Journal of Dynamics and Differential Equations

, Volume 17, Issue 4, pp 737–758 | Cite as

On Invariant Tori for a Stochastic Ito System

  • A. M. Samoilenko
  • O. M. Stanzhyts’kyj
  • A. M. Ateiwi
Article

Abstract

By using the Green function, we obtain conditions implying existence of invariant sets for Ito systems that are extensions of dynamical systems on a torus.

Keywords

Invariant set torus Green function stability 

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References

  1. 1.
    Arnold, L. 1998Random Dynamical SystemsSpringer-VerlagBerlin-Heidelberg-New YorkGoogle Scholar
  2. 2.
    Gikhman, I. I., Skorokhod, A. V. 1982Stochastic Differential Equations and their ApplicationsKievNaukova Dumka (Russian)Google Scholar
  3. 3.
    Samoilenko A.M. (1991). Elements of the mathematical theory of multi-frequence oscillations. Moscow, Nauka, 1987 (Russian). Translated into Englsih:Samojlenko, A.M. Elements of the mathematical theory of multi-frequency oscillations. Mathematics and Its Applications, Soviet Series, 71. Dordrecht etc.: Kluwer Academic Publishers.Google Scholar
  4. 4.
    Sell, F.R. 1979Bifurcation of higher dimensional toriArch. Ration. Mech Anal.69199230CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Stanzhyts’kyj, O.M. 2000A study of stability of invariant sets using local coordinatesNonlinear Oscill3226270(Ukrainian)Google Scholar
  6. 6.
    Tsar’kov, E.F. 1989Random Peturbations of Functional-Differential EquationsRigaZinatne(Russian)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. M. Samoilenko
    • 1
  • O. M. Stanzhyts’kyj
    • 2
  • A. M. Ateiwi
    • 3
  1. 1.Institute of Mathematics of the National Academy of Sciences of UkraineKyivUkraine
  2. 2.Taras Shevchenko Kyiv UniversityKyivUkraine
  3. 3.Mathematics Department, Department of MathematicsAL-Hussein Bin Talal UniversityMa’anJordan

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