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On binary differential equations and umbilics

  • J.W. Bruce (a1) and D.L. Fidal (a1)

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.

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1 V. I. Arnold . Geometrical methods in the theory of ordinary differential equations (Berlin: Springer, 1983).

3 Th. Brocker and L. C. Lander . Differentiable germs and Catastrophes, London Math. Soc. Lecture Note Series 17 (Cambridge: Cambridge University Press, 1975).

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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