Catastrophes and partial differential equations
Annales de l'Institut Fourier, Tome 23 (1973) no. 2, pp. 31-59.

On explore ici le rapport entre la théorie des catastrophes de Thom et la théorie Hamilton-Jacobi des équations différentielles de premier ordre. La représentation des solutions d’une équation aux dérivées partielles du premier ordre comme variétés lagrangiennes permet d’étudier la structure locale de leurs singularités. La structure des singularités génériques est près du concept de Thom de catastrophe élémentaire associée à une singularité. On discute trois notions de la stabilité d’une singularité.

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.

@article{AIF_1973__23_2_31_0,
     author = {Guckenheimer, John},
     title = {Catastrophes and partial differential equations},
     journal = {Annales de l'Institut Fourier},
     pages = {31--59},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {23},
     number = {2},
     year = {1973},
     doi = {10.5802/aif.455},
     mrnumber = {51 #1879},
     zbl = {0271.35006},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.455/}
}
TY  - JOUR
AU  - Guckenheimer, John
TI  - Catastrophes and partial differential equations
JO  - Annales de l'Institut Fourier
PY  - 1973
SP  - 31
EP  - 59
VL  - 23
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.455/
DO  - 10.5802/aif.455
LA  - en
ID  - AIF_1973__23_2_31_0
ER  - 
%0 Journal Article
%A Guckenheimer, John
%T Catastrophes and partial differential equations
%J Annales de l'Institut Fourier
%D 1973
%P 31-59
%V 23
%N 2
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.455/
%R 10.5802/aif.455
%G en
%F AIF_1973__23_2_31_0
Guckenheimer, John. Catastrophes and partial differential equations. Annales de l'Institut Fourier, Tome 23 (1973) no. 2, pp. 31-59. doi : 10.5802/aif.455. http://archive.numdam.org/articles/10.5802/aif.455/

[1] V. I. Arnold, Characteristic class entering in quantization conditions, Functional Anal. Appl., 1 (1967), 1-13. | Zbl

[2] Darboux, Mémoire sur les solutions singulières des équations aux dérivées partielles du premier ordre, Mémoires de l'Institut Sav. Étrangers, (1883).

[3] J. Guckenheimer, Bifurcation and catastrophe, to appear. | Zbl

[4] L. Hörmander, Fourier integral operators I (especially section 3. 1), Acta Mathematica, v. 127, (1971), 79-183. | Zbl

[5] L. Hörmander and Duistermaat, Fourier integral operators II, Acta Mathematica, v. 128, (1972), 183-270. | MR | Zbl

[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. | MR | Zbl

[7] J. Mather, Stability of mappings I - VI

J. Mather, I. Annals of Mathematics, v. 87, (1968), 89-104. | Zbl

J. Mather, II. Annals of Mathematics, v. 89, (1969), 254-291. | Zbl

J. Mather, III. Publ. Math. IHES, n° 35, (1968), 127-156. | Numdam | Zbl

J. Mather, IV. Publ. Math. IHES, n° 37, (1969), 223-248. | Numdam | Zbl

J. Mather, V. Advances in Mathematics, v. 4, (1970), 301-336. | Zbl

J. Mather, VI. Proceedings of Liverpool Singularities Symposium, Lecture Notes in Math., v. 192, pp. 207-253. | Zbl

[8] I. Porteous, Normal singularities of submanifolds, J. Diff. Geo., v. 5, (1971), 543-564. | MR | Zbl

[9] R. Thom, Stabilité Structurelle et Morphogenèse, to appear. | Zbl

[10] R. Thom and H. Levine, Lecture notes on singularities, Proceedings of Liverpool Singularities Symposium, Lecture Notes in Math., v. 192.

[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. | MR | Zbl

[12] A. Weinstein, Singularities of families of functions, Berichte aus den Mathematischen Forschungsinstitut, Band 4, (1971), 323-330. | MR | Zbl

[13] A. Weinstein, Lagrangean manifolds, Advances in Math., v. 6, (1971), 329-346. | MR | Zbl

Cité par Sources :