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May  2015, 35(5): 1921-1932. doi: 10.3934/dcds.2015.35.1921

Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem

Jifeng Chu 1, , Pedro J. Torres 2, and  Feng Wang 1,

1. 

Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China, China
2. 

Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada

Received  January 2014 Revised  October 2014 Published  December 2014

For the Gylden-Meshcherskii-type problem with a periodically cha-nging gravitational parameter, we prove the existence of radially periodic solutions with high angular momentum, which are Lyapunov stable in the radial direction.
Keywords: Gylden-Meshcherskii-type problem, Radial stability, twist., periodic solutions.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34D2.
Citation: Jifeng Chu, Pedro J. Torres, Feng Wang. Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1921-1932. doi: 10.3934/dcds.2015.35.1921
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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