Open Systems & Information Dynamics

, Volume 3, Issue 1, pp 23–95 | Cite as

Fundamental concepts of classical chaos I

  • Tassos Bountis


In this review, some of the fundamental concepts of the theory of classical chaos are presented in a pedagogical style. The paper is directed primarily to readers without any prerequisite knowledge of the subject matter apart from a good background orientation in undergraduate mathematics. Our approach is to a large extent geometrical and is motivated by specific problems of physical interest and illustrated by examples. Neither the notion of “quantum chaos”, nor the probabilistic aspects of chaos requiring certain familiarity with statistical physics are treated here. In particular, the following questions are addressed: (a) What are the fundamental scientific issues that the science of chaos aims to resolve? (b) What are the main new ideas and methods that chaos puts forward to achieve these goals? (c) How successful has chaos been so far in providing solutions to concrete problems?


Statistical Physic Mechanical Engineer System Theory Subject Matter Specific Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Barnsley, M. F.,Fractals Everywhere, Academic Press, San Diego, 1988.Google Scholar
  2. [2]
    Bazzani, A., P. Mazzani, G. Servizi, and G. Turchetti, Il Nuovo Cim.102B(1), 51 (1988).Google Scholar
  3. [3]
    Bergé, P., Y. Pommeau, and C. Vidal,Order Within Chaos, J. Wiley & Sons, New York, 1986.Google Scholar
  4. [4]
    Bollt, E. M. and J. D. Meiss, Physica66D, 282 (1993).Google Scholar
  5. [5]
    Bountis, T., Physica3D, 577 (1981).Google Scholar
  6. [6]
    Bountis, T. and L. Drossos, Phys. Lett.143A(8), 379 (1989).Google Scholar
  7. [7]
    Bountis, T. and M. Kollmann, Physica71D(1, 2), 122 (1994).Google Scholar
  8. [8]
    Chirikov, B. V.,A Universal Instability of Many Dimensional Oscillator Systems, Phys. Rep.52(5), 264 (1979).CrossRefGoogle Scholar
  9. [9]
    Collet, P. and J.-P. Eckmann,Iterated Maps of the Interval as Dynamical Systems, Birkhäuser, Boston, 1980.Google Scholar
  10. [10]
    Collet, P., J.-P. Eckmann, and O. Lanford, Commun. Math. Phys.76, 211 (1980).CrossRefGoogle Scholar
  11. [11]
    Contopoulos, G., Cel. Mech.38, 1 (1986).CrossRefGoogle Scholar
  12. [12]
    Cvitanovič, P., ed.,Universality in Chaos, 2nd edition, Adam Hilger, Bristol, 1989.Google Scholar
  13. [13]
    Devaney, R.,An Introduction to Chaotic Dynamical Systems, Benjamin-Cummings, Menlo Park, California, 1986.Google Scholar
  14. [14]
    Eckmann, J.-P. and D. Ruelle, Rev. Mod. Phys.57, 617 (1985).CrossRefGoogle Scholar
  15. [15]
    Escande, D. F.,Stochasticity in Classical Hamiltonian Systems: Universal Aspects, Phys. Rep.121(3, 4), 166 (1985).CrossRefGoogle Scholar
  16. [16]
    Feigenbaum, M. J., J. Stat. Phys.19, 25 (1978).CrossRefGoogle Scholar
  17. [17]
    Feigenbaum, M. J., J. Stat. Phys.21, 669 (1979).CrossRefGoogle Scholar
  18. [18]
    Feigenbaum, M. J., Commun. Math. Phys.77, 65 (1980).CrossRefGoogle Scholar
  19. [19]
    Feigenbaum, M. J., L. P. Kadanoff, and S. Shenker, Physica5D, 370 (1982).Google Scholar
  20. [20]
    Gleick, J.,Chaos: Making of a New Science, Viking, New York, 1987.Google Scholar
  21. [21]
    Greene, J., J. Math. Phys.20, 1183 (1979).CrossRefGoogle Scholar
  22. [22]
    Greene, J., R. S. MacKay, F. Vivaldi, and M. J. Feigenbaum, Physica3D, 468 (1981).Google Scholar
  23. [23]
    Guckenheimer, J. and P. J. Holmes,Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983.Google Scholar
  24. [24]
    Helleman, R. H. G., [in:]Fundamental Problems in Statistical Mechanics, vol.5, ed. E. G. D. Cohen, North Holland, Amsterdam, 1980).Google Scholar
