# American Institute of Mathematical Sciences

September  2011, 31(3): 847-875. doi: 10.3934/dcds.2011.31.847

## Frequency locking of modulated waves

 1 Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany 2 Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska St., 01601 Kiev, Ukraine, Ukraine, Ukraine 3 Institute of Mathematics, Humboldt University of Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

Received  June 2010 Revised  September 2010 Published  August 2011

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs.
Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.
Citation: Lutz Recke, Anatoly Samoilenko, Alexey Teplinsky, Viktor Tkachenko, Serhiy Yanchuk. Frequency locking of modulated waves. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 847-875. doi: 10.3934/dcds.2011.31.847
##### References:
 [1] U. Bandelow, L. Recke and B. Sandstede, Frequency regions for forced locking of self-pulsating multi-section DFB lasers, Opt. Commun., 147 (1998), 212-218. doi: 10.1016/S0030-4018(97)00570-1. [2] N. N. Bogoliubov and Y. A. Mitropolsky, "Asymptotic Methods in the Theory of Non-linear Oscillations," International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961. [3] C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition, Texts in Applied Mathematics, 34, Springer, New York, 2006. [4] D. Chillingworth, Generic multiparameter bifurcation from a manifold, Dyn. Stab. Syst., 15 (2000), 101-137. [5] B. P. Demidovich, "Lectures on Stability Theory," Nauka, Moscow, 1967. [6] U. Feiste, D. J. As and A. Erhardt, 18 GHz all-optical frequency locking and clock recovery using a self-pulsating two-section laser, IEEE Photon. Technol. Lett., 6 (1994), 106-108. doi: 10.1109/68.265905. [7] M. Lichtner, M. Radziunas and L. Recke, Well-posedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics, Math. Methods Appl. Sci., 30 (2007), 931-960. doi: 10.1002/mma.816. [8] M. Nizette, T. Erneux, A. Gavrielides and V. Kovanis, Stability and bifurcations of periodically modulated, optically injected laser diodes, Phys. Rev. E, 63 (2001), Paper number 026212. [9] D. Peterhof and B. Sandstede, All-optical clock recovery using multisection distributed-feedback lasers, J. Nonlinear Sci., 9 (1999), 575-613. doi: 10.1007/s003329900079. [10] M. Radziunas, Numerical bifurcation analysis of the traveling wave model of multisection semiconductor lasers, Physica D, 213 (2006), 98-112. doi: 10.1016/j.physd.2005.11.003. [11] L. Recke, Forced frequency locking of rotating waves, Ukraīn. Math. J, 50 (1998), 94-101. [12] L. Recke and D. Peterhof, Abstract forced symmetry breaking and forced frequency locking of modulated waves, J. Differential Equations, 144 (1998), 233-262. [13] A. M. Samoilenko, "Elements of the Mathematical Theory of Multi-Frequency Oscillations," Mathematics and its Applications (Soviet Series), 71, Kluwer Acad. Publ. Group, Dordrecht, 1991. [14] A. M. Samoilenko and L. Recke, Conditions for synchronization of one oscillatory system, Ukrain. Math. J., 57 (2005), 1089-1119. doi: 10.1007/s11253-005-0250-3. [15] B. Sartorius, C. Bornholdt, O. Brox, H. J. Ehrke, D. Hoffmann, R. Ludwig and M. Möhrle, All-optical clock recovery module based on self-pulsating DFB laser, Electronics Letters, 34 (1998), 1664-1665. doi: 10.1049/el:19981152. [16] K. R. Schneider, Entrainment of modulation frequency: A case study, Int. J. Bifurc. Chaos Appl. Sci. Eng., 15 (2005), 3579-3588. doi: 10.1142/S0218127405014234. [17] J. Sieber, Numerical bifurcation analysis for multisection semiconductor lasers, SIAM J. Appl. Dyn. Syst., 1 (2002), 248-270. doi: 10.1137/S1111111102401746. [18] S. Wieczorek, B. Krauskopf, T. B. Simpson and D. Lenstra, The dynamical complexity of optically injected semiconductor lasers, Phys. Rep., 416 (2005), 1-128. doi: 10.1016/j.physrep.2005.06.003. [19] Y. F. Yi, Stability of integral manifold and orbital attraction of quasi-periodic motion, J. Differential Equation, 103 (1993), 278-322. [20] Y. F. Yi, A generalized integral manifold theorem, J. Differential Equation, 102 (1993), 153-187.

