Synergetics pp 105-146 | Cite as


Old Structures Give Way to New Structures
  • Hermann Haken
Part of the Springer Series in Synergetics book series (SSSYN, volume 1)


This chapter deals with completely deterministic processes. The question of stability of motion plays a central role. When certain parameters change, stable motion may become unstable and completely new types of motion (or structures) appear. Though many of the concepts are derived from mechanics, they apply to many disciplines.


Singular Point Static Instability Excited Atom Stable Limit Cycle Potential Curve 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. N. N. Bogoliubov, Y. A. Mitropolsky: Asymptotic Methods in the Theory of Nonlinear Oscillations (Hindustan Publ. Corp., Delhi 1961)Google Scholar
  2. N. Minorski: Nonlinear Oscillations (Van Nostrand, Toronto 1962)Google Scholar
  3. A. Andronov, A. Vitt, S. E. Khaikin: Theory of Oscillators (Pergamon Press, London-Paris 1966)MATHGoogle Scholar
  4. D. H. Sattinger In Lecture Notes in Mathematics, Vol. 309: Topics in Stability and Bifurcation Theory, ed. by A. Dold, B. Eckmann (Springer, Berlin-Heidelberg-New York 1973)Google Scholar
  5. M. W. Hirsch, S. Smale: Differential Equations, Dynamical Systems, and Linear Algebra (Academic Press, New York-London 1974)MATHGoogle Scholar
  6. V. V. Nemytskii, V. V. Stepanov: Qualitative Theory of Deferential Equations (Princeton Univ. Press, Princeton, N.J. 1960)Google Scholar
  7. H. Poincaré: Oeuvres, Vol. 1 (Gauthiers-Villars, Paris 1928)Google Scholar
  8. H. Poincaré: Sur l’equilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math. 7 (1885)Google Scholar
  9. H. Poincaré: Figures d’equilibre d’une masse fluide (Paris 1903)Google Scholar
  10. H. Poincaré: Sur le problème de trois corps et les équations de la dynamique. Acta Math. 13 (1890)Google Scholar
  11. H. Poincaré: Les méthodes nouvelles de la méchanique céleste (Gauthier-Villars, Paris 1892–1899)Google Scholar


  1. J. La Salle, S. Lefshetz: Stability by Ljapunov’s Direct Method with Applications (Academic Press, New York-London 1961)Google Scholar
  2. W. Hahn: Stability of Motion. In Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 138 (Springer, Berlin-Heidelberg-New York 1967)Google Scholar
  3. D. D. Joseph: Stability of Fluid Motions I and II, Springer Tracts in Natural Philosophy, Vols. 27, 28 (Springer, Berlin-Heidelberg-New York 1976)Google Scholar
  4. Exercises 5.3: F. Schlögl: Z. Phys. 243, 303 (1973)Google Scholar

Examples and Exercises on Bifurcation and Stability

  1. A. Lotka: Proc. Nat. Acad. Sci. (Wash.) 6, 410 (1920)ADSCrossRefGoogle Scholar
  2. V. Volterra: Leçons sur la théorie mathematiques de la lutte pour la vie (Paris 1931)Google Scholar
  3. N. S. Goel, S. C. Maitra, E. W. Montroll: Rev. Mod. Phys. 43, 231 (1971)MathSciNetADSCrossRefGoogle Scholar
  4. B. van der Pol: Phil. Mag. 43, 6, 700 (1922);Google Scholar
  5. B. van der Pol: Phil. Mag. 2, 7, 978 (1926);Google Scholar
  6. B. van der Pol: Phil. Mag. 3, 7, 65 (1927)Google Scholar
  7. H. T. Davis: Introduction to Nonlinear Differential and Integral Equations (Dover Publ. Inc., New York 1962)MATHGoogle Scholar
  8. G. Ioss, D. D. Joseph: Elementary Stability and Bifurcation Theory (Springer, Berlin, Heidelberg, New York 1980)Google Scholar

Classification of Static Instabilities, or an Elementary Approach to Thorn’s Theory of Catastrophes

  1. R. Thorn: Structural Stability and Morphogenesis (W. A. Benjamin, Reading, Mass. 1975) Thorn’s book requires a good deal of mathematical background. Our “pedestrian’s” approach provides a simple access to Thorn’s classification of catastrophes. Our interpretation of how to apply these results to natural sciences, for instance biology is, however, entirely different from Thorn’s.Google Scholar
  2. T. Poston, I. Steward: Catastrophe Theory and its Applications (Pitman, London 1978)MATHGoogle Scholar
  3. E. C. Zeeman: Catastrophe Theory (Addison-Wesley, New York 1977)MATHGoogle Scholar
  4. P. T. Sounders: An Introduction to Catastrophe Theory (Cambridge University Press, Cambridge 1980)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Hermann Haken
    • 1
  1. 1.Institut für Theoretische PhysikUniversität StuttgartStuttgart 80Fed. Rep. of Germany

Personalised recommendations