Theoretical and Mathematical Physics

, Volume 181, Issue 2, pp 1339–1348 | Cite as

Synchronization in a nonisochronous nonautonomous system

  • L. A. KalyakinEmail author


We study a model system of nonautonomous nonlinear differential equations arising in magnetodynamics theory. We find constraints on the parameters such that Lyapunov-stable solutions with a stabilized phase exist. These solutions describe the synchronization phenomenon in a nonisochronous system with slowly varying parameters.


nonlinear oscillation asymptotic behavior synchronization stability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge Nonlin. Sci. Ser., Vol. 12), Cambridge Univ. Press, Cambridge (2001).CrossRefGoogle Scholar
  2. 2.
    L. A. Kalyakin, Russ. Math. Surveys, 63, 791–857 (2008).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    L. A. Kalyakin and M. A. Shamsutdinov, Theor. Math. Phys., 160, 960–967 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    L. A. Kalyakin and O. A. Sultanov, Differ. Equ., 49, 267–281 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    N. N. Bogoliubov and Yu. A. Mitropolskiy, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974); English transl., Gordon and Breach, New York (1961).Google Scholar
  6. 6.
    V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt, “Mathematical aspects of classical and celestial mechanics,” in: Dynamical Systems — 3 [in Russian] (Sovrem. Probl. Mat. Fund. Naprav., Vol. 3), VINITI, Moscow (1985), pp. 5–290.Google Scholar
  7. 7.
    R. Adler, Proc. IRE, 34, 351–357 (1946).CrossRefGoogle Scholar
  8. 8.
    A. Slavin and V. Tiberkevich, IEEE Transaction on Magnetics, 45, 1875–1918 (2009).ADSCrossRefGoogle Scholar
  9. 9.
    L. A. Kalyakin, Comput. Math. Math. Phys., 50, 1338–1349 (2010).CrossRefMathSciNetGoogle Scholar
  10. 10.
    A. N. Kuznetsov, Funct. Anal. Appl., 6, No. 2, 119–127 (1972).CrossRefzbMATHGoogle Scholar
  11. 11.
    M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations [in Russian], Nauka, Moscow (1983); English transl.: Asymptotic Analysis: Linear Ordinary Differential Equations, Springer, Berlin (1993).zbMATHGoogle Scholar
  12. 12.
    A. M. Lyapunov, The General Problem of the Stability of Motion [in Russian], Gostekhizdat, Moscow (1950); English transl., Taylor and Francis, London (1992).zbMATHGoogle Scholar
  13. 13.
    N. N. Krasovskii, Some Problems in the Theory of the Stability of Motion [in Russian], Fizmatgiz, Moscow (1959).Google Scholar
  14. 14.
    V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations (Princeton Math. Ser., Vol. 22), Princeton Univ. Press, Princeton, N. J. (1960).zbMATHGoogle Scholar
  15. 15.
    J. G. Malkin, Theory of the Stability of Motion [in Russian], Gostekhizdat, Moscow (1952); German transl.: Theorie der Stabilität einer Bewegung, R. Oldenbourg, München (1959).Google Scholar
  16. 16.
    O. A. Sultanov, Trudy Inst. Mat. i Mekh. UrO RAN, 19, 274–283 (2013).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Institute of Mathematics with Computer CenterRASUfaRussia

Personalised recommendations