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Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem
1. | Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China, China |
2. | Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada |
References:
[1] |
A. A. Bekov, Periodic solutions of the Gylden-Meshcherskii problem, Astron. Rep., 37 (1993), 651-654. |
[2] |
J. Chu and M. Li, Twist periodic solutions of second order singular differential equations, J. Math. Anal. Appl., 355 (2009), 830-838.
doi: 10.1016/j.jmaa.2009.02.033. |
[3] |
J. Chu, P. J. Torres and M. Zhang, Periodic solutions of second order non-autonomous singular dynamical systems, J. Differential Equations, 239 (2007), 196-212.
doi: 10.1016/j.jde.2007.05.007. |
[4] |
J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst., 21 (2008), 1071-1094.
doi: 10.3934/dcds.2008.21.1071. |
[5] |
E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations, 6 (1994), 631-637.
doi: 10.1007/BF02218851. |
[6] |
C. De Coster and P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, in Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations, edited by F. Zanolin, CISM-CICMS 371 (Springer-Verlag, New York, 1996), pp. 1-78. |
[7] |
M. A. del Pino and R. F. Manásevich, Infinitely many $T$-periodic solutions for a problem arising in nonlinear elasticity, J. Differential Equations, 103 (1993), 260-277.
doi: 10.1006/jdeq.1993.1050. |
[8] |
A. Deprit, The secular accelerations in Gylden's problem, Celestial Mechanics, 31 (1983), 1-22.
doi: 10.1007/BF01272557. |
[9] |
A. Fonda and R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach, J. Differential Equations, 244 (2008), 3235-3264.
doi: 10.1016/j.jde.2007.11.005. |
[10] |
A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth, Nonlinear Anal., 74 (2011), 2485-2496.
doi: 10.1016/j.na.2010.12.004. |
[11] |
A. Fonda and R. Toader, Periodic orbits of radially symmetric systems with a singularity: The repulsive case, Adv. Nonlinear Stud., 11 (2011), 853-874. |
[12] |
A. Fonda and R. Toader, Periodic solutions of radially symmetric perturbations of Newtonian systems, Proc. Amer. Math. Soc., 140 (2012), 1331-1341.
doi: 10.1090/S0002-9939-2011-10992-4. |
[13] |
A. Fonda, R. Toader and F. Zanolin, Periodic solutions of singular radially symmetric systems with superlinear growth, Ann. Mat. Pura Appl., 191 (2012), 181-204.
doi: 10.1007/s10231-010-0178-6. |
[14] |
A. Fonda and A. J. Ureña, Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force, Discrete Contin. Dyn. Syst., 29 (2011), 169-192.
doi: 10.3934/dcds.2011.29.169. |
[15] |
D. Jiang, J. Chu and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. Differential Equations, 211 (2005), 282-302.
doi: 10.1016/j.jde.2004.10.031. |
[16] |
A. C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc., 99 (1987), 109-114.
doi: 10.1090/S0002-9939-1987-0866438-7. |
[17] |
J. Lei, X. Li, P. Yan and M. Zhang, Twist character of the least amplitude periodic solution of the forced pendulum, SIAM J. Math. Anal., 35 (2003), 844-867.
doi: 10.1137/S003614100241037X. |
[18] |
J. Lei, P. J. Torres and M. Zhang, Twist character of the fourth order resonant periodic solution, J. Dynam. Differential Equations, 17 (2005), 21-50.
doi: 10.1007/s10884-005-2937-4. |
[19] |
Q. Liu and D. Qian, Nonlinear dynamics of differential equations with attractive-repulsive singularities and small time-dependent coefficients, Math. Methods Appl. Sci., 36 (2013), 227-233.
doi: 10.1002/mma.2594. |
[20] |
R. Ortega, Periodic solution of a Newtonian equation: Stability by the third approximation, J. Differential Equations, 128 (1996), 491-518.
doi: 10.1006/jdeq.1996.0103. |
[21] |
A. Pal, D. Selaru, V. Mioc and C. Cucu-Dumitrescu, The Gylden-type problem revisited: More refined analytical solutions, Astron. Nachr., 327 (2006), 304-308.
doi: 10.1002/asna.200510537. |
[22] |
I. Rachunková, M. Tvrdý and I. Vrkoč, Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. Differential Equations, 176 (2001), 445-469.
doi: 10.1006/jdeq.2000.3995. |
[23] |
J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Applied Math. Sci., 59, Springer, New York, 1985.
doi: 10.1007/978-1-4757-4575-7. |
[24] |
W. C. Saslaw, Motion around a source whose luminosity changes, The Astrophysical Journal, 226 (1978), 240-252.
doi: 10.1086/156603. |
[25] |
D. Selaru, C. Cucu-Dumitrescu and V. Mioc, On a two-body problem with periodically changing equivalent gravitational parameter, Astron. Nachr., 313 (1992), 257-263.
