# American Institute of Mathematical Sciences

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

## 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

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.
Citation: Jifeng Chu, Pedro J. Torres, Feng Wang. Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar [2] J. Chu and M. Li, Twist periodic solutions of second order singular differential equations,, J. Math. Anal. Appl., 355 (2009), 830. doi: 10.1016/j.jmaa.2009.02.033. Google Scholar [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. doi: 10.1016/j.jde.2007.05.007. Google Scholar [4] J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions,, Discrete Contin. Dyn. Syst., 21 (2008), 1071. doi: 10.3934/dcds.2008.21.1071. Google Scholar [5] E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions,, J. Dynam. Differential Equations, 6 (1994), 631. doi: 10.1007/BF02218851. Google Scholar [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, (1996), 1. Google Scholar [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. doi: 10.1006/jdeq.1993.1050. Google Scholar [8] A. Deprit, The secular accelerations in Gylden's problem,, Celestial Mechanics, 31 (1983), 1. doi: 10.1007/BF01272557. Google Scholar [9] A. Fonda and R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach,, J. Differential Equations, 244 (2008), 3235. doi: 10.1016/j.jde.2007.11.005. Google Scholar [10] A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth,, Nonlinear Anal., 74 (2011), 2485. doi: 10.1016/j.na.2010.12.004. Google Scholar [11] A. Fonda and R. Toader, Periodic orbits of radially symmetric systems with a singularity: The repulsive case,, Adv. Nonlinear Stud., 11 (2011), 853. Google Scholar [12] A. Fonda and R. Toader, Periodic solutions of radially symmetric perturbations of Newtonian systems,, Proc. Amer. Math. Soc., 140 (2012), 1331. doi: 10.1090/S0002-9939-2011-10992-4. Google Scholar [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. doi: 10.1007/s10231-010-0178-6. Google Scholar [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. doi: 10.3934/dcds.2011.29.169. Google Scholar [15] D. Jiang, J. Chu and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations,, J. Differential Equations, 211 (2005), 282. doi: 10.1016/j.jde.2004.10.031. Google Scholar [16] A. C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities,, Proc. Amer. Math. Soc., 99 (1987), 109. doi: 10.1090/S0002-9939-1987-0866438-7. Google Scholar [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. doi: 10.1137/S003614100241037X. Google Scholar [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. doi: 10.1007/s10884-005-2937-4. Google Scholar [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. doi: 10.1002/mma.2594. Google Scholar [20] R. Ortega, Periodic solution of a Newtonian equation: Stability by the third approximation,, J. Differential Equations, 128 (1996), 491. doi: 10.1006/jdeq.1996.0103. Google Scholar [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. doi: 10.1002/asna.200510537. Google Scholar [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. doi: 10.1006/jdeq.2000.3995. Google Scholar [23] J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems,, Applied Math. Sci., 59 (1985). doi: 10.1007/978-1-4757-4575-7. Google Scholar [24] W. C. Saslaw, Motion around a source whose luminosity changes,, The Astrophysical Journal, 226 (1978), 240. doi: 10.1086/156603. Google Scholar [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. doi: 10.1002/asna.2113130408. Google Scholar [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. Google Scholar [27] D. Selaru, V. Mioc and C. Cucu-Dumitrescu, The periodic Gyldén-type problem in Astrophysics,, AIP Conf. Proc., 895 (2007), 163. Google Scholar [28] C. Siegel and J. Moser, Lectures on Celestial Mechanics,, Springer-Verlag, (1971). Google Scholar [29] S. Solimini, On forced dynamical systems with a singularity of repulsive type,, Nonlinear Anal., 14 (1990), 489. doi: 10.1016/0362-546X(90)90037-H. Google Scholar [30] P. J. Torres, Twist solutions of a Hill's equations with singular term,, Adv. Nonlinear Stud., 2 (2002), 279. Google Scholar [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. doi: 10.1016/S0022-0396(02)00152-3. Google Scholar [32] P. J. Torres, Weak singularities may help periodic solutions to exist,, J. Differential Equations, 232 (2007), 277. doi: 10.1016/j.jde.2006.08.006. Google Scholar [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. doi: 10.1017/S0308210505000739. Google Scholar [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. doi: 10.1002/mana.200310033. Google Scholar [35] P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations,, Nonlinear Anal., 56 (2004), 591. doi: 10.1016/j.na.2003.10.005. Google Scholar [36] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Universitext, (1996). doi: 10.1007/978-3-642-61453-8. Google Scholar [37] P. Yan and M. Zhang, Higher order nonresonance for differential equations with singularities,, Math. Methods Appl. Sci., 26 (2003), 1067. doi: 10.1002/mma.413. Google Scholar [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. doi: 10.1112/S0024610702003939. Google Scholar [39] M. Zhang, Periodic solutions of equations of Ermakov-Pinney type,, Adv. Nonlinear Stud., 6 (2006), 57. Google Scholar

