Archive for Rational Mechanics and Analysis

, Volume 49, Issue 3, pp 172–186 | Cite as

Necessary and sufficient conditions for complex balancing in chemical kinetics



In a recent publication (Horn & Jackson [1]) it was shown that complex balancing together with mass action type rate laws ensures certain stability properties of a kinetic system, thereby precluding sustained oscillations, bistability and other types of irregular dynamics. In this paper a necessary condition for complex balancing in general kinetics and necessary and sufficient conditions for complex balancing in mass action systems are derived. A theorem is stated which excludes the occurence of equilibria in certain composition regions of general kinetic systems. For mass action systems it is shown that it is sometimes true that the algebraic structure of the reactions suffices to ensure complex balancing, while in other cases complex balancing occurs only if certain relations between the rate constants are satisfied. The number of these relations, called the deficiency of the mass action system is determined by the algebraic structure of the set of reactions underlying that system.


Neural Network Complex System Nonlinear Dynamics Electromagnetism Mass Action 
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  1. 1.
    Horn, F., & R. Jackson, General mass action kinetics. Arch. Rational Mech. Anal. 47, 81–116 (1972).Google Scholar
  2. 2.
    Feinberg, M., Complex balancing in general kinetic systems. Arch. Rational Mech. Anal. 49, 187–194 (1972).Google Scholar
  3. 3.
    Horn, F., Stability and Complex Balancing in Mass Action Kinetics with Three Short Complexes. In Preparation.Google Scholar
  4. 4.
    Gantmacher, F. R., The Theory of Matrices, Chapter XIII. New York, N.Y.: Chelsea Publishing Co. 1964.Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • F. Horn
    • 1
  1. 1.University of RochesterNew York

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