doi: 10.1002/asna.2113130408. |
[26] |
D. Selaru and V. Mioc, Le probleme de Gyldén du point de vue de la théorie KAM, C. R. Acad. Sci. Paris, 325 (1997), 487-490. |
[27] |
D. Selaru, V. Mioc and C. Cucu-Dumitrescu, The periodic Gyldén-type problem in Astrophysics, AIP Conf. Proc., 895 (2007), 163-170. |
[28] |
C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971. |
[29] |
S. Solimini, On forced dynamical systems with a singularity of repulsive type, Nonlinear Anal., 14 (1990), 489-500.
doi: 10.1016/0362-546X(90)90037-H. |
[30] |
P. J. Torres, Twist solutions of a Hill's equations with singular term, Adv. Nonlinear Stud., 2 (2002), 279-287. |
[31] |
P. J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190 (2003), 643-662.
doi: 10.1016/S0022-0396(02)00152-3. |
[32] |
P. J. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations, 232 (2007), 277-284.
doi: 10.1016/j.jde.2006.08.006. |
[33] |
P. J. Torres, Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity, Proc. Royal Soc. Edinburgh Sect. A., 137 (2007), 195-201.
doi: 10.1017/S0308210505000739. |
[34] |
P. J. Torres and M. Zhang, A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nachr., 251 (2003), 101-107.
doi: 10.1002/mana.200310033. |
[35] |
P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations, Nonlinear Anal., 56 (2004), 591-599.
doi: 10.1016/j.na.2003.10.005. |
[36] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, 1996.
doi: 10.1007/978-3-642-61453-8. |
[37] |
P. Yan and M. Zhang, Higher order nonresonance for differential equations with singularities, Math. Methods Appl. Sci., 26 (2003), 1067-1074.
doi: 10.1002/mma.413. |
[38] |
M. Zhang, The best bound on the rotations in the stability of periodic solutions of a Newtonian equation, J. London Math. Soc., 67 (2003), 137-148.
doi: 10.1112/S0024610702003939. |
[39] |
M. Zhang, Periodic solutions of equations of Ermakov-Pinney type, Adv. Nonlinear Stud., 6 (2006), 57-67. |
show all references
References:
[1] |
A. A. Bekov, Periodic solutions of the Gylden-Meshcherskii problem, Astron. Rep., 37 (1993), 651-654. |
[2] |
J. Chu and M. Li, Twist periodic solutions of second order singular differential equations, J. Math. Anal. Appl., 355 (2009), 830-838.
doi: 10.1016/j.jmaa.2009.02.033. |
[3] |
J. Chu, P. J. Torres and M. Zhang, Periodic solutions of second order non-autonomous singular dynamical systems, J. Differential Equations, 239 (2007), 196-212.
doi: 10.1016/j.jde.2007.05.007. |
[4] |
J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst., 21 (2008), 1071-1094.
doi: 10.3934/dcds.2008.21.1071. |
[5] |
E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations, 6 (1994), 631-637.
doi: 10.1007/BF02218851. |
[6] |
C. De Coster and P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, in Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations, edited by F. Zanolin, CISM-CICMS 371 (Springer-Verlag, New York, 1996), pp. 1-78. |
[7] |
M. A. del Pino and R. F. Manásevich, Infinitely many $T$-periodic solutions for a problem arising in nonlinear elasticity, J. Differential Equations, 103 (1993), 260-277.
doi: 10.1006/jdeq.1993.1050. |
[8] |
A. Deprit, The secular accelerations in Gylden's problem, Celestial Mechanics, 31 (1983), 1-22.
doi: 10.1007/BF01272557. |
[9] |
A. Fonda and R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach, J. Differential Equations, 244 (2008), 3235-3264.
doi: 10.1016/j.jde.2007.11.005. |
[10] |
A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth, Nonlinear Anal., 74 (2011), 2485-2496.
doi: 10.1016/j.na.2010.12.004. |
[11] |
A. Fonda and R. Toader, Periodic orbits of radially symmetric systems with a singularity: The repulsive case, Adv. Nonlinear Stud., 11 (2011), 853-874. |
[12] |
A. Fonda and R. Toader, Periodic solutions of radially symmetric perturbations of Newtonian systems, Proc. Amer. Math. Soc., 140 (2012), 1331-1341.
doi: 10.1090/S0002-9939-2011-10992-4. |
[13] |
A. Fonda, R. Toader and F. Zanolin, Periodic solutions of singular radially symmetric systems with superlinear growth, Ann. Mat. Pura Appl., 191 (2012), 181-204.
