Advertisement

Differential Equations

, Volume 49, Issue 1, pp 126–131 | Cite as

Weak periodic shift operator and generalized-periodic motions

  • A. P. Afanas’ev
  • S. M. Dzyuba
Short Communications

Abstract

We introduce the notions of a weak periodic system and its generalized-periodic motions. Such systems include systems generated by ordinary differential equations, equations with retarded argument, Volterra type equations, etc. We present a criterion for the existence of generalized-periodic motions and establish their main properties.

Keywords

Periodic Solution Periodic Motion Periodic System Shift Operator Nonautonomous System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Massera, J.L., The Existence of Periodic Solutions of Systems of Differential Equations, Duke Math. J., 1950, vol. 17, pp. 457–475.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Afanas’ev, A.P. and Dzyuba, S.M., Ustoichivost’ po Puassonu v dinamicheskikh i nepreryvnykh periodicheskikh sistemakh (Poisson Stability in Dynamical and Continuous Periodic Systems), Moscow, 2007.Google Scholar
  3. 3.
    Millionshchikov, V.M., Recurrent and Almost Periodic Limit Solutions of Nonautonomous Systems, Differ. Uravn., 1968, vol. 4, no. 9, pp. 1555–1559.zbMATHGoogle Scholar
  4. 4.
    Shcherbakov, B.A., Multidimensional Dynamical Systems, Differ. Uravn., 1994, vol. 30, no. 5, pp. 739–747.MathSciNetGoogle Scholar
  5. 5.
    Bronshtein, I.U. and Chernii, V.F., Extensions of Dynamical Systems with Uniformly Asymptotically Stable Points, Differ. Uravn., 1974, vol. 10, no. 7, pp. 1225–1230.Google Scholar
  6. 6.
    Gerko, A.I., Asymptotically Recurrent Solutions of β-Differential Equations, Mat. Zametki, 2000, vol. 67, no. 6, pp. 837–851.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cheban, D.N., Asymptotically Almost Periodic Solutions of Differential Equations, New York, 2009.zbMATHCrossRefGoogle Scholar
  8. 8.
    Krasnosel’skii, M.A., Operator sdviga po traektoriyam differentsial’nykh uravnenii (Shift Operators along Trajectories of Differential Equations), Moscow: Nauka, 1966.Google Scholar
  9. 9.
    Hale, J., Theory of Functional Differential Equations, New York: Springer, 1977. Translated under the title Teoriya funktsional’no-differentsial’nykh uravnenii, Moscow: Mir, 1984.zbMATHCrossRefGoogle Scholar
  10. 10.
    Afanas’ev, A.P. and Dzyuba, S.M., Quasiperiodic Processes in Control Problems, Izv. Akad. Nauk Teor. Sist. Upr., 2001, no. 2, pp. 22–28.Google Scholar
  11. 11.
    Schwartz, L., Analiz (Analysis), Moscow, 1972, vol. 2.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • A. P. Afanas’ev
    • 1
    • 2
    • 3
  • S. M. Dzyuba
    • 1
    • 2
    • 3
  1. 1.Institute of Systems AnalysisRussian Academy of SciencesMoscowRussia
  2. 2.Moscow Institute of Physics and TechnologyMoscowRussia
  3. 3.Tambov State Technical UniversityTambovRussia

Personalised recommendations