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Ergodic theory approach to chaos: remarks and computational aspects

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the previously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota's conjecture concerning nontrivial ergodic properties of the model.
Rocznik
Strony
259--267
Opis fizyczny
Bibliogr. 59 poz., rys., wykr.
Twórcy
autor
  • Faculty of Electrical Engineering, Automatics, Computer Science and Electronics, AGH University of Science and Technology, al. Mickiewicza 30/B-1, 30-059 Cracow, Poland, pawel.mitkowski@gmail.com
Bibliografia
  • [1] Anosov, D. V. (1963). Ergodic properties of geodesic flows on closed Riemanian manifolds of negative curvature, Soviet Mathematics-Doklady 4: 1153-1156.
  • [2] Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics, 2nd Edn., Springer-Verlag, New York, NY, (translation from Russian).
  • [3] Auslander, J. and Yorke, J. A. (1980). Interval maps, factors of maps and chaos. Tohoku Mathematical Journal. II. Series 32: 177-188.
  • [4] Bass, J. (1974). Stationary functions and their applications to the theory of turbulence, Journal of Mathematical Analysis and Applications 47: 354-399.
  • [5] Birkhoff, G. D. (1931a). Proof of a recurrence theorem for strongly transitive systems, Proceedings of the National Academy of Sciences of the United States of America 17: 650-655.
  • [6] Birkhoff, G. D. (1931b). Proof of the ergodic theorem, Proceedings of the National Academy of Sciences of the United States of America 17: 656-660.
  • [7] Birkhoff, G. D. and Koopman, B. O. (1932). Recent contributions to the ergodic theory, Mathematics: Proceedings of the National Academy of Sciences 18: 279-282.
  • [8] Bronsztejn, I. N., Siemiendiajew, K. A., Musiol, G. and Muhlig, H. (2004). Modern Compendium of Mathematics, PWN, Warsaw, (in Polish, translation from German).
  • [9] Dawidowicz, A. L. (1992). On invariant measures supported on the compact sets II, Universitatis Iagellonicae Acta Mathematica 29: 25-28.
  • [10] Dawidowicz, A. L. (1992). A method of construction of an invariant measure, Annales Polonici Mathematici LVII(3): 205-208.
  • [11] Dawidowicz, A. L. (2007). On the Avez method and its generalizations, Matematyka Stosowana 8: 46-55, (in Polish).
  • [12] Dawidowicz, A. L., Haribash, N. and Poskrobko, A. (2007). On the invariant measure for the quasi-linear Lasota equation. Mathematical Methods in the Applied Sciences 30: 779-787.
  • [13] Devaney, R. L. (1987). An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company, New York, NY.
  • [14] Dorfman, J. R. (2001). Introduction to Chaos in Nonequilibrium Statistical Mechanics, PWN, Warsaw, (in Polish, translation from English).
  • [15] Foias, C. (1973). Statistical study of Navier-Stokes equations II, Rendiconti del Seminario Matematico della Universita di Padova 49: 9-123.
  • [16] Fomin, S. W., Kornfeld, I. P. and Sinaj, J. G. (1987). Ergodic Theory, PWN, Warsaw, (in Polish, translation from Russian).
  • [17] Górnicki, J. (2001). Fundamentals of nonlinear ergodic theory, Wiadomosci Matematyczne 37: 5-16, (in Polish).
  • [18] Gurney, W. S. C., Blythe, S. P. and Nisbet, R. M. (1980). Nicholson's blowflies revisited, Nature 287: 17-21.
  • [19] Kudrewicz, J. (1991). Dynamics of Phase-Locked Loops, WNT, Warsaw, (in Polish).
  • [20] Kudrewicz, J. (1993, 2007). Fractals and Chaos, WNT,Warsaw, (in Polish).
  • [21] Landau, L. D., Lifszyc, J. M. (2007). Mechanics, PWN, Warsaw, (in Polish, translation from Russian).
  • [22] de Larminat, P. and Thomas, Y. (1983). Automatic Control-Linear Systems. Vol. 1: Signals and Systems, WNT, Warsaw, (in Polish, translation from French).
  • [23] Lasota, A. (1977). Ergodic problems in biology, Société Mathematique de France, Asterisque 50: 239-250.
  • [24] Lasota, A. (1979). Invariant measures and a linear model of turbulence, Rediconti del Seminario Matematico della Universita di Padova 61: 39-48.
  • [25] Lasota, A. (1981). Stable and chaotic solutions of a first order partial differential equation, Nonlinear Analysis Theory, Methods & Applications 5(11): 1181-1193.
  • [26] Lasota, A. and Mackey, M. C. (1994). Chaos, Fractals, and Noise Stochastic Aspects of Dynamics, Springer-Verlag, New York, NY.
  • [27] Lasota, A., Mackey, M. C. and Wazewska-Czyzewska, M. (1981). Minimazing theraupetically induced anemia, Journal of Mathematical Biology 13: 149-158.
  • [28] Lasota, A. and Myjak, J. (2002). On a dimension of measures, Bulletin of the Polish Academy of Sciences: Mathematics 50(2): 221-235.
  • [29] Lasota, A. and Szarek, T. (2002). Dimension of measures invariant with respect to the Wazewska partial differential equation, Journal of Differential Equations 196: 448-465.
