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On binary differential equations and umbilics

Published online by Cambridge University Press:  14 November 2011

J.W. Bruce  and
D.L. Fidal
J.W. Bruce
Affiliation:
Department of Mathematics, University of Newcastle upon Tyne, Newcastle NE1 7RU, U.K.
D.L. Fidal
Affiliation:
Department of Mathematics, University of Newcastle upon Tyne, Newcastle NE1 7RU, U.K.
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Synopsis

In this paper we give the local classification of solution curves of bivalued direction fields determined by the equation

where a and b are smooth functions which we suppose vanish at 0 ∈ ℝ2. Such fields arise on surfaces in Euclidean space, near umbilics, as the principal direction fields, and also in applications of singularity theory to the structure of flow fields and monochromatic-electromagnetic radiation. We give a classification up to homeomorphism (there are three types) but the methods furnish much additional information concerning the fields, via a crucial blowing-up construction.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 111 , Issue 1-2 , 1989 , pp. 147 - 168
Copyright
Copyright © Royal Society of Edinburgh 1989

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References

1Arnold, V. I.. Geometrical methods in the theory of ordinary differential equations (Berlin: Springer, 1983).CrossRefGoogle Scholar
2Berry, M. V. and Hannay, J. H..Umbilic points on Gaussian random surfaces. J. Phys. A 10 (1977), 1809–1821.CrossRefGoogle Scholar
3Brocker, Th. and Lander, L. C.. Differentiable germs and Catastrophes, London Math. Soc. Lecture Note Series 17 (Cambridge: Cambridge University Press, 1975).CrossRefGoogle Scholar
4Bruce, J. W.. A note on first order differential equations of degree greater than one and wavefront evolution. Bull. London Math. Soc. 16 (1984), 139–144.CrossRefGoogle Scholar
5Dara, L.. Singularités génériques des equations differentielles multiforms. Bol. Soc. Brasil. Math. 6 (1975), 95–128.CrossRefGoogle Scholar
6Darboux, G.. Lemons sur la thèorie générale des surfaces, Vol. 4 (Paris: Gauthiers Villars, 1896).Google Scholar
7Davydov, A. A.. Normal form of a differential equation, not solvable for the derivative, in a neighbourhood of a singular point. Functional Anal. Appl. 19 (1985), 81–89.CrossRefGoogle Scholar
8Irwin, M. C.. On the stable manifold theorem. Bull. London Math. Soc. 2 (1970), 196–198.CrossRefGoogle Scholar
9Hartman, P.. On local homeomorphisms of Euclidean spaces. Bol. Soc. Math. Mexicana 5 (1960), 220–241.Google Scholar
10Porteous, I. R.. Normal singularities of surfaces in R3. Proc. Sympos. Pure Math. 40 (1983), 379–393.CrossRefGoogle Scholar
11Sotomayer, J. and Guttierrez, C.. Structurally stable configurations of lines of principal curvature. Astirisque 98–99 (1982).Google Scholar
12Thom, R.. Sur les equations differentielles multiformes et leur integrals singulieres. Bol. Soc. Brasil. Math. 3 (1971), 1–11.Google Scholar
13Thorndike, A. S., Cooley, C. R. and Nye, J. F.. The structure and evolution of flowfields and other vector fields. J. Phys. A 11 (1978), 1455–1490.CrossRefGoogle Scholar
14Nye, J. F. and Hajnal, J. V.. The wave structure of monochromatic electromagnetic radiation. Proc. Roy. Soc. London Ser. A 409 (1987), 21–36.Google Scholar