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

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.

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.

@article{AIF_1973__23_2_31_0,
     author = {Guckenheimer, John},
     title = {Catastrophes and partial differential equations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {23},
     number = {2},
     year = {1973},
     pages = {31-59},
     doi = {10.5802/aif.455},
     zbl = {0271.35006},
     mrnumber = {51 \#1879},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1973__23_2_31_0}
}
Guckenheimer, John. Catastrophes and partial differential equations. Annales de l'Institut Fourier, Volume 23 (1973) no. 2, pp. 31-59. doi : 10.5802/aif.455. http://www.numdam.org/item/AIF_1973__23_2_31_0/

[1] V. I. Arnold, Characteristic class entering in quantization conditions, Functional Anal. Appl., 1 (1967), 1-13. | Zbl 0175.20303

[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 0287.58005

[4] L. Hörmander, Fourier integral operators I (especially section 3. 1), Acta Mathematica, v. 127, (1971), 79-183. | Zbl 0212.46601

[5] L. Hörmander and Duistermaat, Fourier integral operators II, Acta Mathematica, v. 128, (1972), 183-270. | MR 52 #9300 | Zbl 0232.47055

[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 39 #7617 | Zbl 0184.48501

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

J. Mather, I. Annals of Mathematics, v. 87, (1968), 89-104. | Zbl 0159.24902

J. Mather, II. Annals of Mathematics, v. 89, (1969), 254-291. | Zbl 0177.26002

J. Mather, III. Publ. Math. IHES, n° 35, (1968), 127-156. | Numdam | Zbl 0159.25001

J. Mather, IV. Publ. Math. IHES, n° 37, (1969), 223-248. | Numdam | Zbl 0202.55102

J. Mather, V. Advances in Mathematics, v. 4, (1970), 301-336. | Zbl 0207.54303

J. Mather, VI. Proceedings of Liverpool Singularities Symposium, Lecture Notes in Math., v. 192, pp. 207-253. | Zbl 0211.56105

[8] I. Porteous, Normal singularities of submanifolds, J. Diff. Geo., v. 5, (1971), 543-564. | MR 45 #1179 | Zbl 0226.53010

[9] R. Thom, Stabilité Structurelle et Morphogenèse, to appear. | Zbl 0365.92001

[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 44 #2244 | Zbl 0211.56104

[12] A. Weinstein, Singularities of families of functions, Berichte aus den Mathematischen Forschungsinstitut, Band 4, (1971), 323-330. | MR 54 #11382 | Zbl 0221.58008

[13] A. Weinstein, Lagrangean manifolds, Advances in Math., v. 6, (1971), 329-346. | MR 44 #3351 | Zbl 0213.48203