Catastrophes and Bifurcations in Variational Problems

  • G. Dangelmayr
Part of the Springer Series in Synergetics book series (SSSYN, volume 4)


Using an extension of catastrophe theory to infinite dimensional spaces, the bifurcation properties of the solutions of nonlinear operator equations, deriving from variational problems, are shown to be determined by those of the stationary points of unfoldings of singularities and so are classifiable in terms of catastrophe polynomials. A stability analysis of the bifurcating solutions is carried out. The results are applied to the buckling of columns and lead here to a nonversal unfolding of a double-cusp. Implications of the techniques for nonequilibrium thermodynamics are indicated.


Stationary Point Variational Problem Bifurcation Diagram Catastrophe Theory Reflexive Banach Space 
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  1. 1.
    R. Thom, “Structural Stability and Morphogenesis”, Benjamin, New York 1975.MATHGoogle Scholar
  2. 2.
    V.I. Arnold, “Critical Points of Smooth Functions”, in “Proceedings of the International Congress of Mathematicians”, Vol. I, Vancouver 1972.Google Scholar
  3. 3.
    E.C. Zeeman, “Catastrophe Theory, Selected Papers 1972–1977”, Advanced Book Program, Reading, Addison-Wesley 1977.Google Scholar
  4. 3.
    T. Poston and I. Stewart, “Catastrophe Theory and its Applications”, Pitman 1978.MATHGoogle Scholar
  5. 4.
    M.A. Krasnoselskii, “Topological Methods in the Theory of Nonlinear Integral Equations”, Pergamon Press, Oxford 1964.Google Scholar
  6. 4.
    R.J. Magnus, “Applications of Topological Degree to the Theory of Branching”, Math. Report 90, Battelle, Geneva 1974.Google Scholar
  7. 5.
    J.P. Keener and H.B. Keller, “Perturbed Bifurcation Theory”, Arch. Rat. Mech. Anal. 50 (1973), 159.MathSciNetMATHCrossRefGoogle Scholar
  8. 5.
    J.P. Keener, “Perturbed Bifurcation Theory at Multiple Eigenvalues”, Arch. Rat. Mech. Anal. 56 (1974), 348.MathSciNetMATHCrossRefGoogle Scholar
  9. 6.
    M. Vainberg and V. Trenogin, “Theory of Branching of Solutions of Nonlinear Equations”, Noordhoff 1974.Google Scholar
  10. 7.
    S. Chow, J. Hale and J. Mallet-Paret, “Application of Generic Bifurcation”, Arch. Rat. Mech. Anal. 59 (1975), 159 and Arch. Rat. Mech. Anal. 62 (1976), 209.MathSciNetMATHCrossRefGoogle Scholar
  11. 7.
    J. Hale, “Generic Bifurcations with Applications”, in “Nonlinear analysis and mechanics: Heriot-Watt Symposium Vol. I” (R.J. Knops, Ed.) Research Notes in Math. 17, Pitman 1977.Google Scholar
  12. 8.
    R.J. Magnus, “On Universal Unfoldings of Certain Real Functions on a Banach Space”, Math. Rep. 100, Battelle, Geneva 1976.Google Scholar
  13. 8.
    R.J. Magnus, “Determinacy in a Certain Class of Germs on a Reflexive Banach Space”, Math. Rep. 103, Battelle, Geneva 1976.Google Scholar
  14. 8.
    R.J. Magnus, “Universal Unfoldings in Banach Spaces: Reduction and Stability”, Math. Rep. 103, Battelle, Geneva 1977.Google Scholar
  15. 9.
    J.P. Keener, “Buckling Imperfection Sensitivity of Columns and Spherical Caps”, Quarterly of Appl. Math., July 1974.Google Scholar
  16. 10.
    S. Lang, “Differential Manifolds”, Addison-Wesley 1972.MATHGoogle Scholar
  17. 11.
    D. Gromoll and W. Meyer, “On Differentiable Functions with Isolated Critical Points”, Topology 8 (1969), 361.MathSciNetMATHCrossRefGoogle Scholar
  18. 12.
    G. Dangelmayr, Thesis, University of Tübingen 1978.Google Scholar
  19. 13.
    R.J. Magnus and T. Poston, “Infinite Dimensions and the Fold Catastrophe”, this volume.Google Scholar
  20. 14.
    T. Poston and I.N. Stewart, “Taylor Expansions and Catastrophes”, Research Notes in Math. 7, Pitman 1976.Google Scholar
  21. 15.
    I. Segal, “Nonlinear Semigroups”, Ann. Math. 78 (1963), 339.MATHCrossRefGoogle Scholar
  22. 15.
    P.R. Chernoff and J.E. Marsden, “Properties of Infinite Dimensional Hamiltonian Systems”, Lecture Notes in Math. 425, Springer 1974.Google Scholar
  23. 16.
    L. Landau und E. Lifschitz, “Theoretische Physik” Bd. VII (Elastizitätstheorie), Akademie Verlag Berlin 1970.Google Scholar
  24. 17.
    J. Thompson and G. Hunt, “A General Theory of Elastic Stability”, Wiley- Interscience 1973.MATHGoogle Scholar
  25. 18.
    L. Bauer, H. Keller and E.L. Reiss, “Multiple Eigenvalues lead to Secondary Bifurcation”, SIAM Rev. 17 (1975), 101.MathSciNetCrossRefGoogle Scholar
  26. 19.
    E.C. Zeeman, “The Umbilic Bracelett and the Double-Cusp Catastrophe”, in “Structural Stability, the Theory of Catastrophes and Applications in the Sciences”, (P. Hilton, Ed.) Lecture Notes in Math. 525, Springer 1976.Google Scholar
  27. 20.
    J. Callahan, “Special bifurcations of the double cusp”, University of Warwick, Preprint, 1978.Google Scholar
  28. 21.
    R. Magnus and T. Poston, “On the Full Unfolding of the Von Karman Equations at a Double Eigenvalue”, Math. Rep. 109, Battelle, Geneva 1977.Google Scholar
  29. 22.
    P. Glansdorff and I. Prigogine, “Thermodynamic Theory of Structures, Stability and Fluctuations”, Wiley-Interscience 1971.Google Scholar
  30. 23.
    G. Nicolis, “Dissipative Structures with Applications to Chemical Reactions”, in “Cooperative Effects, Progress in Synergetics”, (H. Haken, Ed.), North-Holland 1974.Google Scholar
  31. 24.
    F. Schlögl, “Fluctuations in Thermodynamic Non Equilibrium States”, Z. Physik 249 (1971), 1.ADSCrossRefGoogle Scholar
  32. 25.
    H.K. Janssen, “Stability of Transport”, Z. Physik 253 (1972), 176.ADSCrossRefGoogle Scholar
  33. 26.
    I. Prigogine, “Introduction to Thermodynamics of Irreversible Processes”, Wiley Interscience 1967.Google Scholar
  34. 27.
    S.R. DeGroot and P. Mazur, “Non-Equilibrium Thermodynamics”, North-Holland 1962.Google Scholar
  35. 28.
    D. Edelen, “A Nonlinear Onsager Theory of Irreversibility”, Int. J. Engng. Sci. 10 (1972), 481.MathSciNetMATHCrossRefGoogle Scholar
  36. 29.
    E. Pfaffelhuber, “Information-Theoretic Stability and Evolution Criteria in Irreversible Thermodynamics”, J. Stat. Phys. 16 (1977), 69.MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • G. Dangelmayr
    • 1
  1. 1.Institute for Information SciencesUniversity of TübingenTübingenFed. Rep. of Germany

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