Skip to main content
SpringerLink
Log in

Principal Asymptotics in the Problem on the Andronov–Hopf Bifurcation and Their Applications

  • Ordinary Differential Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

New formulas are obtained for the principal asymptotics of bifurcation solutions in the problem on the Andronov–Hopf bifurcation, leading to new algorithms for studying bifurcations in the general setting. The approach proposed in the paper allows one to consider not only the classical problems about bifurcations of codimension one but also some problems concerning bifurcations of codimension two. A new approach to the analysis of bifurcations of cycles in systems with homogeneous nonlinearities is proposed. As an application, we consider the problem on the bifurcation of periodic solutions of the van der Pol equation.

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. Marsden, J.E. and McCracken, M., The Hopf Bifurcation and Its Applications, New York: Springer-Verlag, 1976. Translated under the title Bifurkatsiya rozhdeniya tsikla i ee prilozheniya, Moscow: Mir, 1980.

    MATH  Google Scholar 

  2. Hassard, B.D., Kazarinoff, N.D., and Wang, Y.-H., Theory and Applications of the Hopf Bifurcation, Cambridge: Cambridge Univ. Press, 1980. Translated under the title Teoriya i prilozheniya bifurkatsii rozhdeniya tsikla, Moscow: Mir, 1985.

    MATH  Google Scholar 

  3. Kuznetsov, Yu.A., Elements of Applied Bifurcation Theory, New York: Springer-Verlag, 1998.

    MATH  Google Scholar 

  4. Shil’nikov, L.P., Shil’nikov, A.L., Turaev, D.V., and Chua, L.O., Methods of Qualitative Theory in Nonlinear Dynamics, Singapore: World Sci., 2001, part 2.

    Book  Google Scholar 

  5. Arnol’d, V.I., Geometricheskie metody v teorii obyknovennykh differentsial’nykh uravnenii (Geometric Methods for Ordinary Differential Equations), Moscow; Izhevsk: Inst. Komp. Issled., 2000.

    Google Scholar 

  6. Guckenheimer, J. and Holmes, P.J., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, New York: Springer-Verlag, 1983.

    Book  MATH  Google Scholar 

  7. Bryuno, A.D., The analytic form of differential equations, Tr. Mosk. Mat. Obs., 1971, vol. 25, pp. 119–262; 1972, vol. 26, pp. 199–239.

    MathSciNet  MATH  Google Scholar 

  8. Chow, S.-N., Li, C., and Wang, D., Normal Forms and Bifurcation of Planar Vector Fields, Cambridge: Cambridge Univ. Press, 1994.

    Book  MATH  Google Scholar 

  9. Rachinskii, D. and Schneider, K.R., Dynamic Hopf bifurcations generated by nonlinear terms, J. Differential Equations, 2005, vol. 210, no. 1, pp. 65–86.

    Article  MathSciNet  MATH  Google Scholar 

  10. Krasnosel’skii, M.A., Kuznetsov, N.A., and Yumagulov, M.G., Parameter functionability and asymptotic behavior of cycles in Hopf’s bifurcation, Autom. Remote Control, 1996, vol. 57, no. 11, part 1, pp. 1548–1553.

    MATH  Google Scholar 

  11. Krasnosel’skii, M.A., Kuznetsov, N.A., and Yumagulov, M.G., An operator method for investigating the stability of cycles in Hopf bifurcation, Autom. Remote Control, 1996, vol. 57, no. 12, part 1, pp. 1701–1708.

    MATH  Google Scholar 

  12. Vyshinskii, A.A., Ibragimova, L.S., Murtazina, S.A., and Yumagulov, M.G., An operator method for approximate investigation of a regular bifurcation in multiparameter dynamical systems, Ufim. Mat. Zh., 2010, vol. 2, no. 4, pp. 3–26.

    MATH  Google Scholar 

  13. Kozyakin, V.S. and Krasnosel’skii, M.A., The method of parameter functionalization in the Hopf bifurcation problem, Nonlinear Anal. Theory Methods Appl., 1987, vol. 11, no. 2, pp. 149–161.

    Article  MathSciNet  MATH  Google Scholar 

  14. Krasnosel’skii, M.A., Vainikko, G.M., Zabreiko, P.P., et al., Priblizhennoe reshenie operatornykh uravnenii (Approximate Solution of Operator Equations), Moscow: Nauka, 1969.

    Google Scholar 

  15. Krasnosel’skii, M.A. and Yumagulov, M.G., The parameter functionalization method in the eigenvalue problem, Dokl. Math., 1999, vol. 59, no. 2, pp. 204–206.

    MathSciNet  Google Scholar 

  16. Kuznetsov, N.A., Yumagulov, M.G., and Sharafutdinov, I.V., The investigation algorithm of stability of periodic oscillations in the problem for the Andronov–Hopf bifurcation, Autom. Remote Control, 2008, vol. 69, no. 12, pp. 2033–2038.

    Article  MathSciNet  MATH  Google Scholar 

  17. Krasnosel’skii, A.M., Mennicken, R., and Rachinskii, D.I., Cycle stability for Hopf bifurcation generated by sublinear terms, Math. Nachr., 2002, vol. 233–234, pp. 171–195.

    Article  MathSciNet  Google Scholar 

  18. Diamond, P., Rachinskii, D., and Yumagulov, M., Stability of large cycles in a nonsmooth problem with Hopf bifurcation at infinity, Nonlinear Anal. Theory Methods Appl., 2000, vol. 42, no. 6, pp. 1017–1031.

    Article  MathSciNet  MATH  Google Scholar 

  19. Krasnosel’skii, A. and Rachinskii, D., Hopf bifurcations from infinity, generated by bounded nonlinear terms, Funct. Differ. Equ., 1999, vol. 6, no. 3–4, pp. 357–374.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bashkir State University, Ufa, 450076, Russia

    M. G. Yumagulov & E. S. Imangulova

  2. Bashkir State Agrarian University, Ufa, 450001, Russia

    L. S. Ibragimova

Authors
  1. M. G. Yumagulov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. S. Ibragimova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. S. Imangulova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. G. Yumagulov.

Additional information

Original Russian Text © M.G. Yumagulov, L.S. Ibragimova, E.S. Imangulova, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 12, pp. 1627–1642.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yumagulov, M.G., Ibragimova, L.S. & Imangulova, E.S. Principal Asymptotics in the Problem on the Andronov–Hopf Bifurcation and Their Applications. Diff Equat 53, 1578–1594 (2017). https://doi.org/10.1134/S0012266117120060

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266117120060

Search

Navigation