Ukrainian Mathematical Journal

, Volume 57, Issue 7, pp 1089–1119 | Cite as

Conditions for Synchronization of One Oscillation System

  • A. M. Samoilenko
  • L. Recke


Using methods of perturbation theory, we investigate the global behavior of trajectories on a toroidal attractor and in its neighborhood for a system of differential equations that arises in the study of synchronization of oscillations in the mathematical model of an optical laser.


Differential Equation Mathematical Model Perturbation Theory Oscillation System Global Behavior 
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  1. 1.
    S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wunsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev., 69, 016206 (2004).Google Scholar
  2. 2.
    B. Tromborg, H. E. Lassen, and H. Olesen, “Travelling wave analysis of semiconductor lasers,” IEEE J. Quant. El., 30, No.5, 939–956 (1994).Google Scholar
  3. 3.
    J. Sieber, “Numerical bifurcation analysis for multisection semiconductor lasers,” SIAM J. Appl. Dynam. Syst., 1, No.2, 248–270 (2002).MATHMathSciNetGoogle Scholar
  4. 4.
    B. Sartorius, C. Bornholdt, O. Brox, H. J. Ehrke, D. Hoffmann, R. Ludwig, and M. Mohrle, “All-optical clock recovery module based on self-pulsating DFB laser,” Electron. Lett., 34, No.17, 1664–1665 (1998).CrossRefGoogle Scholar
  5. 5.
    L. Recke and D. Peterhof, “Abstract forced symmetry breaking and forced frequency locking of modulated waves,” J. Different. Equat., 144, No.2, 233–262 (1998).MathSciNetMATHGoogle Scholar
  6. 6.
    U. Bandelow, L. Recke, and B. Sandstede, “Frequency regions for forced locking of self-pulsating multisection DFB lasers,” Opt. Commun., 147, No.1–3, 212–218 (1998).Google Scholar
  7. 7.
    D. Peterhof and B. Sandstede, “All-optical clock recovery using multisection distributed-feedback lasers,” J. Nonlin. Sci., 9, No.5, 575–613 (1999).CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    L. Recke, “Forced frequency locking of rotating waves,” Ukr. Mat. Zh., 50, No.1, 94–101 (1998).MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    B. Krauskopf and S. M. Wieczorek, “Accumulating regions of winding periodic orbits in optically driven lasers,” Physica D, 173, No.1–2, 97–113 (2002).MathSciNetMATHGoogle Scholar
  10. 10.
    L. Recke, K. R. Schneider, and J. Sieber, “Dynamics of multisection semiconductor lasers,” Sovr. Mat. Fundam. Napr., 1, No.1, 1–12 (2003).Google Scholar
  11. 11.
    N. N. Bogolyubov and Yu. A. Mitropol'skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).MATHGoogle Scholar
  12. 12.
    J. Moser, “A rapidly convergent iteration method and nonlinear differential equations,” Ann. Scuola Norm. Super. Pisa, 20, No.3, 499–535 (1966).MATHGoogle Scholar
  13. 13.
    M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant Manifolds, Springer, Berlin (1977).MATHGoogle Scholar
  14. 14.
    A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Oscillations, Kluwer, Dordrecht (1991).Google Scholar
  15. 15.
    Yu. A. Mitropol'skii and O. B. Lykova, Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow (1973).Google Scholar
  16. 16.
    N. N. Bogoliubov, Yu. A. Mitropolsky, and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer, Berlin (1976).Google Scholar
  17. 17.
    A. M. Samoilenko, Investigation of a Dynamical System in the Neighborhood of a Quasiperiodic Trajectory [in Russian], Preprint No. 90.35, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1990).Google Scholar
  18. 18.
    A. M. Samoilenko and R. Petryshyn, Multifrequency Oscillations of Nonlinear Systems, Kluwer, Dordrecht (2004).MATHGoogle Scholar
  19. 19.
    A. Denjoy, “Sur les courbes definies par les equations differentielles a la surface du tore,” J. Math. Pure Appl., Ser. 9, 11, No.4, 333–375 (1932).MATHGoogle Scholar
  20. 20.
    V. A. Pliss, Nonlocal Problems in the Theory of Oscillations [in Russian], Nauka, Moscow (1964).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. M. Samoilenko
    • 1
  • L. Recke
    • 2
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKievUkraine
  2. 2.Humboldt UniversityBerlinGermany

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