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##### References:
 [1] U. Bandelow, L. Recke and B. Sandstede, Frequency regions for forced locking of self-pulsating multi-section DFB lasers, Opt. Commun., 147 (1998), 212-218. doi: 10.1016/S0030-4018(97)00570-1. [2] N. N. Bogoliubov and Y. A. Mitropolsky, "Asymptotic Methods in the Theory of Non-linear Oscillations," International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961. [3] C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition, Texts in Applied Mathematics, 34, Springer, New York, 2006. [4] D. Chillingworth, Generic multiparameter bifurcation from a manifold, Dyn. Stab. Syst., 15 (2000), 101-137. [5] B. P. Demidovich, "Lectures on Stability Theory," Nauka, Moscow, 1967. [6] U. Feiste, D. J. As and A. Erhardt, 18 GHz all-optical frequency locking and clock recovery using a self-pulsating two-section laser, IEEE Photon. Technol. Lett., 6 (1994), 106-108. doi: 10.1109/68.265905. [7] M. Lichtner, M. Radziunas and L. Recke, Well-posedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics, Math. Methods Appl. Sci., 30 (2007), 931-960. doi: 10.1002/mma.816. [8] M. Nizette, T. Erneux, A. Gavrielides and V. Kovanis, Stability and bifurcations of periodically modulated, optically injected laser diodes, Phys. Rev. E, 63 (2001), Paper number 026212. [9] D. Peterhof and B. Sandstede, All-optical clock recovery using multisection distributed-feedback lasers, J. Nonlinear Sci., 9 (1999), 575-613. doi: 10.1007/s003329900079. [10] M. Radziunas, Numerical bifurcation analysis of the traveling wave model of multisection semiconductor lasers, Physica D, 213 (2006), 98-112. doi: 10.1016/j.physd.2005.11.003. [11] L. Recke, Forced frequency locking of rotating waves, Ukraīn. Math. J, 50 (1998), 94-101. [12] L. Recke and D. Peterhof, Abstract forced symmetry breaking and forced frequency locking of modulated waves, J. Differential Equations, 144 (1998), 233-262. [13] A. M. Samoilenko, "Elements of the Mathematical Theory of Multi-Frequency Oscillations," Mathematics and its Applications (Soviet Series), 71, Kluwer Acad. Publ. Group, Dordrecht, 1991. [14] A. M. Samoilenko and L. Recke, Conditions for synchronization of one oscillatory system, Ukrain. Math. J., 57 (2005), 1089-1119. doi: 10.1007/s11253-005-0250-3. [15] B. Sartorius, C. Bornholdt, O. Brox, H. J. Ehrke, D. Hoffmann, R. Ludwig and M. Möhrle, All-optical clock recovery module based on self-pulsating DFB laser, Electronics Letters, 34 (1998), 1664-1665. doi: 10.1049/el:19981152. [16] K. R. Schneider, Entrainment of modulation frequency: A case study, Int. J. Bifurc. Chaos Appl. Sci. Eng., 15 (2005), 3579-3588. doi: 10.1142/S0218127405014234. [17] J. Sieber, Numerical bifurcation analysis for multisection semiconductor lasers, SIAM J. Appl. Dyn. Syst., 1 (2002), 248-270. doi: 10.1137/S1111111102401746. [18] S. Wieczorek, B. Krauskopf, T. B. Simpson and D. Lenstra, The dynamical complexity of optically injected semiconductor lasers, Phys. Rep., 416 (2005), 1-128. doi: 10.1016/j.physrep.2005.06.003. [19] Y. F. Yi, Stability of integral manifold and orbital attraction of quasi-periodic motion, J. Differential Equation, 103 (1993), 278-322. [20] Y. F. Yi, A generalized integral manifold theorem, J. Differential Equation, 102 (1993), 153-187.
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