Differential Equations

, Volume 44, Issue 10, pp 1406–1411 | Cite as

Time symmetry preserving perturbations of systems, and Poincaré mappings

  • V. I. Mironenko
  • V. V. Mironenko
Ordinary Differential Equations


In the present paper, we obtain necessary and sufficient conditions under which two differential systems have the same symmetries described by a reflecting function. Under these conditions, the systems in question have a common shift operator along solutions of these systems on a symmetric time interval [−ω, ω]. Therefore, the mappings over the period [−ω, ω] coincide for such systems provided that these systems are 2ω-periodic.


General Solution Periodic Solution Vector Function Entire Space Continuous Matrix 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Krasnosel’skii, M.A., Operator sdviga po traektoriyam differentsial’nykh uravnenii (Operator of Shift Along Trajectories of Differential Equations), Moscow: Nauka, 1996.Google Scholar
  2. 2.
    Mironenko, V.I., Differ. Uravn., 1980, vol. 16, no. 11, pp. 1985–1994.zbMATHMathSciNetGoogle Scholar
  3. 3.
    Mironenko, V.I., Otrazhayushchaya funktsiya i periodicheskie resheniya differentsial’nykh uravnenii (Reflection Function and Periodic Solutions of Differential Equations), Minsk: Universitetskoe, 1986.Google Scholar
  4. 4.
    Al’sevich, L.A., Differ. Uravn., 1983, vol. 19, no. 8, pp. 1446–1449.zbMATHMathSciNetGoogle Scholar
  5. 5.
    Veresovich, P.P., Differ. Uravn., 1998, vol. 34, no. 10, pp. 1420–1422.MathSciNetGoogle Scholar
  6. 6.
    Musafirov, E.V., Differ. Uravn., 2002, vol. 38, no. 4, pp. 570–572.MathSciNetGoogle Scholar
  7. 7.
    Zhengxin Zhou, Nonlinear Analysis, 2003, vol. 53, no. 6, pp. 733–741.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Zhengxin Zhou, J. Math. Anal. Appl., 2003, vol. 278, no. 1, pp. 18–26.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Mironenko, V.V., Differ. Uravn., 2004, vol. 40, no. 10, pp. 1325–1332.MathSciNetGoogle Scholar
  10. 10.
    Mironenko, V.I., Otrazhayushchaya funktsiya i issledovanie mnogomernykh differentsial’nykh sistem (Reflecting Function and the Investigation of Multidimensional Differential Systems), Gomel’: Gomel’skii Gos. Univ., 2004.zbMATHGoogle Scholar

Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  • V. I. Mironenko
    • 1
  • V. V. Mironenko
    • 1
  1. 1.Gomel State UniversityGomelBelarus

Personalised recommendations