Advertisement

Journal of Mathematical Sciences

, Volume 233, Issue 6, pp 875–904 | Cite as

Differential Equations with Degenerate Operators at the Derivative Depending on an Unknown Function

  • B. V. Loginov
  • Yu. B. Rousak
  • L. R. Kim-Tyan
Article
  • 6 Downloads

Abstract

We develop the theory of generalized Jordan chains of multiparameter operator functions A(λ) : E1 → E2, λ ∈ Λ, dimΛ = k, dimE1 = dimE2 = n, where A0 = A(0) is an irreversible operator. For simplicity, in Secs. 1–3, the geometric multiplicity of λ0 is equal to one, i.e., dimN(A0) = 1, N(A0) = span{φ}, dimN*(\( {A}_0^{\ast } \)) = 1, N*(\( {A}_0^{\ast } \)) = span{ψ}, and it is assumed that the operator function A(λ) is linear with respect to λ. In Sec. 4, the polynomial dependence of A(λ) is linearized. However, the results of existence theorems for bifurcations are obtained for the case where there are several Jordan chains. Applications to degenerate differential equations of the form [A0 + R, x)]x′= Bx are provided.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. I. Arnol’d, Geometric Methods in the Theory of Ordinary Differential Equations [in Russian], MTsNMO, Moscow (2002).Google Scholar
  2. 2.
    J. Carr, “Application of center manifold theory,” Appl. Math. Sci., 35 (1981).Google Scholar
  3. 3.
    M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities [Russian translation], Mir, Moscow (1977).zbMATHGoogle Scholar
  4. 4.
    Ph. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York–London–Sydney (1964).zbMATHGoogle Scholar
  5. 5.
    M. A. Krasnosel’skij, Topological Methods in the Theory of Nonlinear Integral Equations [in Russian], GITTL, Moscow (1956).Google Scholar
  6. 6.
    M. A. Krasnosel’skij, A. I. Perov, A. I. Povolotskij, and P. P. Zabreyko, Vector Fields in a Plane [in Russian], Fizmatlit, Moscow (1963).Google Scholar
  7. 7.
    M. A. Krasnosel’skij and P. P. Zabreyko, Geometric Methods of Nonlinear Analysis and Its Applications [in Russian], Nauka, Moscow (1975).Google Scholar
  8. 8.
    B. V. Loginov, “Branching equation in the root subspaces,” Nonlinear Anal., 32, No. 3, 439–448 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    B. V. Loginov and Yu. B. Rusak, “Generalized Jordan structures in the branching theory,” In: Direct and Inverse Problems for Partial Differential Equations and Their Applications, FAN, Tashkent (1978), pp. 113–148.Google Scholar
  10. 10.
    B. V. Loginov and Yu. B. Rousak, “Generalized Jordan structure in the problem of stability of bifurcation equations,” Nonlinear Anal., 17, No. 3, 219–231 (1991).MathSciNetCrossRefGoogle Scholar
  11. 11.
    B. V. Loginov, Yu. B. Rusak, and L. R. Kim-Tyan, “Normal forms of differential equations with degenerate matrix at the derivative when a Jordan chain of maximal length exists,” Proc. of Sci. Conf. “Hertsen readings,” Saint-Petersburg, 93–109 (2013).Google Scholar
  12. 12.
    B. V. Loginov, Yu. B. Rusak, and L. R. Kim-Tyan, “Differential equations with an operator that is degenerate and linearly dependent on unknown functions at derivatives,” Abstr. of Int. Conf. on Differential Equations and Dynamic Systems, Suzdal’, 106-107 (2014).Google Scholar
  13. 13.
    B. V. Loginov, Yu. B. Rousak, and L. R. Kim-Tyan, “Differential equations with an operator that is degenerate and linearly dependent on unknowns at the derivative,” In: Current Trends in Analysis and Its Applications, Birkhäuser, Basel (2015), pp. 101–108.Google Scholar
  14. 14.
    T. Ma and Sh. Wang, Bifurcation Theory and Applications, World Scientific, Hackensack (2005).CrossRefzbMATHGoogle Scholar
  15. 15.
    R. J. Magnus, “A generalization of multiplicity and the problem of bifurcation,” Proc. Lond. Math. Soc. (3), 32, 251–278 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    W. Marszalek, “Fold points and singularity induced bifurcation in inviscid transonic flow,” Phys. Lett. A, 376, No. 28-29, 2032–2037 (2012).CrossRefzbMATHGoogle Scholar
  17. 17.
    L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Inst. Math. Sci., New York Univ., New York (1974).Google Scholar
  18. 18.
    M. M. Postnikov, Introduction to the Morse Theory [in Russian], Nauka, Moscow (1971).Google Scholar
  19. 19.
    Yu. B. Rusak, Generalized Jordan Structures in Branching Theory, PhD Thesis, Tashkent, 1979.Google Scholar
  20. 20.
    N. A. Sidorov, General Issues of Regularization in Problems of the Branching Theory [in Russian], Irkutsk State Univ., Irkutsk (1982).Google Scholar
  21. 21.
    N. Sidorov, B. Loginov, A. Sinitsyn, and M. V. Falaleev, Lyapunov–Schmidt Methods in Nonlinear Analysis and Applications, Kluwer Academic Publ., Dordrecht (2012).zbMATHGoogle Scholar
  22. 22.
    V. A. Trenogin, Functional Analysis [in Russian], Fizmatlit, Moscow (2002).Google Scholar
  23. 23.
    V. A. Trenogin and A. F. Filippov (eds.), Nonlinear Analysis and Nonlinear Differential Equations [in Russian], Fizmatlit, Moscow (2003).Google Scholar
  24. 24.
    V. A. Trenogin and N. A. Sidorov, “An investigation of the bifurcation points and nontrivial branches of the solutions of nonlinear equations,” In: Differential and Integral Equations, No. 1, Irkutsk State Univ., Irkutsk, 216–247 (1972).Google Scholar
  25. 25.
    M. M. Vaynberg and V. A. Trenogin, Branching Theory of Solutions to Nonlinear Equations [in Russian], Nauka, Moscow (1969).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • B. V. Loginov
    • 1
  • Yu. B. Rousak
    • 2
  • L. R. Kim-Tyan
    • 3
  1. 1.Ulyanovsk State Technical UniversityUlyanovskRussia
  2. 2.Department of Social ServiceCanberraAustralia
  3. 3.National University of Science and Technology “MISiS”MoscowRussia

Personalised recommendations