SSSYN, volume 52)

Abstract

The mathematical models known as discrete maps are closely related to dynamical systems with continuous time. They can arise naturally in problems where the state of a system is allowed to change only at some prescribed instants in time. In fact, discrete maps are a special case of an automaton with instantaneous states described by continuous variables.

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References

  1. 5.1
    M. Henon: Commun. Math. Phys. 50, 69–77 (1976)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 5.2
    N.V. Butenin, Yu.L Neimark, N.A. Fufayev: Introduction to the Theory of Nonlinear Oscillations (Nauka, Moscow 1987, in Russian)zbMATHGoogle Scholar
  3. 5.3
    P. Collet, J.P. Eckmann: Iterated Maps of the Interval as Dynamical Systems (Birkhäuser, Boston 1980)Google Scholar
  4. 5.4
    A.N. Sharkovskii: Ukr. Mat. Zh. 1, 61–71 (1964)Google Scholar
  5. 5.5
    A.N. Sharkovskii, Yu.A. Maistrenko, E.Yu. Romanenko: Difference Equations and Their Applications (Naukova Dumka, Kiev 1986, in Russian)Google Scholar
  6. 5.6
    T.Y. Li, J.A. Yorke: Amer. Math. Mon. 82, 985–992 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 5.7
    M.J. Feigenbaum: J. Stat. Phys. 19, 25–52 (1978)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 5.8
    M.J. Feigenbaum: J. Stat. Phys. 21, 669–706 (1979)MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 5.9
    M.J. Feigenbaum: Los Alamos Science 1, 4–27 (1980); Physica D 7, 16–39 (1983)MathSciNetGoogle Scholar
  10. 5.10
    S. Grossmann, S. Thomae: Z. Naturforsch. 32a, 1353–1363 (1977)MathSciNetADSGoogle Scholar
  11. 5.11
    B. Hu: “Functional renormalization-group equation approach to the transition to chaos”, in Chaos and Statistical Methods, ed. by Y. Kuramoto (Springer, Berlin, Heidelberg 1984)Google Scholar
  12. 5.12
    O.E. Lanford: Bull. Amer. Math. Soc. 6, 427–434 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 5.13
    P. Collet, J.P. Eckmann, O.E. Lanford: Commun. Math. Phys. 76, 211–254 (1980)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 5.14
    H.G. Schuster: Deterministic Chaos (Physik-Verlag, Weinheim 1984)zbMATHGoogle Scholar
  15. 5.15
    E.B. Vul, Ya.G. Sinai, K.M. Khanin: Usp. Mat. Nauk 39, 3–37 (1984)MathSciNetGoogle Scholar
  16. 5.16
    J.A. Yorke, C. Grebogi, E. Ott, L. Tedeschini-Lalli: Phys. Rev. Lett. 54, 1093 (1985)MathSciNetADSCrossRefGoogle Scholar
  17. 5.17
    M. Bucher: Phys. Rev. 33A, 3544–3546 (1986)ADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Alexander S. Mikhailov
    • 1
    • 2
  • Alexander Yu. Loskutov
    • 1
  1. 1.Department of PhysicsMoscow State UniversityMoscowUSSR
  2. 2.Institut für Theoretische Physik und SynergetikUniversität StuttgartStuttgart 80Fed. Rep. of Germany

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