Skip to main content
SpringerLink
Log in

Multidimensional Poincaré construction and singularities of lifted fields for implicit differential equations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

This 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 studies the phase portraits and renormal forms of such fields in a neighborhood of their singular points. In conclusion, this paper considers the lifted vectors fields generated by Euler-Lagrange and Euler-Poisson equations and fast-slow systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Abraham, J. E. Marsden, and M. Ratiu, Manifolds, Tensor Analysis and Applications, Addison Wesley, Reading (1983).

    MATH  Google Scholar 

  2. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  3. V. I. Arnol’d, “Wavefront evolution and the equivariant Morse lemma,” Commun. Pure Appl. Math., 29, No. 6, 557–582 (1976).

    Article  MATH  Google Scholar 

  4. V. I. Arnol’d, Additional Chapters of Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  5. V. I. Arnol’d, Geometric Methods in Theory of Ordinary Differential Equations [in Russian], Udmurtsk State University, Izhevsk (2000).

    Google Scholar 

  6. V. I. Arnol’d, Contact Structure, Relaxed Oscillations, and Singular Points of Ordinary Differential Equations [in Russian], Fazis, Moscow (1997).

    Google Scholar 

  7. V. I. Arnol’d and Yu. S. Il’yashenko, “Ordinary differential Equations,” in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 1, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1985), pp. 7–149.

    Google Scholar 

  8. V. I. Arnol’d, V. S. Afraimovich, Y. S. Il’yashenko, and L. P. Shil’nikov, “Bifurcation theory,” in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 5, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1986).

    Google Scholar 

  9. V. I. Arnol’d, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Mappings, Classification of Critical Points, Caustics, and Wave Fronts [in Russian], Nauka, Moscow (1982).

    MATH  Google Scholar 

  10. J. Basto-Gonçalves, “Singularities of Euler equations and implicit Hamilton equations,” in: Real and Complex Singularities, Pitman Research Notes in Math., Vol. 333, Longman, (1995), pp. 203–212.

  11. M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A, 10, 1809–1821 (1977).

    Article  Google Scholar 

  12. Th. Bröker and L. Lander, Differential Germs and Catastrophes, Cambridge University Press (1975).

  13. A. D. Bryuno, Local Methods of Nonlinear Analysis [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  14. J. W. Bruce, “A note on first-order differential equations of degree greater than one and wavefront evolution,” Bull. London Math. Soc., 16, 139–144 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  15. J. W. Bruce and D. L. Fidal, “On binary differential equations and umbilics,” Proc. Roy. Soc. Edinburg, 111A, 147–168 (1989).

    MathSciNet  Google Scholar 

  16. J. W. Bruce and F. Tari, “On binary differential equations,” Nonlinearity, 8, 255–271 (1989).

    Article  MathSciNet  Google Scholar 

  17. J. W. Bruce and F. Tari, “Implicit differential equations from the singularity theory viewpoint,” Banach Center Publ., 33, 23–38 (1996).

    MathSciNet  Google Scholar 

  18. J. F. Bruce and F. Tari, “Generic 1-parameter families of binary differential equations,” Discr. Continuous Dynam. Systems, 3, 79–90 (1997).

    MATH  MathSciNet  Google Scholar 

  19. J. W. Bruce and F. Tari, “On the multiplicity of implicit differential equations,” J. Differ. Equations, 148, 122–147 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  20. J. F. Bruce, G. J. Fletcher, and F. Tari, “Bifurcations of binary differential equations,” Proc. Roy. Soc. Edinburg, 130A, 485–506 (2000).

    MathSciNet  Google Scholar 

  21. J. W. Bruce, G. J. Fletcher, and F. Tari, “Zero curves of families of curve congruences,” Contemp. Math., 354, 1–18 (2004).

    MathSciNet  Google Scholar 

  22. J. W. Bruce and F. Tari, “Duality and implicit differential equations,” Nonlinearity, 13, No. 3, 791–812 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  23. J. F. Cariñena, “Theory of singular Lagrangians,” Fortschr. Phys., 38, No. 9, 641–679 (1990).

    Article  MathSciNet  Google Scholar 

  24. M. Cibrario, “Sulla reduzione a forma canonica delle equazioni lineari alle derivative parzialy di secondo ordine di tipo misto,” Rend. Lombardo, 65, 889–906 (1932).

