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
B. Mandelbrot. Fractals. Form, Chance and Dimension. Freeman, San Francisco, 1977.
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.
D. K. Umbergerf J. D. Farmer. Phys.Rev.Lett., 1985, 55, 7, p.661.
R. Bowen. Entropy of Group Diffeomorphisms and Homogeneous Spaces. Trans. AMS, 1971, 153, p.401–414.
F. Takens. Detecting Strange Attractors in Turbulence. Lect.Notes in Math., Springer, Berlin, 1980, 898, p.336–382.
R. Adler, G. Konheim, M. H. McAndrew. Topological Entropy. Trans. AMS, 1965, 114, p. 309–319.
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.
J. Milnor, W. Thurston. In Iterated Maps of the Interval. Preprint 1977
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).
M. I. Malkin. On Continuity of Entropy of Discontinuous Maps for a Segment. Ibid, p.35–47.
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.
J. P. Eckmann, D. Ruelle. Ergodic Theory of Chaos and Strange Attractors Rev.Mod.Phys., 1985, 57, 3, p.617–656.
J. P. Eckmann, S. O. Kamphorst, D. Ruelle, S. Ciliberto. Lyapunov Exponents from Time Series. Reprint, 1986.
M. P. Grassberger, I. Procaccia. Characterization of Strange Attractors. Phys. Rev.Lett., 1983, 50, 5, p.346.
P. Grassberger, I. Procaccia. Measuring the Strangeness of Strange Attractors. Physica 90. 1983, p.189–208.
L. Young. Dimension, Entropy and Lyapunov Exponents. Ergod. Theory and Dyn. Syst., 1982, 2, p.109–124.
A. Ben-Mizrachi et al. Measuring the Dimension of Experimental (Noisy) Attractors. Phys.Rev.A, 1984, 29, 2, p. 975.
I. S. Aranson, A. M. Reiman and V. G. Shekhov. Measurement Methods for Correlation Dimension in Experiment. In the present book.
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.
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).
A. Brandstäter et al. Characterizing Turbulent Channel Flow. In: Dimensions and Entropies in Chaotic Sytems, p. 150–157.
R. Badii, A. Politi. On the Fractal Dimension of Filtered Chaotic Signals. In: Dimensions and Entropies in Chaotic Systems, p.67–73.
A. Brandstäter et al. Low-Dimensional Chaos in a Hydrodynamic System. Phys.Rev.Lett., 1983, 51, p.1442–1444.
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).
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).
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.
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.
A. Babloyantz. Evidence of chaotic Dynamics of Brain Activity During the Sleep Cycle. In: Dimensions and Entropies in Chaotic Systems, p.241–256.
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).
P. Tabeling. Phys. Rev. A, 1985, 31, p.3460–3462.
J. Holzfuss, G. Mayer Kress. In: Dimensions and Entropies in Chaotic Systems, p.114–122.
I. S. Aranson, M. I. Rabinovich, I. M. Starobinets. In: Nonlinear and Turbulent Processes in Physics, New York, 1984, 3, p.1139.
J. P. Crutchfield, B. S. McNamara. Equations of Motion from a Data Series. J. of Complex Systems, 1987, 1.
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).
M. A. Shereshevsky. Funktsional’nyi Analiz i Ego Prilozheniya, 1987, 21, 1, p.88–89 (in Russian).
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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
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