show all references

##### References:
 [1] A. A. Bekov, Periodic solutions of the Gylden-Meshcherskii problem,, Astron. Rep., 37 (1993), 651. Google Scholar [2] J. Chu and M. Li, Twist periodic solutions of second order singular differential equations,, J. Math. Anal. Appl., 355 (2009), 830. doi: 10.1016/j.jmaa.2009.02.033. Google Scholar [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. doi: 10.1016/j.jde.2007.05.007. Google Scholar [4] J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions,, Discrete Contin. Dyn. Syst., 21 (2008), 1071. doi: 10.3934/dcds.2008.21.1071. Google Scholar [5] E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions,, J. Dynam. Differential Equations, 6 (1994), 631. doi: 10.1007/BF02218851. Google Scholar [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, (1996), 1. Google Scholar [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. doi: 10.1006/jdeq.1993.1050. Google Scholar [8] A. Deprit, The secular accelerations in Gylden's problem,, Celestial Mechanics, 31 (1983), 1. doi: 10.1007/BF01272557. Google Scholar [9] A. Fonda and R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach,, J. Differential Equations, 244 (2008), 3235. doi: 10.1016/j.jde.2007.11.005. Google Scholar [10] A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth,, Nonlinear Anal., 74 (2011), 2485. doi: 10.1016/j.na.2010.12.004. Google Scholar [11] A. Fonda and R. Toader, Periodic orbits of radially symmetric systems with a singularity: The repulsive case,, Adv. Nonlinear Stud., 11 (2011), 853. Google Scholar [12] A. Fonda and R. Toader, Periodic solutions of radially symmetric perturbations of Newtonian systems,, Proc. Amer. Math. Soc., 140 (2012), 1331. doi: 10.1090/S0002-9939-2011-10992-4. Google Scholar [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. doi: 10.1007/s10231-010-0178-6. Google Scholar [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. doi: 10.3934/dcds.2011.29.169. Google Scholar [15] D. Jiang, J. Chu and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations,, J. Differential Equations, 211 (2005), 282. doi: 10.1016/j.jde.2004.10.031. Google Scholar [16] A. C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities,, Proc. Amer. Math. Soc., 99 (1987), 109. doi: 10.1090/S0002-9939-1987-0866438-7. Google Scholar [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. doi: 10.1137/S003614100241037X. Google Scholar [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. doi: 10.1007/s10884-005-2937-4. Google Scholar [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. doi: 10.1002/mma.2594. Google Scholar [20] R. Ortega, Periodic solution of a Newtonian equation: Stability by the third approximation,, J. Differential Equations, 128 (1996), 491. doi: 10.1006/jdeq.1996.0103. Google Scholar [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. doi: 10.1002/asna.200510537. Google Scholar [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. doi: 10.1006/jdeq.2000.3995. Google Scholar [23] J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems,, Applied Math. Sci., 59 (1985). doi: 10.1007/978-1-4757-4575-7. Google Scholar [24] W. C. Saslaw, Motion around a source whose luminosity changes,, The Astrophysical Journal, 226 (1978), 240. doi: 10.1086/156603. Google Scholar [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. doi: 10.1002/asna.2113130408. Google Scholar [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. Google Scholar [27] D. Selaru, V. Mioc and C. Cucu-Dumitrescu, The periodic Gyldén-type problem in Astrophysics,, AIP Conf. Proc., 895 (2007), 163. Google Scholar [28] C. Siegel and J. Moser, Lectures on Celestial Mechanics,, Springer-Verlag, (1971). Google Scholar [29] S. Solimini, On forced dynamical systems with a singularity of repulsive type,, Nonlinear Anal., 14 (1990), 489. doi: 10.1016/0362-546X(90)90037-H. Google Scholar [30] P. J. Torres, Twist solutions of a Hill's equations with singular term,, Adv. Nonlinear Stud., 2 (2002), 279. Google Scholar [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. doi: 10.1016/S0022-0396(02)00152-3. Google Scholar [32] P. J. Torres, Weak singularities may help periodic solutions to exist,, J. Differential Equations, 232 (2007), 277. doi: 10.1016/j.jde.2006.08.006. Google Scholar [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. doi: 10.1017/S0308210505000739. Google Scholar [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. doi: 10.1002/mana.200310033. Google Scholar [35] P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations,, Nonlinear Anal., 56 (2004), 591. doi: 10.1016/j.na.2003.10.005. Google Scholar [36] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Universitext, (1996). doi: 10.1007/978-3-642-61453-8. Google Scholar [37] P. Yan and M. Zhang, Higher order nonresonance for differential equations with singularities,, Math. Methods Appl. Sci., 26 (2003), 1067. doi: 10.1002/mma.413. Google Scholar [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. doi: 10.1112/S0024610702003939. Google Scholar [39] M. Zhang, Periodic solutions of equations of Ermakov-Pinney type,, Adv. Nonlinear Stud., 6 (2006), 57. Google Scholar
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