Periodicity and Chaos in Biological Systems: New Tools for the Study of Attractors

  • J. Demongeot
  • C. Jacob
  • P. Cinquin
Part of the NATO ASI Series book series (NSSA, volume 138)


In many biological systems, the localisation of attractors of the dynamics and the quantification of the degree of synchrony is very difficult to obtain from experimental data. We propose here new notions of confiners and entropy to solve this problem and we give some examples of application (an “academic example” concerning the noised van der Pol oscillator and two realistic possible fields of application, namely respiratory physiology and population dynamics).


Spline Curve Respiratory Physiology Fourier Power Spectrum Random Oscillation Malthusian Parameter 
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  1. [1]
    P. Baconnier, G. Benchetrit, J. Demongeot and T. Pham Dinh, Lect. Notes in Biomaths 49: 2–16 (1983).CrossRefGoogle Scholar
  2. [2]
    M. Barra, C. Bruni and G. Koch, Lect. Notes in Biomaths 32: 140–154 (1979).CrossRefGoogle Scholar
  3. [3]
    M.S. Bartlett, Proc. Third Berkeley Symp. Math. Stat. and Prob. 4: 81–109 (1956).Google Scholar
  4. [4]
    G. Benchetrit, P. Baconnier and J. Demongeot, “Concepts and formalization of breathing”, Manchester Univ. Press (1987).Google Scholar
  5. [5]
    A.B. Budgor, K. Lindenberg and K.E. Shuler, J. Stat. Phys. 15: 375391 (1976).Google Scholar
  6. [6]
    P. Cinquin, M. Cosnard, J. Demongeot and C. Jacob, in “Analyse numérique des attracteurs étranges”, eds. M. Cosnard and C. Mira, Editions du CNRS, Paris (1987).Google Scholar
  7. [7]
    M. Cosnard, J. Demongeot and A. Le Breton, Lect. Notes in -Biomaths 49, Springer Verlag, New York (1983).Google Scholar
  8. [8]
    M.Cosnard and J. Demongeot, C.R. Acad. Sci. 300: 551–556 (1985).Google Scholar
  9. [9]
    M. Cosnard and J. Demongeot, Lect. Notes in Maths 1163: 23–32 (1985).CrossRefGoogle Scholar
  10. [10]
    L. Demetrius and J. Demongeot, “Etude démographique de la France entre 1860 et 1965”, submitted.Google Scholar
  11. [11]
    J. Demongeot, M. Cosnard and C. Jacob, in “Dynamical systems - a renewal of mechanism”, eds. Diner et al., World Sc. Publ., Singapore (1986).Google Scholar
  12. [12]
    J. Demongeot and C. Jacob, “Confineurs: une approche stochastique”, C.R. Acad Sc., to appear.Google Scholar
  13. [13]
    J. Demongeot, C. Jacob and P. Cinquin, in “Formalisation en biologie et en économie”, eds. J. Demongeot and P. Malgrange, Presses Un. de Dijon (1987).Google Scholar
  14. [14]
    J.L. Doob, “Stochastic processes”, J. Wiley, New York 18953).Google Scholar
  15. [15]
    W. Ebeling and H. Engel-Hebert, Physica 104A: 378–396 (1980).CrossRefGoogle Scholar
  16. [16]
    A. Friedman, “Stochastic differential equations and applications”, Academic Press, New York (1976).Google Scholar
  17. [17]
    D.T. Gillespie, J. Phys. Chem. 81: 2340–2361 (1977).CrossRefGoogle Scholar
  18. [18]
    C. Godbillon, “Eléments de topologie algébrique”, Hermann, Paris (1971).Google Scholar
  19. [19]
    D. Henry, Lect. Notes in Maths 840, Springer Verlag, New York (1981).Google Scholar
  20. [20]
    C. Jacob, in “Time series analysis: theory and practice”, ed. O.D. Anderson, North Holland, Amsterdam (1985).Google Scholar
  21. [21]
    C. Jacob, “Stochastic limit cycles and confiners: definitions and comparative study in the case of Markovian processes”, Lect. Notes in Maths, to appear.Google Scholar
  22. [22]
    W. Jager and J.D. Murray, Lect. Notes in Biomaths 55, Springer Verlag, Berling (1984).Google Scholar
  23. [23]
    C. Jeffries, Lect. Notes in Biomaths 2: 123–131 (1974).CrossRefGoogle Scholar
  24. [24]
    M.A. Krasnosel’skii, Translation operator along trajectories of differential equations. Transi. of Math. Monographs, AMS, Providence (1968).Google Scholar
  25. [25]
    H.J. Kushner, “Stochastic stability and control,” Ac. Press, London (1967).Google Scholar
  26. [26]
    G. Matheron, “Random sets and integral geometry”, J. Wiley, New York (1975).Google Scholar
  27. [27]
    R.M. May, “Stability and complexity in model ecosystems”, Princeton Un. Press, Princeton (1974).Google Scholar
  28. [28]
    J. Neveu, “Bases Mathématiques cu calcul des probabilités”, Masson, Paris (1970).Google Scholar
  29. [29]
    S. Orey, “Lecture notes on limit theorems for Markov chain transition probabilities”, van Nortrand Rheinhold Math. Studies, London (1971).Google Scholar
  30. [30]
    R.A. Parker, Lect. Notes in Biomaths 2: 174–183 (1974).Google Scholar
  31. [31]
    T. Pham Dinh, J. Demongeot, P. Baconnier and G. Benchetrit, J. Theor. Biol. 103: 113–132 (1983).PubMedCrossRefGoogle Scholar
  32. [32]
    W.M. Schaffer, S. Ellner and M. Kot, “Effects of noise on some dynamical models in ecology”, J. Math. Biol., to appear.Google Scholar
  33. [33]
    H. Tong, Lect. Notes in Stats. 21, Springer, Berlin (1983).Google Scholar

Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • J. Demongeot
    • 1
  • C. Jacob
    • 2
  • P. Cinquin
    • 1
  1. 1.TIM3-IMAGUniversity of Grenoble ISt. Martin d’Hères CédexFrance
  2. 2.Laboratoire de biométrieINRA-CNRZJouy-en-JosasFrance

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