Stochastic Description of Various Types of Bifurcations in Chemical Systems

  • H. Lemarchand
  • A. Fraikin
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 27)


When a nonlinear system evolues under far-from-equilibrium conditions in the vicinity of a bifurcation point, a purely deterministic description often proved to be incomplete. The fluctuations of the dynamical variables can play an essential role and obstruct the observation of a transition expected by a deterministic analysis. In the framework of the deterministic approach, the stability of the different states according to the values of the control parameters is studied through a mathematical analysis of the velocity field. In particular, the theory of normal forms leads to the determination of the various kinds of attractors [1,2]. As far as we are concerned with the stochastic approach, the master equation has been widely used to analyze bifurcations of homogeneous or spatially ordered steady states or of limit cycles [3,4]. Our aim in the present contribution is to insist on the generality of the method to analyze various kinds of bifurcations in nonlinear nonequilibrium systems. The general procedure proposed to obtain a local description of the probability, which allows us to determine the system’s attractors, turns out to display marked analogies with the theory of normal forms.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. ARMOLD, Chapitres Supplémentaires de la Théorie des Equations Différentielles Ordiraires (Mir, Moscou 1980)Google Scholar
  2. 2.
    S.N. CHOW and J.K. HALE, Methods of Bifurcation Theory (Berlin, Heidelberg, New York, 1982)MATHCrossRefGoogle Scholar
  3. 3.
    M. MALEK MANSOUR, C. VAN DEN BROECK, G. NICOLIS, and J.W. TURNER, Ann. Phys. (New York) 131, 283 (1981)ADSMATHCrossRefGoogle Scholar
  4. 4.
    H. LEMARCHAND and G. NICOLIS, J.Stat. Phys. (Submitted)Google Scholar
  5. 5.
    H. LEMARCHAND, Bull. Acad. Roy. Belg. (in press)Google Scholar
  6. 6.
    D. WALGRAEF, G. DEWEL and P. BORCKMANS, Adv. Chem. Phys. 49,311 (1982)CrossRefGoogle Scholar
  7. 7.
    D. WALGRAEF, G. DEWEL and P. BORCKMANS, J.Chem. Phys. 78 3043 (1983)ADSCrossRefGoogle Scholar
  8. 8.
    A. FRAIKIN and H. LEMARCHAND, J. Stat. Phys. (Submitted)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • H. Lemarchand
    • 1
  • A. Fraikin
    • 1
  1. 1.Laboratoire de Chimie GénéraleUniversité P. et M. CurieTour 55Paris Cédex 05France

Personalised recommendations