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]

Astron. Rep., 37 (1993), 651-654. Google Scholar

[2]

J. Math. Anal. Appl., 355 (2009), 830-838. doi: 10.1016/j.jmaa.2009.02.033.  Google Scholar

[3]

J. Differential Equations, 239 (2007), 196-212. doi: 10.1016/j.jde.2007.05.007.  Google Scholar

[4]

Discrete Contin. Dyn. Syst., 21 (2008), 1071-1094. doi: 10.3934/dcds.2008.21.1071.  Google Scholar

[5]

J. Dynam. Differential Equations, 6 (1994), 631-637. doi: 10.1007/BF02218851.  Google Scholar

[6]

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]

J. Differential Equations, 103 (1993), 260-277. doi: 10.1006/jdeq.1993.1050.  Google Scholar

[8]

Celestial Mechanics, 31 (1983), 1-22. doi: 10.1007/BF01272557.  Google Scholar

[9]

J. Differential Equations, 244 (2008), 3235-3264. doi: 10.1016/j.jde.2007.11.005.  Google Scholar

[10]

Nonlinear Anal., 74 (2011), 2485-2496. doi: 10.1016/j.na.2010.12.004.  Google Scholar

[11]

Adv. Nonlinear Stud., 11 (2011), 853-874.  Google Scholar

[12]

Proc. Amer. Math. Soc., 140 (2012), 1331-1341. doi: 10.1090/S0002-9939-2011-10992-4.  Google Scholar

[13]

Ann. Mat. Pura Appl., 191 (2012), 181-204. doi: 10.1007/s10231-010-0178-6.  Google Scholar

[14]

Discrete Contin. Dyn. Syst., 29 (2011), 169-192. doi: 10.3934/dcds.2011.29.169.  Google Scholar

[15]

J. Differential Equations, 211 (2005), 282-302. doi: 10.1016/j.jde.2004.10.031.  Google Scholar

[16]

Proc. Amer. Math. Soc., 99 (1987), 109-114. doi: 10.1090/S0002-9939-1987-0866438-7.  Google Scholar

[17]

SIAM J. Math. Anal., 35 (2003), 844-867. doi: 10.1137/S003614100241037X.  Google Scholar

[18]

J. Dynam. Differential Equations, 17 (2005), 21-50. doi: 10.1007/s10884-005-2937-4.  Google Scholar

[19]

Math. Methods Appl. Sci., 36 (2013), 227-233. doi: 10.1002/mma.2594.  Google Scholar

[20]

J. Differential Equations, 128 (1996), 491-518. doi: 10.1006/jdeq.1996.0103.  Google Scholar

[21]

Astron. Nachr., 327 (2006), 304-308. doi: 10.1002/asna.200510537.  Google Scholar

[22]

J. Differential Equations, 176 (2001), 445-469. doi: 10.1006/jdeq.2000.3995.  Google Scholar

[23]

Applied Math. Sci., 59, Springer, New York, 1985. doi: 10.1007/978-1-4757-4575-7.  Google Scholar

[24]

The Astrophysical Journal, 226 (1978), 240-252. doi: 10.1086/156603.  Google Scholar

[25]

Astron. Nachr., 313 (1992), 257-263. doi: 10.1002/asna.2113130408.  Google Scholar

[26]

C. R. Acad. Sci. Paris, 325 (1997), 487-490. Google Scholar

[27]

AIP Conf. Proc., 895 (2007), 163-170. Google Scholar

[28]

Springer-Verlag, Berlin, 1971.  Google Scholar

[29]

Nonlinear Anal., 14 (1990), 489-500. doi: 10.1016/0362-546X(90)90037-H.  Google Scholar

[30]

Adv. Nonlinear Stud., 2 (2002), 279-287.  Google Scholar

[31]

J. Differential Equations, 190 (2003), 643-662. doi: 10.1016/S0022-0396(02)00152-3.  Google Scholar

[32]

J. Differential Equations, 232 (2007), 277-284. doi: 10.1016/j.jde.2006.08.006.  Google Scholar

[33]

Proc. Royal Soc. Edinburgh Sect. A., 137 (2007), 195-201. doi: 10.1017/S0308210505000739.  Google Scholar

[34]

Math. Nachr., 251 (2003), 101-107. doi: 10.1002/mana.200310033.  Google Scholar

[35]

Nonlinear Anal., 56 (2004), 591-599. doi: 10.1016/j.na.2003.10.005.  Google Scholar

[36]

Universitext, Springer, 1996. doi: 10.1007/978-3-642-61453-8.  Google Scholar

[37]

Math. Methods Appl. Sci., 26 (2003), 1067-1074. doi: 10.1002/mma.413.  Google Scholar

[38]

J. London Math. Soc., 67 (2003), 137-148. doi: 10.1112/S0024610702003939.  Google Scholar

[39]

Adv. Nonlinear Stud., 6 (2006), 57-67.  Google Scholar

show all references

References:
[1]

