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Journal of Dynamical and Control Systems

, Volume 18, Issue 1, pp 135–158 | Cite as

On a class of vector fields with discontinuities of divide-by-zero type and its applications to geodesics in singular metrics

  • R. GhezziEmail author
  • A. O. Remizov
Article

We study phase portraits at singular points of vector fields of the special type where all components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. Also we assume some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three applications in differential geometry: singularities of geodesic flows in various singular metrics on two-dimensional manifolds.

Key words and phrases

Vector fields singular points discontinuity divide-by-zero resonances singular metrics geodesic lines 

2000 Mathematics Subject Classification

34C05 34C20 37C15 53B30 53C22 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.CMAP École PolytechniquePalaiseau CedexFrance
  2. 2.SISSA/ISASTriesteItaly

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