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Lyapunov stability of singular planarsystems related to dispersion-managedsolitons in optical fiber. (English) Zbl 1511.34065

In this paper, the authors study the Lyapunov stability of singular planar systems related to dispersion-managed solitons in optical fiber. Using the method of the third order approximation, the upper-lower solutions method and the averaging method, they prove the existence and Lyapunov stability of the periodic solutions of singular planar systems. Morover, they establish a formula of the first twist coefficient and a stability criterion of a nonlinear differential equation.

MSC:

34D20 Stability of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
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[1] Anderson, D., Variational approach to nonlinear pulse propagation in optical fibers, Phys. Rev. A, 27, 3135-3145 (1983) · doi:10.1103/PhysRevA.27.3135
[2] Carretero-González, R.; Frantzeskakis, D.; Kevrekidis, P., Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques, Nonlinearity, 21, 139-202 (2008) · Zbl 1216.82023 · doi:10.1088/0951-7715/21/7/R01
[3] Cheng, Z.; Cui, X., Positive periodic solution for generalized Basener-Ross model, Discrete Contin. Dyn. Syst. Ser. B, 25, 4361-4382 (2020) · Zbl 1458.34084
[4] Chu, J.; Ding, J.; Jiang, Y., Lyapunov stability of elliptic periodic solutions of nonlinear damped equations, J. Math. Anal. Appl., 396, 294-301 (2012) · Zbl 1264.34112 · doi:10.1016/j.jmaa.2012.06.024
[5] Chu, J.; Lei, J.; Zhang, M., The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, J. Differ. Equ., 247, 530-542 (2009) · Zbl 1175.34053 · doi:10.1016/j.jde.2008.11.013
[6] Chu, J.; Xia, T., Lyapunov stability for the linear and nonlinear damped oscillator, Abstr. Appl. Anal., 286040, 1-12 (2010) · Zbl 1204.34073
[7] Chu, J.; Torres, PJ; Wang, F., Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem, Discrete Contin. Dyn. Syst., 35, 1921-1932 (2015) · Zbl 1323.34055 · doi:10.3934/dcds.2015.35.1921
[8] Chu, J.; Torres, PJ; Wang, F., Twist periodic solutions for differential equations with a combined attractive-repulsive singularity, J. Math. Anal. Appl., 437, 1070-1083 (2016) · Zbl 1347.34065 · doi:10.1016/j.jmaa.2016.01.057
[9] Chu, J.; Liang, Z.; Torres, PJ; Zhou, Z., Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass, Discrete Contin. Dyn. Syst. Ser. B, 22, 2669-2685 (2017) · Zbl 1425.70040
[10] Chu, J.; Zhang, M., Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst., 21, 1071-1094 (2008) · Zbl 1161.37041 · doi:10.3934/dcds.2008.21.1071
[11] Dancer, EN; Ortega, R., The index of Lyapunov stable fixed points in two dimensions, J. Dyn. Differ. Equ., 6, 631-637 (1994) · Zbl 0811.34018 · doi:10.1007/BF02218851
[12] De Coster, C., Habets, P.: Two-point boundary value problems: lower and upper solutions. In: Mathematics in Science and Engineering, vol. 205. Elsevier B. V., Amsterdam (2006) · Zbl 1330.34009
[13] Fonda, A.; Sfecci, A., Multiple periodic solutions of Hamiltonian systems confined in a box, Discrete Contin. Dyn. Syst., 37, 1425-1436 (2017) · Zbl 1373.37147 · doi:10.3934/dcds.2017059
[14] Hakl, R.; Torres, PJ, A combined variational-topological approach for dispersion-managed solitons in optical fibers, Z. Angew. Math. Phys., 62, 245-266 (2010) · Zbl 1232.35153 · doi:10.1007/s00033-010-0084-1
[15] Hale, JK, Ordinary Differential Equations (1980), New York: Robert E. Krieger Publishing Co., New York · Zbl 0433.34003
[16] Hasegawa, A.; Kodama, Y., Solitons in Optical Communications (1995), Oxford: Oxford University Press, Oxford · Zbl 0840.35092
[17] Kunze, M., Periodic solutions of a singular Lagrangian system related to dispersion-managed fiber communication devices, Nonlinear Dyn. Syst. Theory, 1, 159-167 (2001) · Zbl 1041.78004
[18] Liu, Q.; Qian, D., Nonlinear dynamics of differential equations with attractive-repulsive singularities and small time-dependent coefficients, Math. Methods Appl. Sci., 36, 227-233 (2013) · Zbl 1264.34082 · doi:10.1002/mma.2594
