Abstract
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).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. Baconnier, G. Benchetrit, J. Demongeot and T. Pham Dinh, Lect. Notes in Biomaths 49: 2–16 (1983).
M. Barra, C. Bruni and G. Koch, Lect. Notes in Biomaths 32: 140–154 (1979).
M.S. Bartlett, Proc. Third Berkeley Symp. Math. Stat. and Prob. 4: 81–109 (1956).
G. Benchetrit, P. Baconnier and J. Demongeot, “Concepts and formalization of breathing”, Manchester Univ. Press (1987).
A.B. Budgor, K. Lindenberg and K.E. Shuler, J. Stat. Phys. 15: 375391 (1976).
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).
M. Cosnard, J. Demongeot and A. Le Breton, Lect. Notes in -Biomaths 49, Springer Verlag, New York (1983).
M.Cosnard and J. Demongeot, C.R. Acad. Sci. 300: 551–556 (1985).
M. Cosnard and J. Demongeot, Lect. Notes in Maths 1163: 23–32 (1985).
L. Demetrius and J. Demongeot, “Etude démographique de la France entre 1860 et 1965”, submitted.
J. Demongeot, M. Cosnard and C. Jacob, in “Dynamical systems - a renewal of mechanism”, eds. Diner et al., World Sc. Publ., Singapore (1986).
J. Demongeot and C. Jacob, “Confineurs: une approche stochastique”, C.R. Acad Sc., to appear.
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).
J.L. Doob, “Stochastic processes”, J. Wiley, New York 18953).
W. Ebeling and H. Engel-Hebert, Physica 104A: 378–396 (1980).
A. Friedman, “Stochastic differential equations and applications”, Academic Press, New York (1976).
D.T. Gillespie, J. Phys. Chem. 81: 2340–2361 (1977).
C. Godbillon, “Eléments de topologie algébrique”, Hermann, Paris (1971).
D. Henry, Lect. Notes in Maths 840, Springer Verlag, New York (1981).
C. Jacob, in “Time series analysis: theory and practice”, ed. O.D. Anderson, North Holland, Amsterdam (1985).
C. Jacob, “Stochastic limit cycles and confiners: definitions and comparative study in the case of Markovian processes”, Lect. Notes in Maths, to appear.
W. Jager and J.D. Murray, Lect. Notes in Biomaths 55, Springer Verlag, Berling (1984).
C. Jeffries, Lect. Notes in Biomaths 2: 123–131 (1974).
M.A. Krasnosel’skii, Translation operator along trajectories of differential equations. Transi. of Math. Monographs, AMS, Providence (1968).
H.J. Kushner, “Stochastic stability and control,” Ac. Press, London (1967).
G. Matheron, “Random sets and integral geometry”, J. Wiley, New York (1975).
R.M. May, “Stability and complexity in model ecosystems”, Princeton Un. Press, Princeton (1974).
J. Neveu, “Bases Mathématiques cu calcul des probabilités”, Masson, Paris (1970).
S. Orey, “Lecture notes on limit theorems for Markov chain transition probabilities”, van Nortrand Rheinhold Math. Studies, London (1971).
R.A. Parker, Lect. Notes in Biomaths 2: 174–183 (1974).
T. Pham Dinh, J. Demongeot, P. Baconnier and G. Benchetrit, J. Theor. Biol. 103: 113–132 (1983).
W.M. Schaffer, S. Ellner and M. Kot, “Effects of noise on some dynamical models in ecology”, J. Math. Biol., to appear.
H. Tong, Lect. Notes in Stats. 21, Springer, Berlin (1983).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer Science+Business Media New York
About this chapter
Cite this chapter
Demongeot, J., Jacob, C., Cinquin, P. (1987). Periodicity and Chaos in Biological Systems: New Tools for the Study of Attractors. In: Degn, H., Holden, A.V., Olsen, L.F. (eds) Chaos in Biological Systems. NATO ASI Series, vol 138. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9631-5_28
Download citation
DOI: https://doi.org/10.1007/978-1-4757-9631-5_28
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-9633-9
Online ISBN: 978-1-4757-9631-5
eBook Packages: Springer Book Archive