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Ergodic theory approach to chaos: Remarks and computational aspects Cover

Ergodic theory approach to chaos: Remarks and computational aspects

International Journal of Applied Mathematics and Computer Science
Volume 22 (2012): Issue 2 (June 2012)
By:  and    
Open Access
|Jun 2012
References

References

  1. Anosov, D. V. (1963). Ergodic properties of geodesic flows on closed Riemanian manifolds of negative curvature, Soviet Mathematics—Doklady 4: 1153-1156.
    Search in Google Scholar
  2. Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics, 2nd Edn., Springer-Verlag, New York, NY, (translation from Russian).
    Search in Google Scholar
  3. Auslander, J. and Yorke, J. A. (1980). Interval maps, factors of maps and chaos. Tohoku Mathematical Journal. II. Series 32: 177-188.
    Search in Google Scholar
  4. Bass, J. (1974). Stationary functions and their applications to the theory of turbulence, Journal of Mathematical Analysis and Applications 47: 354-399.10.1016/0022-247X(74)90026-2
    Search in Google Scholar
  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: 650-655.10.1073/pnas.17.12.650
    Search in Google Scholar
  6. Birkhoff, G. D. (1931b). Proof of the ergodic theorem, Proceedings of the National Academy of Sciences of the United States of America 17: 656-660.10.1073/pnas.17.2.656
    Search in Google Scholar
  7. Birkhoff, G. D. and Koopman, B. O. (1932). Recent contributions to the ergodic theory, Mathematics: Proceedings of the National Academy of Sciences 18: 279-282.10.1073/pnas.18.3.279
    Search in Google Scholar
  8. Bronsztejn, I. N., Siemiendiajew, K. A., Musiol, G. and Muhlig, H. (2004). Modern Compendium of Mathematics, PWN, Warsaw, (in Polish, translation from German).
    Search in Google Scholar
  9. Dawidowicz, A. L. (1992). On invariant measures supported on the compact sets II, Universitatis Iagellonicae Acta Mathematica 29: 25-28.
    Search in Google Scholar
  10. Dawidowicz, A. L. (1992). A method of construction of an invariant measure, Annales Polonici Mathematici LVII(3): 205-208.10.4064/ap-57-3-205-208
    Search in Google Scholar
  11. Dawidowicz, A. L. (2007). On the Avez method and its generalizations, Matematyka Stosowana 8: 46-55, (in Polish).
    Search in Google Scholar
  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.
    Search in Google Scholar
  13. Devaney, R. L. (1987). An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company, New York, NY.
    Search in Google Scholar
  14. Dorfman, J. R. (2001). Introduction to Chaos in Nonequilibrium Statistical Mechanics, PWN, Warsaw, (in Polish, translation from English).
    Search in Google Scholar
  15. Foias, C. (1973). Statistical study of Navier-Stokes equations II, Rendiconti del Seminario Matematico della Universita di Padova 49: 9-123.
    Search in Google Scholar
  16. Fomin, S. W., Kornfeld, I. P. and Sinaj, J. G. (1987). Ergodic Theory, PWN, Warsaw, (in Polish, translation from Russian).
    Search in Google Scholar
  17. Górnicki, J. (2001). Fundamentals of nonlinear ergodic theory, Wiadomosci Matematyczne 37: 5-16, (in Polish).
    Search in Google Scholar
  18. Gurney, W. S. C., Blythe, S. P. and Nisbet, R. M. (1980). Nicholson's blowflies revisited, Nature 287: 17-21.10.1038/287017a0
    Search in Google Scholar
  19. Kudrewicz, J. (1991). Dynamics of Phase-Locked Loops, WNT, Warsaw, (in Polish).
    Search in Google Scholar
  20. Kudrewicz, J. (1993, 2007). Fractals and Chaos, WNT, Warsaw, (in Polish).
    Search in Google Scholar
  21. Landau, L. D., Lifszyc, J. M. (2007). Mechanics, PWN, Warsaw, (in Polish, translation from Russian).
    Search in Google Scholar
  22. de Larminat, P. and Thomas, Y. (1983). Automatic Control—Linear Systems. Vol. 1: Signals and Systems, WNT, Warsaw, (in Polish, translation from French).
    Search in Google Scholar
  23. Lasota, A. (1977). Ergodic problems in biology, Société Mathematique de France, Asterisque 50: 239-250.
    Search in Google Scholar
  24. Lasota, A. (1979). Invariant measures and a linear model of turbulence, Rediconti del Seminario Matematico della Universita di Padova 61: 39-48.
    Search in Google Scholar
  25. Lasota, A. (1981). Stable and chaotic solutions of a first order partial differential equation, Nonlinear Analysis Theory, Methods & Applications 5(11): 1181-1193.10.1016/0362-546X(81)90012-2
    Search in Google Scholar
  26. Lasota, A. and Mackey, M. C. (1994). Chaos, Fractals, and Noise Stochastic Aspects of Dynamics, Springer-Verlag, New York, NY.10.1007/978-1-4612-4286-4
    Search in Google Scholar
  27. Lasota, A., Mackey, M. C. and Wazewska-Czyzewska, M. (1981). Minimazing theraupetically induced anemia, Journal of Mathematical Biology 13: 149-158.10.1007/BF002752107328364
    Search in Google Scholar
  28. Lasota, A. and Myjak, J. (2002). On a dimension of measures, Bulletin of the Polish Academy of Sciences: Mathematics 50(2): 221-235.
    Search in Google Scholar
  29. Lasota, A. and Szarek, T. (2002). Dimension of measures invariant with respect to the Wazewska partial differential equation, Journal of Differential Equations 196: 448-465.10.1016/j.jde.2003.10.005
    Search in Google Scholar
