Abstract
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.
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References
J. Piprek (editor), Optoelectronic Devices, Springer, New York (2005).
J. Sieber, L. Recke, and K. R. Schneider, “Dynamics of multisection semiconductor lasers,” J. Math. Sci., 124, 5298–5309 (2004).
S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers,” Phys. Rev. E, 69, 056221 (2004).
S. Yanchuk, A. Stefanski, T. Kapitaniak, and J. Wojewoda, “Dynamics of an array of coupled semiconductor lasers,” Phys. Rev. E, 73, 016209 (2006).
I. V. Koryukin and P. Mandel, “Two regimes of synchronization in unidirectionally coupled semiconductor lasers,” Phys. Rev. E, 65, 026201 (2002).
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).
A. M. Samoilenko and L. Recke, “Conditions for synchronization of one oscillation system,” Ukr. Math. Zh., 57, No. 7, 1089–1119 (2005).
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).
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).
J. K. White, M. Matus, and J. V. Moloney, “Achronal generalized synchronization in mutually coupled semiconductor lasers,” Phys. Rev. E, 65, 036229 (2002).
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.
L. Recke, M. Wolfrum, and S. Yanchuk, “Analysis and control of complex nonlinear processes,” World Sci. Lect. Notes Complex Systems, 5, 185–212 (2007).
A. F. Glova, “Synchronization of laser radiation for lasers with optical coupling,” Kvant. Élektron., 33, No. 4, 283–306 (2003).
F. Brauer and J. A. Nohel, Qualitative Theory of Ordinary Differential Equations, Benjamin, New York (1969).
M. Farkas, Periodic Motions, Springer (1994).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 426–435, March, 2008.
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Yanchuk, S.V., Schneider, K.R. & Lykova, O.B. Amplitude synchronization in a system of two coupled semiconductor lasers. Ukr Math J 60, 495–507 (2008). https://doi.org/10.1007/s11253-008-0070-3
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DOI: https://doi.org/10.1007/s11253-008-0070-3