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Dimensions and Entropies in Multidimensional Systems

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Nonlinear Waves

Part of the book series: Research Reports in Physics ((RESREPORTS))

Abstract

A strange attractor, which is a mathematical image of the self-oscillatory motion of a real system, shows a variety of properties, each related to a particular characteristic of the real process. Those properties can be described quantitatively by a relevant mathematical quantity number, function, etc. We focus our attention on the attractor dimension and the entropy of dynamic system, which, in our opinion, are most important characteristics of a strange attractor in addition to the common Fourier spectrum. Some other notions and properties associated with stochasticity in dynamic systems are naturally involved.

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References

  1. B. Mandelbrot. Fractals. Form, Chance and Dimension. Freeman, San Francisco, 1977.

    MATH  Google Scholar 

  2. D. K. Umberger, G. Mayer-Kress, E. Jen. Hausdorf Dimension for Sets with Broken Scaling Symmetry. In: Dimensions and Entropies in Chaotic Systems. Ed. G. Mayer-Kress. Springer, Berlin-Heidelberg, 1986, p. 42–53.

    Chapter  Google Scholar 

  3. D. K. Umbergerf J. D. Farmer. Phys.Rev.Lett., 1985, 55, 7, p.661.

    Article  ADS  MathSciNet  Google Scholar 

  4. R. Bowen. Entropy of Group Diffeomorphisms and Homogeneous Spaces. Trans. AMS, 1971, 153, p.401–414.

    Article  MATH  MathSciNet  Google Scholar 

  5. F. Takens. Detecting Strange Attractors in Turbulence. Lect.Notes in Math., Springer, Berlin, 1980, 898, p.336–382.

    Google Scholar 

  6. R. Adler, G. Konheim, M. H. McAndrew. Topological Entropy. Trans. AMS, 1965, 114, p. 309–319.

    Article  MATH  MathSciNet  Google Scholar 

  7. F. Takens. Distinguishing Deterministic and Random Systems. In: Nonlinear Dynamics and Turbulence. Eds. G. I. Barenblatt, G. Iooss, D. D. Joseph. Pitman, 1983, p.314–333.

    Google Scholar 

  8. J. Milnor, W. Thurston. In Iterated Maps of the Interval. Preprint 1977

    Google Scholar 

  9. M. I. Malkin. On Topological Classification of Piecewise Monotonic Map of an Interval. In: Methods of the Qualitative Theory of the Differential Equations. Gorky, 1980, p. 186–1 91.(in Russian).

    Google Scholar 

  10. M. I. Malkin. On Continuity of Entropy of Discontinuous Maps for a Segment. Ibid, p.35–47.

    Google Scholar 

  11. R. Mane. On the Dimension of the Compact Invariant Sets of Certain Nonlinear Maps. Lect.Notes in Math., Springer, New York, 1981, 898, p.230–242.

    MathSciNet  Google Scholar 

  12. J. P. Eckmann, D. Ruelle. Ergodic Theory of Chaos and Strange Attractors Rev.Mod.Phys., 1985, 57, 3, p.617–656.

    Article  ADS  MathSciNet  Google Scholar 

  13. J. P. Eckmann, S. O. Kamphorst, D. Ruelle, S. Ciliberto. Lyapunov Exponents from Time Series. Reprint, 1986.

    Google Scholar 

  14. M. P. Grassberger, I. Procaccia. Characterization of Strange Attractors. Phys. Rev.Lett., 1983, 50, 5, p.346.

    Article  ADS  MathSciNet  Google Scholar 

  15. P. Grassberger, I. Procaccia. Measuring the Strangeness of Strange Attractors. Physica 90. 1983, p.189–208.

    MathSciNet  Google Scholar 

  16. L. Young. Dimension, Entropy and Lyapunov Exponents. Ergod. Theory and Dyn. Syst., 1982, 2, p.109–124.

    Article  MATH  Google Scholar 

  17. A. Ben-Mizrachi et al. Measuring the Dimension of Experimental (Noisy) Attractors. Phys.Rev.A, 1984, 29, 2, p. 975.

    Article  ADS  Google Scholar 

  18. I. S. Aranson, A. M. Reiman and V. G. Shekhov. Measurement Methods for Correlation Dimension in Experiment. In the present book.

