Skip to main content
SpringerLink
Log in

Generic singularities of implicit systems of first order differential equations on the plane

  • Original Article
  • Published:
Japanese Journal of Mathematics Aims and scope

Abstract

For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well-known singularities of generic vector fields on the plane and the singularities described by a generic first order implicit differential equations, there exists only one generic singularity described by the implicit first order equation supplied by Whitney umbrella surface generically embedded to the space of directions on the plane.

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. V.I. Arnold, Contact structure, relaxation oscillations and singular points of implicit differential equations, In: Global Analysis—Studies and Applications, III, Lecture Notes in Math., 1334, Springer-Verlag, 1988, pp. 173–179.

  2. V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of differentiable maps, Vol. I, The classification of critical points, caustics and wave fronts, Monogr. Math., 82, Birkhäuser, Boston, 1986.

  3. V.I. Arnold and Yu.S. Il’yashenko, Ordinary Differential Equations, In: Modern Problems in Mathematics, Dynamical Systems, 1, Springer-Verlag, 1985.

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

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Cibrario, Sulla riduzione a forma canonica delle equazioni lineari alle derivate parziali di secondo ordine di tipo misto, Ist. Lombardo, Rend., II. Ser., 65 (1932), 889–906.

    MATH  Google Scholar 

  6. J. Damon, The unfolding and determinacy theorems for subgroups of \({\fancyscript{A}}\) and \({\fancyscript{K}}\), Mem. Amer. Math. Soc., 50, no. 306, Amer. Math. Soc., 1984.

  7. L. Dara, Singularités générique des équations différentielles multiformes, Bol. Soc. Brasil. Mat., 6 (1975), 95–128.

    Article  MATH  MathSciNet  Google Scholar 

  8. A.A. Davydov, Normal forms of differential equations unresolved with respect to derivatives in a neighbourhood of its singular point, Funct. Anal. Appl., 19 (1985), 1–10.

    Article  MathSciNet  Google Scholar 

  9. A.A. Davydov, The normal form of slow motions of an equation of relaxation type and fibrations of binomial surfaces, Math. USSR-Sb., 60 (1988), 133–141.

    Article  MATH  MathSciNet  Google Scholar 

  10. A.A. Davydov, Local controllability of typical dynamical inequalities on surfaces, Proc. Steklov Inst. Math., 209 (1995), 73–106.

    MathSciNet  Google Scholar 

  11. A.A. Davydov, Whitney umbrella and slow-motion bifurcations of relaxation-type equations, J. Math. Sci. (N. Y.), 126 (2005), 1251–1258.

    MATH  MathSciNet  Google Scholar 

  12. A.A. Davydov and E. Rosales-Gonsales, The complete classification of typical second-order linear partial-differential equations on the plane, Dokl. Math., 54 (1996), 669–672.

    MATH  Google Scholar 

  13. J.-P. Dufour, Modules pour les familles de courbes planes, Ann. Inst. Fourier (Grenoble), 39 (1989), 225–238.

    MATH  MathSciNet  Google Scholar 

  14. C.G. Gibson, Singular points of smooth mappings, Res. Notes in Math., 25, Pitman, 1979.

  15. M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Grad. Texts in Math., 14, Springer-Verlag, 1980.

  16. V.V. Goryunov, Projection of generic surfaces with boundaries, Adv. Soviet Math., 1 (1990), 157–200.

    MathSciNet  Google Scholar 

  17. A. Hayakawa, G. Ishikawa, S. Izumiya and K. Yamaguchi, Classification of generic integral diagrams and first order ordinary differential equations, Internat. J. Math., 5 (1994), 447–489.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Izumiya, On Clairaut-type equations, Publ. Math. Debrecen, 45 (1994), 159–166.

    MATH  MathSciNet  Google Scholar 

  19. S. Izumiya and H. Kurokawa, Holonomic systems of Clairaut type, Differential Geom. Appl., 5 (1995), 219–235.

    Article  MATH  MathSciNet  Google Scholar 

  20. Y. Kurokawa, On functional moduli for first order ordinary differential equations, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 233–238.

    MATH  MathSciNet  Google Scholar 

  21. A.G. Kuz’min, Nonclassical equations of mixed type and their applications in gas dynamics, Internat. Ser. Numer. Math., 109, Birkhäuser, 1992.

  22. J. Martinet, Singularities of smooth functions and maps, London Math. Soc. Lecture Note Ser., 58, Cambridge Univ. Press, 1982.

  23. J.N. Mather, Generic projections, Ann. of Math. (2), 98 (1973), 226–245.

    Article  MathSciNet  Google Scholar 

  24. M. Takahashi, Bifurcations of holonomic systems of general Clairaut type, Hokkaido Math. J., 35 (2006), 905–934.

    MATH  MathSciNet  Google Scholar 

  25. H. Whitney, The singularities of a smooth n-manifold in \((2n-1)\)-space, Ann. of Math. (2), 45 (1944), 247–293.

    Article  MathSciNet  Google Scholar 

  26. H. Whitney, On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane, Ann. of Math. (2), 62 (1955), 374–410.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Vladimir State University, Gorkii street 87, 600026, Vladimir, Russia

    A. A. Davydov

  2. Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

    G. Ishikawa & S. Izumiya

  3. School of Science, Changchun University of Science and Technology, Changchun, 130022, China

    W.-Z. Sun

Authors
  1. A. A. Davydov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Ishikawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Izumiya
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. W.-Z. Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Davydov.

Additional information

Communicated by: Kaoru Ono

A.A. Davydov: Financial support from RFBR 00-01-00343 and Grant-in-Aid for Scientific Research, No. 10304003.

G. Ishikawa: Financial support from Grant-in-Aid for Scientific Research, No. 14340020.

S. Izumiya: Financial support from Grant-in-Aid for Scientific Research, No. 10304003.

W.-Z. Sun: Financial support from Grant-in-Aid for Scientific Research, No. 14604003.

About this article

Cite this article

Davydov, A.A., Ishikawa, G., Izumiya, S. et al. Generic singularities of implicit systems of first order differential equations on the plane. Jpn. J. Math. 3, 93–119 (2008). https://doi.org/10.1007/s11537-008-0664-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11537-008-0664-4

Keywords and phrases

Mathematics Subject Classification (2000)

Search

Navigation