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 CMFD, 2006, Volume 19, Pages 131–170 (Mi cmfd69)

Many-Dimensional Poincaré Construction and Singularities of Lifted Fields For Implicit Differential Equations

A. O. Remizov

University of Porto

Abstract: The paper is devoted to singular points of the so-called lifted vector fields, which arise in studying systems of implicit differential equations by using the method of lifting the equation to a surface, a generalization of the construction used by Poincaré for a single implicit equation. The author study the phase portraits and renormal forms of such fields in a neighborhood of their singular points. In conclusion, the paper considers the lifted vectors fields generated by Euler–Lagrange and Euler–Poisson equations and fast-slow systems.

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English version:
Journal of Mathematical Sciences, 2008, 151:6, 3561–3602

Bibliographic databases:

UDC: 517.922

Citation: A. O. Remizov, “Many-Dimensional Poincaré Construction and Singularities of Lifted Fields For Implicit Differential Equations”, Optimal control, CMFD, 19, PFUR, M., 2006, 131–170; Journal of Mathematical Sciences, 151:6 (2008), 3561–3602

Citation in format AMSBIB
\Bibitem{Rem06} \by A.~O.~Remizov \paper Many-Dimensional Poincar\'e Construction and Singularities of Lifted Fields For Implicit Differential Equations \inbook Optimal control \serial CMFD \yr 2006 \vol 19 \pages 131--170 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd69} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2336476} \zmath{https://zbmath.org/?q=an:1190.34006} \transl \jour Journal of Mathematical Sciences \yr 2008 \vol 151 \issue 6 \pages 3561--3602 \crossref{https://doi.org/10.1007/s10958-008-9043-1} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-49649128824} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. O. Remizov, “Singularities in three-dimensional affine control systems with scalar control”, Russian Math. Surveys, 62:4 (2007), 821–822
2. V. M. Zakalyukin, A. O. Remizov, “Legendre Singularities in Systems of Implicit ODEs and Slow–Fast Dynamical Systems”, Proc. Steklov Inst. Math., 261 (2008), 136–148
3. A. O. Remizov, “Codimension-two singularities in 3D affine control systems with a scalar control”, Sb. Math., 199:4 (2008), 613–627
4. A. O. Remizov, “Geodesics on 2-surfaces with pseudo-Riemannian metric: singularities of changes of signature”, Sb. Math., 200:3 (2009), 385–403
5. A. O. Remizov, “On geodesics in metrics with singularities of the Klein type”, Russian Math. Surveys, 65:1 (2010), 180–182
6. A. O. Remizov, “Singularities of a geodesic flow on surfaces with a cuspidal edge”, Proc. Steklov Inst. Math., 268 (2010), 248–257
7. J. M. Oliver, “On the characteristic curves on a smooth surface”, J. Lond. Math. Soc. (2), 83:3 (2011), 755–767
8. Barlukova A.M., Chupakhin A.P., “Partially Invariant Solutions in Gas Dynamics and Implicit Equations”, J. Appl. Mech. Tech. Phys., 53:6 (2012), 812–824
9. R. Ghezzi, A. O. Remizov, “On a class of vector fields with discontinuities of divide-by-zero type and its applications to geodesics in singular metrics”, J. Dyn. Control Syst., 18:1 (2012), 135–158
10. I. A. Bogaevsky, “Implicit ordinary differential equations: bifurcations and sharpening of equivalence”, Izv. Math., 78:6 (2014), 1063–1078
11. Masatomo Takahashi, “Classifications of completely integrable implicit second order ordinary differential equations”, J. Singul., 10 (2014), 271–285
12. R. A. Chertovskih, A. O. Remizov, “On pleated singular points of first-order implicit differential equations”, J. Dyn. Control Syst., 20:2 (2014), 197–206
13. A. O. Remizov, “On the local and global properties of geodesics in pseudo-Riemannian metrics”, Differential Geometry and its Applications, 39 (2015), 36–58
14. A. O. Remizov, “Geodesics in generalized Finsler spaces: singularities in dimension two”, J. Singul., 14 (2016), 172–193
15. Ugo Boscain, Ludovic Sacchelli, Mario Sigalotti, “Generic singularities of line fields on 2D manifolds”, Differential Geom. Appl., 49 (2016), 326–350
16. Chupakhin A.P., Yanchenko A.A., “Special Vortex in Relativistic Hydrodynamics”: A. Chesnokov, E. Pruuel, V. Shelukhin, All-Russian Conference With International Participation Modern Problems of Continuum Mechanics and Explosion Physics Dedicated to the 60th Anniversary of Lavrentyev Institute of Hydrodynamics SB RAS, Journal of Physics Conference Series, 894, IOP Publishing Ltd, 2017, UNSP 012114
17. N. G. Pavlova, A. O. Remizov, “Completion of the classification of generic singularities of geodesic flows in two classes of metrics”, Izv. Math., 83:1 (2019), 104–123
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