Ukrainian Mathematical Journal

, Volume 67, Issue 10, pp 1527–1536 | Cite as

Construction of Lyapunov Functions in the Theory of Regular Linear Extensions of Dynamical Systems on a Torus

  • V. L. Kulyk
Article
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Lyapunov functions are considered in the form of linear combinations of quadratic forms. We study the conditions under which the linear extensions of dynamic systems on a torus are regular.

Keywords

Cauchy Problem Quadratic Form Green Function Lyapunov Function Naukova Dumka 
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References

  1. 1.
    A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Oscillations [in Russian], Moscow, Nauka (1987).Google Scholar
  2. 2.
    A. M. Samoilenko, “On some problems in the perturbation theory of smooth invariant tori of dynamical systems,” Ukr. Mat. Zh., 46, No. 12, 1665–1699 (1994); English translation: Ukr. Math. J., 46, No. 12, 1848–1889 (1994).Google Scholar
  3. 3.
    A. M. Samoilenko, “On the existence of a unique Green function for the linear extension of a dynamical system on a torus,” Ukr. Mat. Zh., 53, No. 4, 513–521 (2001); English translation: Ukr. Math. J., 53, No. 4, 584–594 (2001).Google Scholar
  4. 4.
    Yu. A. Mitropol’skii, A. M. Samoilenko, and V. L. Kulik, Investigation of the Dichotomy of Linear Systems of Differential Equations with the Help of Lyapunov Functions [in Russian], Naukova Dumka, Kiev (1990).Google Scholar
  5. 5.
    A. A. Boichuk, “A condition for the existence of a unique Green–Samoilenko function for the problem of invariant torus,” Ukr. Mat. Zh., 53, No. 4, 556–559 (2001); English translation: Ukr. Math. J., 53, No. 4, 637–641 (2001).Google Scholar
  6. 6.
    V. L. Kulyk and N. V. Stepanenko, “Alternating Lyapunov functions in the theory of linear extensions of dynamical systems on a torus,” Ukr. Mat. Zh., 59, No. 4, 488–500 (2007); English translation: Ukr. Math. J., 59, No. 4, 546–562 (2007).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • V. L. Kulyk
    • 1
  1. 1.Silesia Technical UniversityGliwicePoland

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