    Google Scholar 

  19. J. P. Caputo et al. Determination of Attractor Dimension and Entropy for Various Flows. An Experimentalist’s Viewpoint. Ins Dimensions and Entropies in Chaotic Systems, p.180–190.

    Google Scholar 

  20. M. I. Rabinovich, A. L. Fabrikant. Stochastic Self-Modulation of Waves in Nonequilibrium Media. Zhurnal Eksperimentalnyi i Teoreticheskoy Fiziki, 1979, 74, p.617 (in Russian).

    ADS  Google Scholar 

  21. A. Brandstäter et al. Characterizing Turbulent Channel Flow. In: Dimensions and Entropies in Chaotic Sytems, p. 150–157.

    Google Scholar 

  22. R. Badii, A. Politi. On the Fractal Dimension of Filtered Chaotic Signals. In: Dimensions and Entropies in Chaotic Systems, p.67–73.

    Google Scholar 

  23. A. Brandstäter et al. Low-Dimensional Chaos in a Hydrodynamic System. Phys.Rev.Lett., 1983, 51, p.1442–1444.

    Article  ADS  MathSciNet  Google Scholar 

  24. V. V. Kozlov et al. Correlation Dimension of a Flow and Spatial Development of Dynamical Chaos in the Boundary Layer. Pis’ma v Zhurnal Eksperimentalnyi i Teoreticheskoy Fiziki, 1987, 13, 16, p.989–991 (in Russian).

    ADS  Google Scholar 

  25. P. R. Gromov et al. Finite-Dimensional Attractors in Shear Flows with a Feedback. Doklady AN SSSR, 1987, 292, 2, p. 284–287. (in Russian).

    ADS  MathSciNet  Google Scholar 

  26. G. A. Held, C. D. Jeffries. Characterization of Chaotic Instabilities in an Electron - Hole Plasma in Germanium. In: Dimensions ans Entropies in Chaotic Systems, p.158–170.

    Google Scholar 

  27. S. Martin, W. Martienssen. Transition form Quasiperiodicity into Chaos in the Periodically Driven Conductivity of BSN Crystal. In: Dimensions and Entropies in Chaotic Systems, p.191–197.

    Google Scholar 

  28. A. Babloyantz. Evidence of chaotic Dynamics of Brain Activity During the Sleep Cycle. In: Dimensions and Entropies in Chaotic Systems, p.241–256.

    Google Scholar 

  29. V. S. Afraimovich. Internal Bifurcation and Crises of Attractors. In: Nonlinear Waves. Structures and Bifurcations. Eds. A. V. Gaponov- Grekhov, M. I. Rabinovich. M. Nauka, 1987, p.189–213 (in Russian).

    Google Scholar 

  30. P. Tabeling. Phys. Rev. A, 1985, 31, p.3460–3462.

    Article  ADS  Google Scholar 

  31. J. Holzfuss, G. Mayer Kress. In: Dimensions and Entropies in Chaotic Systems, p.114–122.

    Google Scholar 

  32. I. S. Aranson, M. I. Rabinovich, I. M. Starobinets. In: Nonlinear and Turbulent Processes in Physics, New York, 1984, 3, p.1139.

    Google Scholar 

  33. J. P. Crutchfield, B. S. McNamara. Equations of Motion from a Data Series. J. of Complex Systems, 1987, 1.

    Google Scholar 

  34. V. S. Afraimovich, M. I. Rabinovich, A. D. Ugodchikov. Pis’ma v Zhurnal Eksperimental’nyi i Teoreticheskoy Fiziki, 1983, 38, 2, p 64–67 (in Russian).

    ADS  Google Scholar 

  35. M. A. Shereshevsky. Funktsional’nyi Analiz i Ego Prilozheniya, 1987, 21, 1, p.88–89 (in Russian).

    Google Scholar 

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© 1989 Springer-Verlag Berlin Heidelberg

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Afraimovich, V.S., Reiman, A.M. (1989). Dimensions and Entropies in Multidimensional Systems. In: Gaponov-Grekhov, A.V., Rabinovich, M.I., Engelbrecht, J. (eds) Nonlinear Waves. Research Reports in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-74366-5_1

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  • DOI: https://doi.org/10.1007/978-3-642-74366-5_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50654-6

  • Online ISBN: 978-3-642-74366-5

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