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, 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. Google Scholar [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.  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), 196-212. 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), 1071-1094. 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), 631-637. 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, edited by F. Zanolin, CISM-CICMS 371 (Springer-Verlag, New York, 1996), pp. 1-78.  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), 260-277. doi: 10.1006/jdeq.1993.1050.  Google Scholar [8] A. Deprit, The secular accelerations in Gylden's problem, Celestial Mechanics, 31 (1983), 1-22. 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), 3235-3264. 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), 2485-2496. 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), 853-874.  Google Scholar [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.  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), 181-204. 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), 169-192. 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), 282-302. 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), 109-114. 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), 844-867. 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), 21-50. 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), 227-233. 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), 491-518. 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), 304-308. 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), 445-469. 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, Springer, New York, 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), 240-252. 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), 257-263. 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), 487-490. Google Scholar [27] D. Selaru, V. Mioc and C. Cucu-Dumitrescu, The periodic Gyldén-type problem in Astrophysics, AIP Conf. Proc., 895 (2007), 163-170. Google Scholar [28] C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971.  Google Scholar [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.  Google Scholar [30] P. J. Torres, Twist solutions of a Hill's equations with singular term, Adv. Nonlinear Stud., 2 (2002), 279-287.  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), 643-662. 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), 277-284. 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), 195-201. 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), 101-107. 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), 591-599. doi: 10.1016/j.na.2003.10.005.  Google Scholar [36] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, 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), 1067-1074. 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), 137-148. doi: 10.1112/S0024610702003939.  Google Scholar [39] M. Zhang, Periodic solutions of equations of Ermakov-Pinney type, Adv. Nonlinear Stud., 6 (2006), 57-67.  Google Scholar

show all references

References:
 [1] A. A. Bekov, Periodic solutions of the Gylden-Meshcherskii problem, Astron. Rep., 37 (1993), 651-654. Google Scholar [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.  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), 196-212. 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), 1071-1094. 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), 631-637. 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, edited by F. Zanolin, CISM-CICMS 371 (Springer-Verlag, New York, 1996), pp. 1-78.  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), 260-277. doi: 10.1006/jdeq.1993.1050.  Google Scholar [8] A. Deprit, The secular accelerations in Gylden's problem, Celestial Mechanics, 31 (1983), 1-22. 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), 3235-3264. 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), 2485-2496. 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), 853-874.  Google Scholar [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.  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), 181-204. 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), 169-192. 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), 282-302. 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), 109-114. 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), 844-867. 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), 21-50. 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), 227-233. 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), 491-518. 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), 304-308. 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), 445-469. 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, Springer, New York, 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), 240-252. 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), 257-263. 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), 487-490. Google Scholar [27] D. Selaru, V. Mioc and C. Cucu-Dumitrescu, The periodic Gyldén-type problem in Astrophysics, AIP Conf. Proc., 895 (2007), 163-170. Google Scholar [28] C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971.  Google Scholar [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.  Google Scholar [30] P. J. Torres, Twist solutions of a Hill's equations with singular term, Adv. Nonlinear Stud., 2 (2002), 279-287.  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), 643-662. 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), 277-284. 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), 195-201. 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), 101-107. 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), 591-599. doi: 10.1016/j.na.2003.10.005.  Google Scholar [36] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, 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), 1067-1074. 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), 137-148. doi: 10.1112/S0024610702003939.  Google Scholar [39] M. Zhang, Periodic solutions of equations of Ermakov-Pinney type, Adv. Nonlinear Stud., 6 (2006), 57-67.  Google Scholar
 [1] Daniel Núñez, Pedro J. Torres. Periodic solutions of twist type of an earth satellite equation. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 303-306. doi: 10.3934/dcds.2001.7.303 [2] Alfonso Castro, Shu-Zhi Song. Infinitely many radial solutions for a super-cubic Kirchhoff type problem in a ball. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3347-3355. doi: 10.3934/dcdss.2020127 [3] Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793 [4] M.I. Gil’. Existence and stability of periodic solutions of semilinear neutral type systems. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 809-820. doi: 10.3934/dcds.2001.7.809 [5] John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047 [6] Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084 [7] Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083 [8] Eudes. M. Barboza, Olimpio H. Miyagaki, Fábio R. Pereira, Cláudia R. Santana. Radial solutions for a class of Hénon type systems with partial interference with the spectrum. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3159-3187. doi: 10.3934/cpaa.2020137 [9] Chia-Yu Hsieh. Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2657-2681. doi: 10.3934/dcdsb.2018269 [10] Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166 [11] Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631 [12] Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences & Engineering, 2007, 4 (2) : 319-338. doi: 10.3934/mbe.2007.4.319 [13] Zongming Guo, Xuefei Bai. On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1091-1107. doi: 10.3934/cpaa.2008.7.1091 [14] Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51 [15] M. Grossi. Existence of radial solutions for an elliptic problem involving exponential nonlinearities. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 221-232. doi: 10.3934/dcds.2008.21.221 [16] Shuangjie Peng, Jing Zhou. Concentration of solutions for a Paneitz type problem. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 1055-1072. doi: 10.3934/dcds.2010.26.1055 [17] Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071 [18] Maria Carvalho, Alexander Lohse, Alexandre A. P. Rodrigues. Moduli of stability for heteroclinic cycles of periodic solutions. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6541-6564. doi: 10.3934/dcds.2019284 [19] Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823 [20] Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure & Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029

2020 Impact Factor: 1.392

Metrics

• HTML views (0)
• Cited by (13)

• on AIMS