Ukrainian Mathematical Journal

, Volume 60, Issue 3, pp 495–507 | Cite as

Amplitude synchronization in a system of two coupled semiconductor lasers

  • S. V. Yanchuk
  • K. R. Schneider
  • O. B. Lykova


We consider a system of ordinary differential equations used to describe the dynamics of two coupled single-mode semiconductor lasers. In particular, we study solutions corresponding to the amplitude synchronization. It is shown that the set of these solutions forms a three-dimensional invariant manifold in the phase space. We study the stability of trajectories on this manifold both in the tangential direction and in the transverse direction. We establish conditions for the existence of globally asymptotically stable solutions of equations on the manifold synchronized with respect to the amplitude.


Periodic Solution Equilibrium Position Semiconductor Laser Invariant Manifold Stable Periodic Solution 
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  1. 1.
    J. Piprek (editor), Optoelectronic Devices, Springer, New York (2005).Google Scholar
  2. 2.
    J. Sieber, L. Recke, and K. R. Schneider, “Dynamics of multisection semiconductor lasers,” J. Math. Sci., 124, 5298–5309 (2004).CrossRefMathSciNetGoogle Scholar
  3. 3.
    S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers,” Phys. Rev. E, 69, 056221 (2004).Google Scholar
  4. 4.
    S. Yanchuk, A. Stefanski, T. Kapitaniak, and J. Wojewoda, “Dynamics of an array of coupled semiconductor lasers,” Phys. Rev. E, 73, 016209 (2006).Google Scholar
  5. 5.
    I. V. Koryukin and P. Mandel, “Two regimes of synchronization in unidirectionally coupled semiconductor lasers,” Phys. Rev. E, 65, 026201 (2002).Google Scholar
  6. 6.
    G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett., 85, 3809–3812 (2000).CrossRefGoogle Scholar
  7. 7.
    A. M. Samoilenko and L. Recke, “Conditions for synchronization of one oscillation system,” Ukr. Math. Zh., 57, No. 7, 1089–1119 (2005).CrossRefMathSciNetGoogle Scholar
  8. 8.
    R. Vicente, Shuo Tang, J. Mulet, C. R. Mirasso, and Jia Ming Liu, “Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling,” Phys. Rev. E, 73, 047201 (2006).Google Scholar
  9. 9.
    I. Wedekind and U. Parlitz, “Synchronization and antisynchronization of chaotic power drop-outs and jump-ups of coupled semiconductor lasers,” Phys. Rev. E, 66, 026218 (2002).Google Scholar
  10. 10.
    J. K. White, M. Matus, and J. V. Moloney, “Achronal generalized synchronization in mutually coupled semiconductor lasers,” Phys. Rev. E, 65, 036229 (2002).Google Scholar
  11. 11.
    E. Wille, M. Peil, I. Fischer, and W. Elsäßer, “Dynamical scenarios of mutually delay-coupled semiconductor lasers in the short coupling regime,” in: D. Lenstra, G. Morthier, T. Erneux, and M. Pessa (editors), Proc. of the SPIE “Semiconductor Lasers and Laser Dynamics,” 5452 (2004), pp. 41–50.Google Scholar
  12. 12.
    L. Recke, M. Wolfrum, and S. Yanchuk, “Analysis and control of complex nonlinear processes,” World Sci. Lect. Notes Complex Systems, 5, 185–212 (2007).MathSciNetGoogle Scholar
  13. 13.
    A. F. Glova, “Synchronization of laser radiation for lasers with optical coupling,” Kvant. Élektron., 33, No. 4, 283–306 (2003).CrossRefGoogle Scholar
  14. 14.
    F. Brauer and J. A. Nohel, Qualitative Theory of Ordinary Differential Equations, Benjamin, New York (1969).Google Scholar
  15. 15.
    M. Farkas, Periodic Motions, Springer (1994).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • S. V. Yanchuk
    • 1
    • 2
    • 3
  • K. R. Schneider
    • 2
  • O. B. Lykova
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKievUkraine
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  3. 3.Institute of MathematicsHumboldt UniversityBerlinGermany

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