  • [30] Lasota, A. and Yorke, J. A. (1973). On the existence of invariant measures for piecewise monotonic transformations, Transactions of the American Mathematical Society 186: 481-488.
  • [31] Lasota, A., and Yorke, J. A. (1977). On the existence of invariant measures for transformations with strictly turbulent trajectories, Bulletin of the Polish Academy of Sciences: Mathematics, Astronomy and Physics 65(3): 233-238.
  • [32] Lebowitz, J. L. and Penrose, O. (1973). Modern ergodic theory, Physics Today 26: 155-175.
  • [33] Liz, E.and Rost, G. (2009). On the global attractor of delay differential equations with unimodal feedback, Discrete and Continuous Dynamical Systems 24(4): 1215-1224.
  • [34] Mackey, M. C. (2007). Adventures in Poland: Having fun and doing research with Andrzej Lasota, Matematyka Stosowana 8: 5-32.
  • [35] Mackey, M. C. and Glass, L. (1977). Oscillations and chaos in physiological control systems, Science, New Series 197(4300): 287-289.
  • [36] Mitkowski, P. J. (2010). Numerical analysis of existence of invariant and ergodic measure in the model of dynamics of red blood cell's production system, Proceedings of the 4th European Conference on Computational Mechanics, Paris, France, pp. 1-2.
  • [37] Mitkowski, P. J. (2011). Chaos in the Ergodic Theory Approach in the Model of Disturbed Erythropoiesis, Ph.D. thesis, AGH University of Science and Technology, Cracow.
  • [38] Mitkowski, W. (2010). Chaos in linear systems, Pomiary, Automatyka, Kontrola 56(5): 381-384, (in Polish).
  • [39] Mitkowski, P. J. and Ogorzałek, M. J. (2010). Ergodic properties of the model of dynamics of blood-forming system, 3rd International Conference on Dynamics, Vibration and Control, Shanghai-Hangzhou, China, pp. 71-74.
  • [40] Myjak, J. (2008). Andrzej Lasota's selected results. Opuscula Mathematica 28(4): 363-394.
  • [41] Myjak, J. and Rudnicki, R. (2002). Stability versus chaos for a partial differential equation, Chaos Solitons & Fractals 14: 607-612.
  • [42] Nadzieja, T. (1996). Individual ergodic theorem from the topological point of view, Wiadomości Matematyczne 32: 27-36, (in Polish).
  • [43] Nicholson, A. J. (1954). An outline of the dynamic of animal population, Australian Journal of Zoology 2: 9-65.
  • [44] Ott, E. (1993). Chaos in Dynamic Systems, WNT, Warsaw, (in Polish, translation from English).
  • [45] Prodi, G. (1960). Teoremi Ergodici per le Equazioni della Idrodinamica, C.I.M.E., Rome.
  • [46] Rudnicki, R. (1985a). Invariant measures for the flow of a first order partial differential equation, Ergodic Theory & Dynamical Systems 5: 437-443.
  • [47] Rudnicki, R. (1985b). Ergodic properties of hyperbolic systems of partial differential equations, Bulletin of the Polish Academy of Sciences: Mathematics 33(11-12): 595-599.
  • [48] Rudnicki, R. (1988). Strong ergodic properties of a first-order partial differential equation, Journal of Mathematical Analysis and Applications 133: 14-26.
  • [49] Rudnicki, R. (2004). Chaos for some infinite-dimensional dynamical systems, Mathematical Methods in the Applied Sciences 27: 723-738.
  • [50] Rudnicki, R. (2009). Chaoticity of the blood cell production system, Chaos: An Interdisciplinary Journal of Nonlinear Science 19(043112): 1-6.
  • [51] Shampine, L. F., Thompson, S. and Kierzenka, J. (2002). Solving delay differential equations with dde23, available at www.mathworks.com/dde_tutorial.
  • [52] Silva, C. E. (2010). Lecture on dynamical systems, Spring School of Dynamical Systems, Będlewo, Poland.
  • [53] Szlenk, W. (1982). Introduction to the Theory of Smooth Dynamical Systems, PWN, Warsaw, (in Polish).
  • [54] Taylor, S. R. (2004), Probabilistic Properties of Delay Differential Equations, Ph.D. thesis, University of Waterloo, Ontario, Canada.
  • [55] Tucker, W. (1999). The Lorenz attractor exists, Comptes Rendus. Mathématique. Académie des Sciences, Paris 328(I): 1197-1202.
  • [56] Ulam, S. M. (1960), A Collection of Mathematical Problems, Interscience Publishers, New York, NY/London.
  • [57] Walther, H. O. (1981). Homoclinic solution and chaos in x˙ (t) = f(x(t − 1)), Nonlinear Analysis: Theory, Methods & Applications 5(7): 775-788.
  • [58] Ważewska-Czyżewska, M. (1983). Erythrokinetics. Radioisotopic Methods of Investigation and Mathematical Approach, National Center for Scientific, Technical and Economic Information, Warsaw.
  • [59] Ważewska-Czyżewska, M. and Lasota, A. (1976). Mathematical problems of blood cells dynamics system, Matematyka Stosowana 6: 23-40, (in Polish).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BPZ7-0001-0019
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