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On Pleated Singular Points of First-Order Implicit Differential Equations

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Abstract

We study phase portraits of a first-order implicit differential equation in a neighborhood of its pleated singular point that is a nondegenerate singular point of the lifted field. Although there is no visible local classification of implicit differential equations at pleated singular points (even in the topological category), we show that there exist only six essentially different phase portraits, which are presented.

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Notes

  1. This approach goes back to Poincaré [1] and Clebsch [2]. The latter paper contains geometric interpretation of differential equations (both ordinary and partial) and related theory of connexes, which is quite similar to the lifting; an account of these ideas is contained also in the famous book Vorlesungen über höhere Geometrie by F. Klein.

  2. The meaning of this name is clear from what follows. Let \(\gamma \) be an integral curve of X, that is, an integral curve of the vector field (2). Suppose that the corresponding solution \(\pi (\gamma )\) of Eq. (1) has an inflection at some point on the \((x,y)\)-plane. Then, the last component, \(-(F_{x}+pF_{y})\), of the vector field (2) vanishes at the corresponding point of the surface \(\cal {F}\).

  3. In [3–5], such points are called regular although being singular points of implicit differential equation. However, we prefer to use another terminology.

  4. The germ of a smooth function is called k-flat at O if its Taylor series at O starts with monomials of degree greater than k.

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Acknowledgments

A.O. Remizov was supported by a grant from FAPESP, proc. 2012/03960-2 for visiting ICMC-USP, São Carlos (Brazil). He expresses deep gratitude to Prof. Farid Tari for the hospitality and useful discussions.

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Authors and Affiliations

  1. Centre for Wind Energy and Atmospheric Flows, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-465, Porto, Portugal

    R. A. Chertovskih

  2. International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Profsoyuznaya str. 84/32, 117997, Moscow, Russia

    R. A. Chertovskih

  3. CMAP École Polytechnique Route de Saclay, 91128, Palaiseau Cedex, France

    A. O. Remizov

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  1. R. A. Chertovskih
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  2. A. O. Remizov
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Chertovskih, R.A., Remizov, A.O. On Pleated Singular Points of First-Order Implicit Differential Equations. J Dyn Control Syst 20, 197–206 (2014). https://doi.org/10.1007/s10883-013-9209-0

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  • DOI: https://doi.org/10.1007/s10883-013-9209-0

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