  25. [25]
    Hénon, M., Commun. Math. Phys.50, 69 (1976).CrossRefGoogle Scholar
  26. [26]
    Hirsch, J. E., B. A. Huberman, and D. J. Scalapino, Phys. Rev.25A, 519 (1982).Google Scholar
  27. [27]
    Hu, B. and J. Rudnick, Phys. Rev. Lett.48, 1645 (1982).CrossRefGoogle Scholar
  28. [28]
    Kadanoff, L. P.,From Order to Chaos, World Scientific, Singapore, 1993.Google Scholar
  29. [29]
    Laskar, J., C. Froeschlé, and A. Celetti, Physica56D, 253 (1992); see also: J. Laskar, Icarus88, 266 (1990); J. Laskar et al., Nature361, 608 and 615 (1993).Google Scholar
  30. [30]
    Libchaber, A. and J. Maurer, J. Phys. (Paris) Coll.41, C3–51 (1980).Google Scholar
  31. [31]
    Libchaber, A. and J. Maurer, [in:]Nonlinear Phenomena at Phase Transitions and Instabilities, ed. T. Riste, Plenum Press, London, 1982.Google Scholar
  32. [32]
    Lichtenberg, A. and M. Lieberman,Regular and Stochastic Motion, Springer, New York, 1983.Google Scholar
  33. [33]
    Lorenz, E. N., J. Atm. Sci.20, 130 (1963).CrossRefGoogle Scholar
  34. [34]
    Ma, S. K,Modern Theory of Critical Phenomena, Benjamin, New York, 1976.Google Scholar
  35. [35]
    MacKay, R. S. and J. D. Meiss, eds.,Hamiltonian Dynamical Systems, Adam Hilger, Bristol, 1987.Google Scholar
  36. [36]
    MacKay, R. S.,Renormalization Approach to Area-Preserving, Twist Maps, World Scientific, Singapore, 1994.Google Scholar
  37. [37]
    Mandelbrot, B. B.,The Fractal Geometry of Nature, Freeman, New York, 1982.Google Scholar
  38. [38]
    Mao, J.-M. and R. H. G. Helleman, Phys. Rev.35A, 1847 (1987).Google Scholar
  39. [39]
    Niven, I.,Rational and Irrational Numbers, Random House, New York, 1963.Google Scholar
  40. [40]
    Peitgen, H.-O. and P. H. Richter,The Beauty of Fractals, Springer-Verlag, Berlin, 1986.Google Scholar
  41. [41]
    Peitgen, H.-O., H. Jürgens, and D. Saupe,Chaos and Fractals: New Frontiers of Science, Springer-Verlag, New York, 1992.Google Scholar
  42. [42]
    Pomeau, Y. and P. Manneville, Commun. Math. Phys.74, 189 (1980).CrossRefGoogle Scholar
  43. [43]
    Rand, D., S. Ostlund, J. Sethna, and E. Siggia, Phys. Rev. Lett.49, 132 (1982).CrossRefGoogle Scholar
  44. [44]
    Rand, D., S. Ostlund, J. Sethna, and E. Siggia, Physica8D, 303 (1983).Google Scholar
  45. [45]
    Rasband, S. N.,Chaotic Dynamics of Nonlinear Systems, J. Wiley, New York, 1990.Google Scholar
  46. [46]
    Reichl, L. E.,The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations, Springer, New York, 1992.Google Scholar
  47. [47]
    Roberts, J. A. G. and G. R. W. Quispel,Chaos and Time-Reversal Symmetry. Order and Chaos in Reversible Dynamical Systems, Phys. Rep.216(2, 3), 64 (1992).CrossRefGoogle Scholar
  48. [48]
    Rudin, W.,Principles of Mathematical Analysis, McGraw-Hill, New York, 1976.Google Scholar
  49. [49]
    Schroeder, M.,Fractals, Chaos and Power Laws, Freeman, New York, 1991.Google Scholar
  50. [50]
    Schuster, H.-G.,Deterministic Chaos: An Introduction, 2nd edition, VCH, Weinheim, 1988.Google Scholar
  51. [51]
    Shenker, S., Physica5D(2, 3), 405 (1982); see also S. Shenker and L. P. Kadanoff, J. Stat. Phys.27, 631 (1982).Google Scholar
  52. [52]
    Sparrow, C.,The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Springer, New York, 1982.Google Scholar
  53. [53]
    Wiggins, S.,Introduction to Applied Dynamical Systems and Chaos, Springer, New York, 1990.Google Scholar

Copyright information

© Nicholas Copernicus University Press 1995

Authors and Affiliations

  • Tassos Bountis
    • 1
  1. 1.Department of MathematicsUniversity of PatrasPatrasGreece

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