  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: 481-488.10.1090/S0002-9947-1973-0335758-1
    Search in Google Scholar
  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.
    Search in Google Scholar
  32. Lebowitz, J. L. and Penrose, O. (1973). Modern ergodic theory, Physics Today 26: 155-175.10.1063/1.3127948
    Search in Google Scholar
  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): 1215-1224.10.3934/dcds.2009.24.1215
    Search in Google Scholar
  34. Mackey, M. C. (2007). Adventures in Poland: Having fun and doing research with Andrzej Lasota, Matematyka Stosowana 8: 5-32.
    Search in Google Scholar
  35. Mackey, M. C. and Glass, L. (1977). Oscillations and chaos in physiological control systems, Science, New Series 197(4300): 287-289.10.1126/science.267326
    Search in Google Scholar
  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.
    Search in Google Scholar
  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.
    Search in Google Scholar
  38. Mitkowski, W. (2010). Chaos in linear systems, Pomiary, Automatyka, Kontrola 56(5): 381-384, (in Polish).
    Search in Google Scholar
  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.
    Search in Google Scholar
  40. Myjak, J. (2008). Andrzej Lasota's selected results. Opuscula Mathematica 28(4): 363-394.
    Search in Google Scholar
  41. Myjak, J. and Rudnicki, R. (2002). Stability versus chaos for a partial differential equation, Chaos Solitons & Fractals 14: 607-612.10.1016/S0960-0779(01)00190-4
    Search in Google Scholar
  42. Nadzieja, T. (1996). Individual ergodic theorem from the topological point of view, Wiadomości Matematyczne 32: 27-36, (in Polish).
    Search in Google Scholar
  43. Nicholson, A. J. (1954). An outline of the dynamic of animal population, Australian Journal of Zoology 2: 9-65.10.1071/ZO9540009
    Search in Google Scholar
  44. Ott, E. (1993). Chaos in Dynamic Systems, WNT, Warsaw, (in Polish, translation from English).
    Search in Google Scholar
  45. Prodi, G. (1960). Teoremi Ergodici per le Equazioni della Idrodinamica, C. I. M. E., Rome.
    Search in Google Scholar
  46. Rudnicki, R. (1985a). Invariant measures for the flow of a first order partial differential equation, Ergodic Theory & Dynamical Systems 5: 437-443.10.1017/S0143385700003059
    Search in Google Scholar
  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.
    Search in Google Scholar
  48. Rudnicki, R. (1988). Strong ergodic properties of a first-order partial differential equation, Journal of Mathematical Analysis and Applications 133: 14-26.10.1016/0022-247X(88)90361-7
    Search in Google Scholar
  49. Rudnicki, R. (2004). Chaos for some infinite-dimensional dynamical systems, Mathematical Methods in the Applied Sciences 27: 723-738.10.1002/mma.498
    Search in Google Scholar
  50. Rudnicki, R. (2009). Chaoticity of the blood cell production system, Chaos: An Interdisciplinary Journal of Nonlinear Science 19(043112): 1-6.10.1063/1.325836420059208
    Search in Google Scholar
  51. Shampine, L. F., Thompson, S. and Kierzenka, J. (2002). Solving delay differential equations with dde23, available at www.mathworks.com/dde_tutorial
    Search in Google Scholar
  52. Silva, C. E. (2010). Lecture on dynamical systems, Spring School of Dynamical Systems, Będlewo, Poland.
    Search in Google Scholar
  53. Szlenk, W. (1982). Introduction to the Theory of Smooth Dynamical Systems, PWN, Warsaw, (in Polish).
    Search in Google Scholar
  54. Taylor, S. R. (2004), Probabilistic Properties of Delay Differential Equations, Ph.D. thesis, University of Waterloo, Ontario, Canada.
    Search in Google Scholar
  55. Tucker, W. (1999). The Lorenz attractor exists, Comptes Rendus. Mathématique. Académie des Sciences, Paris 328(I): 1197-1202.
    Search in Google Scholar
  56. Ulam, S. M. (1960), A Collection of Mathematical Problems, Interscience Publishers, New York, NY/London.
    Search in Google Scholar
  57. Walther, H. O. (1981). Homoclinic solution and chaos in x (t) = f(x(t - 1)), Nonlinear Analysis: Theory, Methods & Applications 5(7): 775-788.10.1016/0362-546X(81)90052-3
    Search in Google Scholar
  58. Ważewska-Czyżewska, M. (1983). Erythrokinetics. Radioisotopic Methods of Investigation and Mathematical Approach, National Center for Scientific, Technical and Economic Information, Warsaw.
    Search in Google Scholar
  59. Ważewska-Czyżewska, M. and Lasota, A. (1976). Mathematical problems of blood cells dynamics system, Matematyka Stosowana 6: 23-40, (in Polish).
    Search in Google Scholar
DOI: https://doi.org/10.2478/v10006-012-0019-4 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X
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Language: English
Page range: 259 - 267
Published on: Jun 28, 2012
Published by: University of Zielona Góra
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year
Keywords:
ergodic theory,
chaos,
invariant measures,
attractors,
delay differential equations
Related subjects:
Mathematics,
Applied mathematics

© 2012 Paweł Mitkowski, Wojciech Mitkowski, published by University of Zielona Góra
This work is licensed under the Creative Commons License.

Volume 22 (2012): Issue 2 (June 2012)