doi: 10.1007/s10231-010-0178-6. |
[14] |
A. Fonda and A. J. Ureña, Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force, Discrete Contin. Dyn. Syst., 29 (2011), 169-192.
doi: 10.3934/dcds.2011.29.169. |
[15] |
D. Jiang, J. Chu and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. Differential Equations, 211 (2005), 282-302.
doi: 10.1016/j.jde.2004.10.031. |
[16] |
A. C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc., 99 (1987), 109-114.
doi: 10.1090/S0002-9939-1987-0866438-7. |
[17] |
J. Lei, X. Li, P. Yan and M. Zhang, Twist character of the least amplitude periodic solution of the forced pendulum, SIAM J. Math. Anal., 35 (2003), 844-867.
doi: 10.1137/S003614100241037X. |
[18] |
J. Lei, P. J. Torres and M. Zhang, Twist character of the fourth order resonant periodic solution, J. Dynam. Differential Equations, 17 (2005), 21-50.
doi: 10.1007/s10884-005-2937-4. |
[19] |
Q. Liu and D. Qian, Nonlinear dynamics of differential equations with attractive-repulsive singularities and small time-dependent coefficients, Math. Methods Appl. Sci., 36 (2013), 227-233.
doi: 10.1002/mma.2594. |
[20] |
R. Ortega, Periodic solution of a Newtonian equation: Stability by the third approximation, J. Differential Equations, 128 (1996), 491-518.
doi: 10.1006/jdeq.1996.0103. |
[21] |
A. Pal, D. Selaru, V. Mioc and C. Cucu-Dumitrescu, The Gylden-type problem revisited: More refined analytical solutions, Astron. Nachr., 327 (2006), 304-308.
doi: 10.1002/asna.200510537. |
[22] |
I. Rachunková, M. Tvrdý and I. Vrkoč, Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. Differential Equations, 176 (2001), 445-469.
doi: 10.1006/jdeq.2000.3995. |
[23] |
J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Applied Math. Sci., 59, Springer, New York, 1985.
doi: 10.1007/978-1-4757-4575-7. |
[24] |
W. C. Saslaw, Motion around a source whose luminosity changes, The Astrophysical Journal, 226 (1978), 240-252.
doi: 10.1086/156603. |
[25] |
D. Selaru, C. Cucu-Dumitrescu and V. Mioc, On a two-body problem with periodically changing equivalent gravitational parameter, Astron. Nachr., 313 (1992), 257-263.
doi: 10.1002/asna.2113130408. |
[26] |
D. Selaru and V. Mioc, Le probleme de Gyldén du point de vue de la théorie KAM, C. R. Acad. Sci. Paris, 325 (1997), 487-490. |
[27] |
D. Selaru, V. Mioc and C. Cucu-Dumitrescu, The periodic Gyldén-type problem in Astrophysics, AIP Conf. Proc., 895 (2007), 163-170. |
[28] |
C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971. |
[29] |
S. Solimini, On forced dynamical systems with a singularity of repulsive type, Nonlinear Anal., 14 (1990), 489-500.
doi: 10.1016/0362-546X(90)90037-H. |
[30] |
P. J. Torres, Twist solutions of a Hill's equations with singular term, Adv. Nonlinear Stud., 2 (2002), 279-287. |
[31] |
P. J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190 (2003), 643-662.
doi: 10.1016/S0022-0396(02)00152-3. |
[32] |
P. J. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations, 232 (2007), 277-284.
doi: 10.1016/j.jde.2006.08.006. |
[33] |
P. J. Torres, Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity, Proc. Royal Soc. Edinburgh Sect. A., 137 (2007), 195-201.
doi: 10.1017/S0308210505000739. |
[34] |
P. J. Torres and M. Zhang, A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nachr., 251 (2003), 101-107.
doi: 10.1002/mana.200310033. |
[35] |
P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations, Nonlinear Anal., 56 (2004), 591-599.
doi: 10.1016/j.na.2003.10.005. |
[36] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, 1996.
doi: 10.1007/978-3-642-61453-8. |
[37] |
P. Yan and M. Zhang, Higher order nonresonance for differential equations with singularities, Math. Methods Appl. Sci., 26 (2003), 1067-1074.
doi: 10.1002/mma.413. |
[38] |
M. Zhang, The best bound on the rotations in the stability of periodic solutions of a Newtonian equation, J. London Math. Soc., 67 (2003), 137-148.
doi: 10.1112/S0024610702003939. |
[39] |
M. Zhang, Periodic solutions of equations of Ermakov-Pinney type, Adv. Nonlinear Stud., 6 (2006), 57-67. |
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