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Catastrophes and partial differential equations

John Guckenheimer

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 2, page 31-59
  • ISSN: 0373-0956

Abstract

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This paper outlines the manner in which Thom’s theory of catastrophes fits into the Hamilton-Jacobi theory of partial differential equations. The representation of solutions of a first order partial differential equation as lagrangian manifolds allows one to study the local structure of their singularities. The structure of generic singularities is closely related to Thom’s concept of the elementary catastrophe associated to a singularity. Three concepts of the stability of a singularity are discussed.

How to cite

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Guckenheimer, John. "Catastrophes and partial differential equations." Annales de l'institut Fourier 23.2 (1973): 31-59. <http://eudml.org/doc/74128>.

@article{Guckenheimer1973,
abstract = {This paper outlines the manner in which Thom’s theory of catastrophes fits into the Hamilton-Jacobi theory of partial differential equations. The representation of solutions of a first order partial differential equation as lagrangian manifolds allows one to study the local structure of their singularities. The structure of generic singularities is closely related to Thom’s concept of the elementary catastrophe associated to a singularity. Three concepts of the stability of a singularity are discussed.},
author = {Guckenheimer, John},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {31-59},
publisher = {Association des Annales de l'Institut Fourier},
title = {Catastrophes and partial differential equations},
url = {http://eudml.org/doc/74128},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Guckenheimer, John
TI - Catastrophes and partial differential equations
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 2
SP - 31
EP - 59
AB - This paper outlines the manner in which Thom’s theory of catastrophes fits into the Hamilton-Jacobi theory of partial differential equations. The representation of solutions of a first order partial differential equation as lagrangian manifolds allows one to study the local structure of their singularities. The structure of generic singularities is closely related to Thom’s concept of the elementary catastrophe associated to a singularity. Three concepts of the stability of a singularity are discussed.
LA - eng
UR - http://eudml.org/doc/74128
ER -

References

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  1. [1] V. I. ARNOLD, Characteristic class entering in quantization conditions, Functional Anal. Appl., 1 (1967), 1-13. Zbl0175.20303
  2. [2] DARBOUX, Mémoire sur les solutions singulières des équations aux dérivées partielles du premier ordre, Mémoires de l'Institut Sav. Étrangers, (1883). 
  3. [3] J. GUCKENHEIMER, Bifurcation and catastrophe, to appear. Zbl0287.58005
  4. [4] L. HÖRMANDER, Fourier integral operators I (especially section 3. 1), Acta Mathematica, v. 127, (1971), 79-183. Zbl0212.46601
  5. [5] L. HÖRMANDER and DUISTERMAAT, Fourier integral operators II, Acta Mathematica, v. 128, (1972), 183-270. Zbl0232.47055MR52 #9300
  6. [6] F. LATOUR, Stabilité des champs d'applications différentiables ; généralisation d'un théorème de J. Mather, C.R. Acad. Sci. Paris, v. 268, (1969), 1331-1334. Zbl0184.48501MR39 #7617
  7. [7] J. MATHER, Stability of mappings I - VI 
  8. J. MATHER, I. Annals of Mathematics, v. 87, (1968), 89-104. Zbl0159.24902
  9. J. MATHER, II. Annals of Mathematics, v. 89, (1969), 254-291. Zbl0177.26002
  10. J. MATHER, III. Publ. Math. IHES, n° 35, (1968), 127-156. Zbl0159.25001
  11. J. MATHER, IV. Publ. Math. IHES, n° 37, (1969), 223-248. Zbl0202.55102
  12. J. MATHER, V. Advances in Mathematics, v. 4, (1970), 301-336. Zbl0207.54303
  13. J. MATHER, VI. Proceedings of Liverpool Singularities Symposium, Lecture Notes in Math., v. 192, pp. 207-253. Zbl0211.56105
  14. [8] I. PORTEOUS, Normal singularities of submanifolds, J. Diff. Geo., v. 5, (1971), 543-564. Zbl0226.53010MR45 #1179
  15. [9] R. THOM, Stabilité Structurelle et Morphogenèse, to appear. Zbl0365.92001
  16. [10] R. THOM and H. LEVINE, Lecture notes on singularities, Proceedings of Liverpool Singularities Symposium, Lecture Notes in Math., v. 192. 
  17. [11] C. T. C. WALL, Lectures on C∞-stability and classification, Proceedings of Liverpool Singularities Symposium, Lecture Notes in Math., v. 192, pp. 178-206. Zbl0211.56104MR44 #2244
  18. [12] A. WEINSTEIN, Singularities of families of functions, Berichte aus den Mathematischen Forschungsinstitut, Band 4, (1971), 323-330. Zbl0221.58008MR54 #11382
  19. [13] A. WEINSTEIN, Lagrangean manifolds, Advances in Math., v. 6, (1971), 329-346. Zbl0213.48203MR44 #3351

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