Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-17T14:16:37.569Z Has data issue: false hasContentIssue false
  • English
  • Français

Catastrophe theory and equations of state: conditions for a butterfly singularity

Published online by Cambridge University Press:  24 October 2008

Ian N. Stewart
Ian N. Stewart
Affiliation:
University of Warwick
Get access Rights & Permissions [Opens in a new window]

Abstract

Parametrized equations of state of the form

where s = (s1, … sr) ∈ Rr, x ∈ R, and F is smooth, are common in applied science. Often there exist multiple states x for given s, which may bifurcate as s varies.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 3 , November 1980 , pp. 429 - 449
Copyright
Copyright © Cambridge Philosophical Society 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Aris, R.Introduction to the analysis of chemical reactors (Englewood Cliffs NJ, Prentice-Hall, 1965).Google Scholar
(2)Arnol'd, V. I.Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, and Ek, and Lagrangian singularities. Funkcional. Anal, i Priložen 6 (1972) 3–25; Functional Anal. Appl. 6 (1972), 254–272.Google Scholar
(3)Bröcker, TH. and Lander, L.Differentiable germs and catastrophes, London Math. Soc. Lecture Notes 17 (Cambridge University Press, 1975).CrossRefGoogle Scholar
(4)Cowan, J. D. and Ermentrout, G. B. Some aspects of the ‘eigenbehaviour’ of neural nets. In Studies in mathematics, vol. 15; Studies in mathematical biology, part i, ed. Levin, S. A., pp. 67–117 (Math. Assoc. America, 1978).Google Scholar
(5)Gibson, C. G.Singular points of smooth mappings, Research Notes in Mathematics, no. 25 (San Francisco and London, Pitman, 1979).Google Scholar
(6)Golubitsky, M.An introduction to catastrophe theory and its applications. SIAM Rev. 28 (1978), 352–387.Google Scholar
(7)Golubitsky, M. and Guillemin, V.Stable mappings and their singularities (Berlin and New York, Springer-Verlag, 1973).Google Scholar
(8)Lu, Y.-C.Singularity theory and an introduction to catastrophe theory. (Berlin and New York, Springer-Verlag, 1976).CrossRefGoogle Scholar
(9)Markus, L. Dynamical systems – five years after. In Dynamical Systems – Warwick 1974, Lecture Notes in Mathematics, no. 468, ed. Manning, A., pp. 354–365 (Berlin and New York, Springer-Verlag, 1975).Google Scholar
(10)Marsden, J. E.Qualitative methods in bifurcation theory, Bull. Amer. Math. Soc. 84 (1978), 1125–1148.CrossRefGoogle Scholar
(11)Martinet, J.Singularités des fonctions et applications differentiables (Lecture Notes, PUC, Rio de Janeiro, 1974).Google Scholar
(12)Poston, T.On deducing the presence of catastrophes. Math. Sci. Humaines. (To appear.)Google Scholar
(13)Poston, T. and Stewart, I. N.Catastrophe theory and its applications. (San Francisco and London, Pitman, 1978).Google Scholar
(14)Rand, D. Arnol'd's classification of simple singularities of smooth functions. Duplicated notes, Mathematics Institute, University of Warwick, 1977.Google Scholar
(15)Smale, S.Topology and mechanics. Invent. Math. 10 (1970), 305–331.CrossRefGoogle Scholar
(16)Thom, R.Structural stability and morphogenesis, (New York, Benjamin-Addison-Wesley, 1975).Google Scholar
(17)Thom, R.Les singularités des applications différentiables, Ann. Inst. Fourier 6 (1955–1956), 43–87.Google Scholar
(18)Wassermann, G.Stability of Unfoldings, Lecture Notes in Mathematics 393 (Berlin and New York, Springer-Verlag, 1974).Google Scholar
(19)Whitney, H.On singularities of mappings of Euclidean spaces, I. Mappings of the plane into the plane. Ann. of Math. 62 (1955), 374–410.CrossRefGoogle Scholar
(20)Wilson, H. R.and Cowan, J. D. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12 (1972), 1–24.CrossRefGoogle ScholarPubMed
(21)Zeeman, E. C.Catastrophe theory: selected papers (1972–1977) (Reading Mass., Addison-Wesley, 1977).Google Scholar
(22)Zeeman, E. C. A boundary value problem involving cusps. (In preparation).Google Scholar