Skip to main content
SpringerLink
Log in

Global solutions and invariant tori of difference-differential equations

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We prove the existence of an m-parameter family of global solutions of a system of difference-differential equations. For difference-differential equations on a torus, we introduce the notion of rotation number. We also consider the problem of perturbation of an invariant torus of a system of difference-differential equations and study the problem of the existence of periodic and quasiperiodic solutions of second-order difference-differential equations.

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. J. Hale, The Theory of Function-Differential Equations [Russian translation], Mir, Moscow (1984).

    Google Scholar 

  2. Yu. A. Mitropol’skii and O. B. Lykova, Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  3. V. I. Fodchuk, “The method of integral manifolds in the theory of differential equations with deviating argument”, in: Problems of the Asymptotic Theory of Nonlinear Oscillations [in Russian] Naukova Dumka, Kiev 1977, pp. 232–237.

    Google Scholar 

  4. Yu. A. Mitropol’skii, A. M. Samoilenko, and D. I. Martynyuk, Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients [in Russian], Naukova Dumka, Kiev (1984).

    Google Scholar 

  5. N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymtotic Methods in the Theory of Nonlinear Oscillations [in Russian], Fizmatgiz, Moscow (1963).

    Google Scholar 

  6. Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  7. D. Henry, Geometrical Theory of Semilinear Parabolic Equations [Russian translation], Mir, Moscow (1984).

    Google Scholar 

  8. E. L. Koddington and N. Levinson, The Theory of Ordinary Differential Equations [Russian translation]. Inostrannaya Literatura, Moscow (1958).

    Google Scholar 

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

    Google Scholar 

  10. W. A. Coppel and K. J. Palmer, “Averaging of integral manifolds,” Bull. Austral. Math. Soc., 197–222 (1970).

  11. E. P. Belan and O. B. Lykova, “Integral manifolds and exponential splitting of linear parabolic equations with rapidly varying coefficients”, Ukr. Mat. Zh., 47, No. 12, 1593–1608 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  12. E. P. Belan and O. B. Lykova, “Theorem on the central manifold of a nonlinear parabolic equation”, Ukr. Mat. Zh., 48, No. 8, 1021–1036 (1996).

    Article  MathSciNet  Google Scholar 

  13. O. V. Anashkin and M. M. Khapaev, “On the stability of nonlinear systems with small parameter,” Differents. Uravn., 29, No. 8, 1300–1307 (1993).

    MathSciNet  Google Scholar 

  14. A. V. Rakhmanov, “On the dimension of the central manifold for semilinear parabolic equations”, Ukr. Mat. Zh., 42, No. 10, 1356–1362 (1990).

    Google Scholar 

  15. V. P. Rubanik, Oscillations of Quasilinear Systems with Delay [in Russian], Nauka, Moscow (1969).

    Google Scholar 

Download references

Authors
  1. E. P. Belan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Simferopol University, Simferopol. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 1, pp. 11–24, January, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belan, E.P. Global solutions and invariant tori of difference-differential equations. Ukr Math J 49, 9–24 (1997). https://doi.org/10.1007/BF02486613

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02486613

Keywords

Search

Navigation