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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)

Abstract

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.

Keywords

Elementary Catastrophe Theory 
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References

  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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