Skip to main content
SpringerLink
Log in

Generic bifurcations of varieties

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We study the zero point set of a parametrized smooth map germ. It is not only a natural generalization of Mather's theory of smooth map germs (cf. [11],[12]), but also it contains the bifurcation theory of stationary solutions of parametrized ordinary differential equations. One of our main results is a classification of parametrized smooth map germs under a certain equivalence relation.

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. DAMON, J.: The classification of discrete algebra types, Preprint

  2. DAMON, J.: The Unfolding and Determinacy theorems for subgroups of A and K. Proceedings of Symposia in Pure Mathematics Vol. 40, Part I, 233–254 A.M.S. (1983)

    Google Scholar 

  3. DEBREU, G.: Excess demand functions. J. Math. Econom. 1, 15–22 (1971)

    Google Scholar 

  4. DUPOUR, J: Sur la stabilité des diagrammes d'applications différentiables. Ann. scient. ec. Norm. Sup. 10, 153–174 (1977)

    Google Scholar 

  5. GIBSON, C. G.: Singular points of smooth mappings. Research Notes in Mathematics 25, Pitman (1979)

  6. GOLUBITSKY, M. amd SCHAEFFER, D.: A theory for imperfect bifurcation via singularity theory. Comm. Pure Appl. Math. 32, 21–98 (1979)

    Google Scholar 

  7. GOMOZOV, E. P.: A versality theorem for a birateral group of change of variables. Funct. Ann. Appl. 9, 332–333 (1975)

    Google Scholar 

  8. GUIMARAES, L. C.: Contact equivalence and bifurcation theory. Preprint

  9. IZUMIYA, S.: Generic bifurcations of varieties. Proceedings of the Japan Academy 58, 337 -340 (1982)

    Google Scholar 

  10. MARTINET, J.: Deploiments versels des applications differentiables et classification des applications stables. In: Singularités d'Applications Differentiables, 11–44, Springer L.N.M. 535, Springer-Verlag (1976)

  11. MATHER, J.: Stability of C∞ mappings in: finitely determined map germs. Publ. Math. I.H.E.S. 35, 127–156 (1969)

    Google Scholar 

  12. MATHER, J.: Stability of C∞ mappings IV: classification of stable germs by R-algebras. Publ. Math. I.H.E.S. 37, 223–248 (1970)

    Google Scholar 

  13. NAKAI, I.: Structural stability of composed mappings. Preprint.

  14. DU PLESSIS, A.: On the determinacy of smooth map germs. Invent. Math. 58, 107–160 (1980)

    Google Scholar 

  15. POSTON, T.: Perturbed bifurcations and crystal spectra. In: Applications of non-linear Analysis in the Physical Sciences (ed Aman, H et al). Pitman, 77–91 (1981)

  16. Wall, C. T. C.: Finite determinacy of smooth map germs. Bull. London Math. Soc. 13, 481–539 (1981)

    Google Scholar 

  17. WASSERMAN, G.: Stability of Unfoldings. Springer L.N.M. 393, Springer-Verlag (1974)

  18. ZAKALYUKIN, V. M.: Lagrangian and Legendrian singularities. Funct. Annl. Appl. 10, 37–45 (1976)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nara Women's University, 630, Nara, Japan

    Shyūichi Izumiya

Authors
  1. Shyūichi Izumiya
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Professor Minoru Nakaoka on his 60th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Izumiya, S. Generic bifurcations of varieties. Manuscripta Math 46, 137–164 (1984). https://doi.org/10.1007/BF01185199

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01185199

Keywords

Search

Navigation