[19] Liu, Q.; Liu, W.; Torres, PJ; Huang, W., Periodic dynamics of a derivative nonlinear Schrödinger equation with variable coefficients, Appl. Anal., 99, 407-427 (2020) · Zbl 1428.35527 · doi:10.1080/00036811.2018.1498971
[20] Liu, Q.; Torres, PJ; Xing, M., Modulated amplitude waves with non-trivial phase of multi-component Bose-Einstein condensates in optical lattices, IMA J. Appl. Math., 84, 145-170 (2019) · Zbl 1484.82025 · doi:10.1093/imamat/hxy053
[21] Liang, Z., Radially stable periodic solutions for radially symmetric Keplerian-like systems, J. Dyn. Control Syst., 23, 363-373 (2017) · Zbl 1373.34065 · doi:10.1007/s10883-016-9327-6
[22] Liang, Z.; Liao, F., Radial stability of periodic orbits of damped Keplerian-like systems, Nonlinear Anal. Real World Appl., 54, 103093 (2020) · Zbl 1436.34038 · doi:10.1016/j.nonrwa.2020.103093
[23] Liang, Z.; Yang, Y., Existence and stability of periodic oscillations of a smooth and discontinuous oscillator, Phys. A, 555, 124511 (2020) · Zbl 1496.34092 · doi:10.1016/j.physa.2020.124511
[24] Montesinos, G.; Perez-Garcia, V.; Torres, PJ, Stabilization of solitons of the multidimensional nonlinear Schrödinger equation: matter-wave breathers, Phys. D, 191, 193-210 (2004) · Zbl 1057.35065 · doi:10.1016/j.physd.2003.12.001
[25] Ortega, R., Periodic solution of a Newtonian equation: stability by the third approximation, J. Differ. Equ., 128, 491-518 (1996) · Zbl 0855.34058 · doi:10.1006/jdeq.1996.0103
[26] Rützel, S.; Lee, S.; Raman, A., Nonlinear dynamics of atomic-force-microscope probes driven in Lennard-Jones potentials, Proc. Roy. Soc. Lond. Ser. A, 459, 1925-1948 (2003) · Zbl 1049.81065 · doi:10.1098/rspa.2002.1115
[27] Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems, vol. 59. Springer, New York (1985) · Zbl 0586.34040
[28] Sfecci, A., Positive periodic solutions for planar differential systems with repulsive singularities on the axes, J. Math. Anal. Appl., 415, 110-120 (2014) · Zbl 1308.34053 · doi:10.1016/j.jmaa.2013.12.068
[29] Siegel, C.; Moser, J., Lectures on Celestial Mechanics (1971), Berlin: Springer, Berlin · Zbl 0312.70017 · doi:10.1007/978-3-642-87284-6
[30] Turitsyn, SK; Mezentsev, VK; Shapiro, EG, Dispersion-managed solitons and optimization of the dispersion management, Opt. Fiber Technol., 4, 384-452 (1998) · doi:10.1006/ofte.1998.0271
[31] Turitsyn, SK; Shapiro, EG, Variational approach to the design of optical communication systems with dispersion management, Opt. Fiber Technol., 4, 151-188 (1998) · doi:10.1006/ofte.1998.0246
[32] Torres, PJ; Zhang, M., A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nachr., 251, 101-107 (2003) · Zbl 1024.34030 · doi:10.1002/mana.200310033
[33] Torres, PJ; Zhang, M., Twist periodic solutions of repulsive singular equations, Nonlinear Anal., 56, 591-599 (2004) · Zbl 1058.34052 · doi:10.1016/j.na.2003.10.005
[34] Torres, PJ, Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity, Proc. Roy. Soc. Edinb. Sect. A, 137, 195-201 (2007) · Zbl 1190.34050 · doi:10.1017/S0308210505000739
[35] Torres, P.J.: Mathematical Models with Singularities-Zoo of Singular Creatures. Atlantis Press, Paris (2015) · Zbl 1305.00097
[36] Wang, F.; Cid, JÁ; Zima, M., Lyapunov stability for regular equations and applications to the Liebau phenomenon, Discrete Contin. Dyn. Syst., 38, 4657-4674 (2018) · Zbl 1477.34067 · doi:10.3934/dcds.2018204
[37] Wang, F.; Cid, JÁ; Li, S.; Zima, M., Lyapunov stability of periodic solutions of Brillouin type equations, Appl. Math. Lett., 101, 106057 (2020) · Zbl 1436.34039 · doi:10.1016/j.aml.2019.106057
[38] Zhang, M., The best bound on the rotations in the stability of periodic solutions of a Newtonian equation, J. Lond. Math. Soc., 67, 137-148 (2003) · Zbl 1050.34075 · doi:10.1112/S0024610702003939
[39] Zhang, M.; Chu, J.; Li, X., Lyapunov stability of periodic solutions of the quadratic Newtonian equation, Math. Nachr., 282, 1354-1366 (2009) · Zbl 1192.37080 · doi:10.1002/mana.200610799
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