    MATH  Google Scholar 

  25. M. Cinquini-Cibrario, “Una propriete degli integrali delle equazioni ellitico-paraboliche del secondo tipo misto,” Reale Accad. D’Italia, Rendiconti Classe di Scienze, Fisiche, Mat. Nat., 7, No. 9 (1952).

  26. A. Clebsch and F. Lindemann, Vorlesungen über Geometrie, Vol. 1 (1876).

  27. A. Clebsch, “Uber ein neues Grundgebilde der analytischen Geometrie der Ebene,” Nachr. Kgl. Ges. Wiss. Göttingen, 429–449 (1872); reprinted in Math. Ann., 6, 203–225 (1873).

  28. L. Dara, “Singularités générique des équations différentielles multiformes,” Bol. Soc. Bras. Math., 6, No. 2, 95–128 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  29. G. Darboux, “Sur la forme des lignes de courbure dans la voisingage d’un ombilic,” Leçons sur la théorie générale des surfaces, vol. IV, Note 7, Gauthier Villars, Paris (1896).

    Google Scholar 

  30. A. A. Davydov, “Normal form of an equation not resolved with respect to derivative in a neighborhood of its singular point,” Funkts. Anal. Prilozh., 19, No. 2, 1–10 (1985).

    MathSciNet  Google Scholar 

  31. A. A. Davydov, Implicit Differential Equations and Qualitative Theory of Control Systems on the Plane [in Russian], Moscow (1993).

  32. A. A. Davydov, Qualitative Theory of Control Systems, Math. Monogr., 141, Amer. Math. Soc., Providence, Rhode Island (1994).

    MATH  Google Scholar 

  33. A. A. Davydov and L. Ortiz-Bobadilla, “Smooth normal forms of folded elementary singular points,” J. Dynam. Control Systems, 1, No. 4, 463–482 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  34. A. A. Davydov and L. Ortiz-Bobadilla, “Nomal form of folded elementary singular points,” Usp. Mat. Nauk, 50, No. 6, 175–177 (1995).

    MathSciNet  Google Scholar 

  35. A. A. Davydov and E. Rosales-Gonsales, “A complete classification of typical linear partial differential equations of the second order on the plane,” Dokl. Russian Akad. Nauk, 350, No. 2, 151–154 (1996).

    MathSciNet  Google Scholar 

  36. A. A. Davydov, G. Ishikawa, S. Izumiya, and W.-Z. Sun, “Generic singularities of implicit systems of first-order differential equations on the plane,” ArXiV: math.DS/0302134 (2003).

  37. P. A. M. Dirac, Lectures on Quantum Mechanics, Yeshiva University, New York (1964).

    Google Scholar 

  38. M. De León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda, and N. Román-Roy, “Singular Lagrangian systems on jet bundles,” Fortschr. Phys., 50, No. 2, 105–169 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  39. A. F. Filippov, “Uniqueness of solutions of a system of differential equations not resolved with respect to derivatives,” Differents. Uravn., 41, No. 1, 87–92 (2005).

    Google Scholar 

  40. X. Gràcia, M. C. Muñoz-Lecanda, and N. Román-Roy, “On some aspects of the geometry of differential equations in physics,” Int. J. Geometric Methods Mod. Phys., 1, 265–284 (2004).

    Article  MATH  Google Scholar 

  41. F. Klein, Higher Geometry [Russian translation], GONTI, Moscow-Leningrad (1939).

    Google Scholar 

  42. A. G. Kuz’min, “On the behavior of characteristics of a mixed-type equation near the line of degeneracy,” Differents. Uravn., 17, No. 11, 2052–2063 (1981).

    MATH  MathSciNet  Google Scholar 

  43. A. G. Kuz’min, Nonclassical Equations of Mixed Type and Their Applications to Gas Dynamics [in Russian], LGU, Leningrad (1990).

    MATH  Google Scholar 

  44. A. G. Kuz’min, On qualitative theory of the equation a(x, y)(y′) 2 − 2b(x, y)y′ + c(x, y) = 0, Deposited at VINITI, No. 4143-81 (1981).

