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Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling
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 |
  Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.
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. |
show all references
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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