Algebraic computations in elementary catastrophe theory

  • K. Millington
  • F. J. Wright
Applications I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 204)


Elementary catastrophe theory describes the behaviour of the stationary points of a typical family of real-valued functions, and thereby provides a model for many different types of "quasi-static" system. Many algebraic problems are posed by applications of the theory: the "classification" problem involves working with ideals of rings of functions; the "mapping" problem involves constructing smooth coordinate transformations. We survey previous applications of computer algebra and describe our contribution to developing programs to solve these two problems.


Elementary Catastrophe Theory 
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  1. Aman J, D'Inverno R, Joly G & MacCallum M 1985 Progress on the equivalence problem — these proceedings.Google Scholar
  2. Archbold J W 1970 Algebra (London: Pitman) 4th edition.Google Scholar
  3. Armbruster D 1985 Bifurcation theory and computer algebra: an initial approach — these proceedings.Google Scholar
  4. Arnol'd V I 1972 Normal forms for functions near degenerate critical points, the Weyl groups A k, D k and E k and Lagrangian singularities. Funkcional Anal. i Priložen 6, 3–25; Functional Anal. Appl. 6, 254–72.Google Scholar
  5. Bröcker Th & Lander L 1975 Differentiable germs and catastrophes. LMS Lecture Notes 17. CUP.Google Scholar
  6. Buchberger B 1984 Gröbner bases: an algorithmic method in polynomial ideal theory. Chap. 6 of Recent trends in multidimensional systems theory, ed. N K Bose (Reidel).Google Scholar
  7. Chester C, Friedman B & Ursell F 1957 An extension of the method of steepest descents. Proc. Camb. Phil. Soc. 53, 599–611.Google Scholar
  8. Connor J N L 1981 Uniform semiclassical evaluation of Franck-Condon factors and inelastic atom-atom scattering amplitudes. J. Chem. Phys. 74(2), 1047–1052.Google Scholar
  9. Connor J N L & Curtis P R 1984a Differential equations for the cuspoid canonical integrals. J. Math. Phys. 25(10), 2895–2902.Google Scholar
  10. Connor J N L, Curtis P R & Farrelly D 1984b The uniform asymptotic swallowtail approximation: practical methods for oscillating integrals with four coalescing saddle points. J. Phys. A17, 283–310.Google Scholar
  11. Dangelmayr G & Wright F J 1984 On the validity of the paraxial eikonal in catastrophe optics. J. Phys. A17, 99–108.Google Scholar
  12. Dangelmayr G & Wright F J 1985 Caustics and diffraction from a line source. Optica Acta, to appear.Google Scholar
  13. Dangelmayr G, Armbruster D & Neveling M 1985 A codimension-three bifurcation for the laser with saturable absorber. Preprint, Institut für Informationsverarbeitung, University of Tübingen, FRG.Google Scholar
  14. Hassard B 1978 Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon. J. Theoret. Biol. 71, 401–420.Google Scholar
  15. Keener J P 1983 Oscillatory coexistence in the chemostat: a codimension two unfolding. SIAM J. Appl. Math. 43, 1005–18.Google Scholar
  16. Mather J 1968 Stability of C-mappings III: Finitely determined map-germs. Publ. Math. IHES 35, 127–56.Google Scholar
  17. Millington K & Wright F J 1985 Practical determination via Taylor coefficients of the right-equivalences used in elementary catastrophe theory: cuspoid unfoldings of univariate functions. In preparation.Google Scholar
  18. Millington K 1985 In preparation.Google Scholar
  19. Peregrine D H & Smith R 1979 Nonlinear effects upon waves near caustics. Phil. Trans. Roy. Soc. Lond. A292, 341–70.Google Scholar
  20. Poston T & Stewart I N 1978 Catastrophe Theory and its Applications (London: Pitman).Google Scholar
  21. Rand R H 1984a Computer Algebra in Applied Mathematics: An Introduction to MACSYMA. (London: Pitman).Google Scholar
  22. Rand R H 1984b Derivation of the Hopf Bifurcation Formula using Lindstedt's Perturbation Method and MACSYMA andGoogle Scholar
  23. Rand R H & Keith W L 1984 Normal Form and Center Manifold Calculations on MACSYMA both in Applications of Computer Algebra, ed. R Pavelle.Google Scholar
  24. Thom R 1972 Stabilité Structurelle et Morphogénèse (Reading, Mass.: Benjamin) (Engl. transl. 1975 Structural Stability and Morphogenesis (Reading, Mass.: Benjamin)).Google Scholar
  25. Uspensky J V 1948 Theory of Equations (New York: McGraw-Hill).Google Scholar
  26. Uzer T & Child M S 1982 Collisions and Umbilic Catastrophes: Direct Determination of the Control Parameters for use in a Uniform S Matrix Approximation. Mol. Phys. 46, 1371–88.Google Scholar
  27. Uzer T & Child M S 1983 Mol. Phys. 50, 247.Google Scholar
  28. Wassermann G 1974 Stability of Unfoldings. Lecture Notes in Mathematics 393 (New York: Springer).Google Scholar
  29. Wright F J & Dangelmayr G 1985a On the exact reduction of a univariate catastrophe to normal form. J. Phys. A18, in press.Google Scholar
  30. Wright F J & Dangelmayr G 1985b Explicit iterative algorithms to reduce a univariate catastrophe to normal form. Computing, to appear.Google Scholar
  31. Zeeman E C 1977 Catastrophe Theory: Selected Papers 1972–1977 (Reading, Mass.: Addison-Wesley).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • K. Millington
    • 1
  • F. J. Wright
    • 1
  1. 1.School of Mathematical Sciences Queen Mary CollegeUniversity of LondonLondonUK

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