Advertisement

Nonlinear Equations. Qualitative Macroscopic Changes

  • Hermann Haken
Part of the Springer Series in Synergetics book series (SSSYN, volume 20)

Abstract

In this and the subsequent chapter we deal with a problem central to synergetics, namely qualitative macroscopic changes of complex systems. Though it is possible to treat the various instabilities under the impact of noise by means of a single approach, for pedagogical reasons we shall deal with the special cases individually. For the same reasons we first start with equations which do not contain fluctuations (noise) and shall treat the corresponding problems only later. The general philosophy of our approach was outlined in Sect. 1.14.

Keywords

Nonlinear Equation Hopf Bifurcation Linear Stability Analysis Period Doubling Transient Solution 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. In this chapter I present an approach initiated in 1962 (H. Haken: Talk at the International Conference on Optical Pumping, Heidelberg 1962), and applied to laser theory including quasi-periodic motion, e.g. bifurcation to tori (see, e.g.Google Scholar
  2. H. Haken, H. Sauermann: Z. Phys. 176, 47 (1963)ADSCrossRefGoogle Scholar
  3. H. Haken: Laser Theory, in Encylopedia of Physics, Vol. XXV, 2c, Light and Matter Ic (Springer, Berlin, Heidelberg, New York 1970) and reprint edition Laser Theory (Springer, Berlin, Heidelberg, New York 1983)Google Scholar
  4. This author’s approach is based on the slaving principle and represents, in modern language, “dynamic bifurcation theory” (which allows one to cope with transients and fluctuations). “Static” bifurcation theory was initiated in the classical papers byGoogle Scholar
  5. H. Poincaré: Les méthodes nouvelles de la mécanique céleste T. 1 (Gauthier-Villars, Paris 1892) H. Poincaré: Acta Math. 7, 1 (1885)zbMATHGoogle Scholar
  6. A. M. Lyapunov: Sur le masse liquide homogène donnée d’un mouvement de rotation. Zap. Acad. Nauk, St. Petersburg 1, 1 (1906)Google Scholar
  7. E. Schmidt: Zur Theorie der linearen und nichtlinearen Integralgleichungen, 3. Teil, Math. Annalen 65, 370 (1908)zbMATHCrossRefGoogle Scholar
  8. While this field seems to have been more or less dormant for a while (with the exception of bifurcation theory in fluid dynamics), the past decade has seen a considerable increase of interest as reflected by recent texts. We mention in particularGoogle Scholar
  9. D. H. Sattinger: Topics in Stability and Bifurcation Theory, Lecture Notes Math., Vol. 309 (Springer, Berlin, Heidelberg, New York 1972)Google Scholar
  10. D. H. Sattinger: Group Theoretic Methods in Bifurcation Theory, Lecture Notes Math., Vol. 762 (Springer, Berlin, Heidelberg, New York 1980)Google Scholar
  11. G. Iooss: Bifurcation of Maps and Applications, Lecture Notes, Mathematical Studies (North-Holland, Amsterdam 1979)Google Scholar
  12. G. Iooss, D. D. Joseph: Elementary Stability and Bifurcation Theory (Springer, Berlin, Heidelberg, New York 1980)zbMATHGoogle Scholar
  13. These authors deal in an elegant fashion with “static” bifurcation theory.Google Scholar
  14. H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983)Google Scholar
  15. H. Haken: Unpublished material References on catastrophe theory areGoogle Scholar
  16. R. Thorn: Structural Stability and Morphogenesis (Benjamin, Reading, MA 1975) Further references on this subject can be found inGoogle Scholar
  17. H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983)Google Scholar
  18. E. Hopf: Abzweigung einer periodischen Lösung eines Differentialsystems. Berichte der Mathematisch-Physikalischen Klasse der Sächsischen Akademie der Wissenschaften zu Leipzig XCIV, 1 (1942)Google Scholar
  19. J. Marsden, M. McCracken: The Hopf Bifurcation and Its Applications. Lecture Notes Appl. Math. Sci., Vol. 18 (Springer, Berlin, Heidelberg, New York 1976)Google Scholar
  20. D. D. Joseph: Stability of Fluids Motion. Springer Tracts Natural Philos., Vols. 27, 28 (Springer, Berlin, Heidelberg, New York 1976)Google Scholar
  21. A. S. Monin, A. M. Yaglom: Statistical Fluid Mechanics, Vol. I (MIT Press, Cambridge, MA 1971)Google Scholar
  22. H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983)Google Scholar
  23. H. Haken: Laser Theory, in Encylopedia of Physics, Vol. XXV, 2c, Light and Matter Ic (Springer, Berlin, Heidelberg, New York 1970) and reprint edition Laser Theory (Springer, Berlin, Heidelberg, New York 1983)Google Scholar
  24. R. L. Stratonovich: Topics in the Theory of Random Noise, Vols. 1, 2 (Gordon and Breach, New York 1963, 1967)zbMATHGoogle Scholar
  25. H. Haken: Z. Phys. B29, 61 (1978)ADSGoogle Scholar
  26. H. Haken: Unpublished materialGoogle Scholar
  27. Compare Sect. 8.7 and H. Haken: Z. Phys. B30, 423 (1978) and unpublished material. For different approaches cf. A. Chenciner, G, Iooss: Arch. Ration. Mech. Anal. 69, 109 (1979)MathSciNetADSGoogle Scholar
  28. G. R. Sell: Arch. Ration. Mech. Anal. 69, 199 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  29. G. R. Sell: In Chaos and Order in Nature, Springer Ser. Synergetics, Vol. 11, ed. by H. Haken (Springer, Berlin, Heidelberg, New York 1980) p. 84Google Scholar
  30. L. D. Landau, E. M. Lifshitz: In Course of Theoretical Physics, Vol. 6, Fluid Mechanics (Pergamon, London, New York 1959)Google Scholar
  31. E. Hopf: Commun. Pure Appl. Math. 1, 303 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  32. D. Ruelle, F. Takens: Commun. Math. Phys. 20, 167 (1971)MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. S. Newhouse, D. Ruelle, F. Takens: Commun. Math. Phys. 64, 35 (1978)MathSciNetADSzbMATHCrossRefGoogle Scholar
  34. S. Grossmann, S. Thomae: Z. Naturforsch. 32A, 1353 (1977)MathSciNetADSGoogle Scholar
  35. M. J. Feigenbaum: J. Stat. Phys. 19, 25 (1978)MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. —: Phys. Lett. 74A, 375 (1979)MathSciNetADSGoogle Scholar
  37. P. Collet, J. P. Eckmann: Iterated Maps on the Interval as Dynamical Systems (Birkhäuser, Boston 1980)zbMATHGoogle Scholar
  38. T. Geisel, J. Nierwetberg: In Evolution of Order and Chaos, Springer Ser. Synergetics, Vol. 17, ed. by H. Haken (Springer, Berlin, Heidelberg, New York 1982) p. 187Google Scholar
  39. Y. Pomeau, P. Manneville: Commun. Math. Phys. 77, 189 (1980)MathSciNetADSCrossRefGoogle Scholar
  40. G. Mayer-Kress, H. Haken: Phys. Lett. 82A, 151 (1981)MathSciNetADSGoogle 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