  45. A. N. Kolmogorov and A. P. Yushkevich (Eds.), Ordinary Differential Equations, Mathematics of the XIX Century [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  46. H. Poincaré, “Mémoire sur les courbes définies par les équations différentielles,” J. Math. Pures Appl., Sér 4, 1, 167–244 (1885).

    Google Scholar 

  47. H. Poincaré, On Curves Defined by Differential Equations [Russian translation], GITL, Moscow-Leningrad (1947).

    Google Scholar 

  48. H. Poincaré, Selected Works [Russian translation], Vol. 3, Nauka, Moscow (1974).

    MATH  Google Scholar 

  49. I. R. Porteous, Geometric Differentiation for the Intelligence of Curves and Surfaces, Cambridge University Press, Cambridge (1994).

    MATH  Google Scholar 

  50. A. V. Pkhakadze and A. A. Shestakov, “On classification of singular points of a differential equation of the first order not resolved with respect to the derivative,” Mat. Sb., 49, No. 1, 3–12 (1959).

    MathSciNet  Google Scholar 

  51. A. D. Piliya and V. I. Fedorov, “Singularities of electromagnetic wave field in a cold plasma with two-dimensional inhomogeneity,” ZhETF, 60, No. 1, 389–399 (1971).

    Google Scholar 

  52. P. K. Rashevskii, A Course of Differential Geometry [in Russian], Gostekhizdat (1956).

  53. P. J. Rabier and W. C. Rheinboldt, “A geometric treatment of implicit differential-algebraic equations,” J. Diff. Equations, 109, 110–146 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  54. S. Reich, “On a geometric interpretation of differential-algebraic equations,” Circuits Syst. Signal Process., 9, 367–382 (1990).

    Article  MATH  Google Scholar 

  55. W. C. Rheinboldt, “Differential-algebraic systems as differential equations on manifolds,” Math. Comp., 43(168), 473–482 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  56. V. S. Samovol, “Equivalence of systems of differential equations in a neighborhood of a singular point,” Tr. Mosk. Mat. Obshsch., 44, 213–234 (1982).

    MATH  MathSciNet  Google Scholar 

  57. A. N. Shoshitaishvili, “On bifurcation of the topological type of singular points of vector fields depending on parameters,” Tr. Semin. I. G. Petrovskogo, 1, 279–309 (1975).

    Google Scholar 

  58. M. M. Smirnov, Equations of Mixed Type [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  59. P. V. Sokolov, “To the paper of A. V. Pkhakadze and A. A. Shestakov ‘On classification of singular points of a differential equation of the first order not resolved with respect to the derivative’,” Mat. Sb., 53(95), No. 4, 541–543 (1961).

    MathSciNet  Google Scholar 

  60. J. Sotomayor, “Historical comments on Monge’s ellipsoid and the configuration of lines of curvature on surfaces immersed in ℝ3,” ArXiV:math.HO/0411403 (2004).

  61. D. J. Struik, Lectures on Classical Differential Geometry, Addison Wesley (1950).

  62. F. Takens, “Constrained equations; a study of implicit differential equations and their discontinuous solutions,” Lect. Notes Math., 525, 143–234 (1976).

    Article  Google Scholar 

  63. F. Takens, “Implicit differential equations: some open problems,” Lect. Notes Math., 535, 237–253 (1976).

    Article  MathSciNet  Google Scholar 

  64. F. Tari, “Two-parameter families of implicit differential equations,” Discrete Continuous Dynamical Systems, 13, No. 1, 139–162 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  65. F. Tari, “Geometric properties of the intergal curves of an implicit differential equations,” Discrete Continuous Dynamical Systems (in press).

  66. F. Tricomi, “Sulle equazioni lineari alle derivate parziali di seconde ordine di tipo misto,” Rendiconti Atti dell’ Accademia Nazionale dei Lincei, Ser. 5, 14, 134–247 (1923).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. U. Porto Ct. Matemática., Rua Do Campo Alegre, 687, 4100-172, Porto, Portugal

    A. O. Remizov

Authors
  1. A. O. Remizov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. O. Remizov.

Additional information

__________

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Remizov, A.O. Multidimensional Poincaré construction and singularities of lifted fields for implicit differential equations. J Math Sci 151, 3561–3602 (2008). https://doi.org/10.1007/s10958-008-9043-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-9043-1

Keywords

Search

Navigation