Astron. Rep., 37 (1993), 651-654. Google Scholar

[2]

J. Math. Anal. Appl., 355 (2009), 830-838. doi: 10.1016/j.jmaa.2009.02.033.  Google Scholar

[3]

J. Differential Equations, 239 (2007), 196-212. doi: 10.1016/j.jde.2007.05.007.  Google Scholar

[4]

Discrete Contin. Dyn. Syst., 21 (2008), 1071-1094. doi: 10.3934/dcds.2008.21.1071.  Google Scholar

[5]

J. Dynam. Differential Equations, 6 (1994), 631-637. doi: 10.1007/BF02218851.  Google Scholar

[6]

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]

J. Differential Equations, 103 (1993), 260-277. doi: 10.1006/jdeq.1993.1050.  Google Scholar

[8]

Celestial Mechanics, 31 (1983), 1-22. doi: 10.1007/BF01272557.  Google Scholar

[9]

J. Differential Equations, 244 (2008), 3235-3264. doi: 10.1016/j.jde.2007.11.005.  Google Scholar

[10]

Nonlinear Anal., 74 (2011), 2485-2496. doi: 10.1016/j.na.2010.12.004.  Google Scholar

[11]

Adv. Nonlinear Stud., 11 (2011), 853-874.  Google Scholar

[12]

Proc. Amer. Math. Soc., 140 (2012), 1331-1341. doi: 10.1090/S0002-9939-2011-10992-4.  Google Scholar

[13]

Ann. Mat. Pura Appl., 191 (2012), 181-204. doi: 10.1007/s10231-010-0178-6.  Google Scholar

[14]

Discrete Contin. Dyn. Syst., 29 (2011), 169-192. doi: 10.3934/dcds.2011.29.169.  Google Scholar

[15]

J. Differential Equations, 211 (2005), 282-302. doi: 10.1016/j.jde.2004.10.031.  Google Scholar

[16]

Proc. Amer. Math. Soc., 99 (1987), 109-114. doi: 10.1090/S0002-9939-1987-0866438-7.  Google Scholar

[17]

SIAM J. Math. Anal., 35 (2003), 844-867. doi: 10.1137/S003614100241037X.  Google Scholar

[18]

J. Dynam. Differential Equations, 17 (2005), 21-50. doi: 10.1007/s10884-005-2937-4.  Google Scholar

[19]

Math. Methods Appl. Sci., 36 (2013), 227-233. doi: 10.1002/mma.2594.  Google Scholar

[20]

J. Differential Equations, 128 (1996), 491-518. doi: 10.1006/jdeq.1996.0103.  Google Scholar

[21]

Astron. Nachr., 327 (2006), 304-308. doi: 10.1002/asna.200510537.  Google Scholar

[22]

J. Differential Equations, 176 (2001), 445-469. doi: 10.1006/jdeq.2000.3995.  Google Scholar

[23]

Applied Math. Sci., 59, Springer, New York, 1985. doi: 10.1007/978-1-4757-4575-7.  Google Scholar

[24]

The Astrophysical Journal, 226 (1978), 240-252. doi: 10.1086/156603.  Google Scholar

[25]

Astron. Nachr., 313 (1992), 257-263. doi: 10.1002/asna.2113130408.  Google Scholar

[26]

C. R. Acad. Sci. Paris, 325 (1997), 487-490. Google Scholar

[27]

AIP Conf. Proc., 895 (2007), 163-170. Google Scholar

[28]

Springer-Verlag, Berlin, 1971.  Google Scholar

[29]

Nonlinear Anal., 14 (1990), 489-500. doi: 10.1016/0362-546X(90)90037-H.  Google Scholar

[30]

Adv. Nonlinear Stud., 2 (2002), 279-287.  Google Scholar

[31]

J. Differential Equations, 190 (2003), 643-662. doi: 10.1016/S0022-0396(02)00152-3.  Google Scholar

[32]

J. Differential Equations, 232 (2007), 277-284. doi: 10.1016/j.jde.2006.08.006.  Google Scholar

[33]

Proc. Royal Soc. Edinburgh Sect. A., 137 (2007), 195-201. doi: 10.1017/S0308210505000739.  Google Scholar

[34]

Math. Nachr., 251 (2003), 101-107. doi: 10.1002/mana.200310033.  Google Scholar

[35]

Nonlinear Anal., 56 (2004), 591-599. doi: 10.1016/j.na.2003.10.005.  Google Scholar

[36]

Universitext, Springer, 1996. doi: 10.1007/978-3-642-61453-8.  Google Scholar

[37]

Math. Methods Appl. Sci., 26 (2003), 1067-1074. doi: 10.1002/mma.413.  Google Scholar

[38]

J. London Math. Soc., 67 (2003), 137-148. doi: 10.1112/S0024610702003939.  Google Scholar

[39]

Adv. Nonlinear Stud., 6 (2006), 57-67.  Google Scholar

[1]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3817-3836. doi: 10.3934/dcds.2021018

[2]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021058

[3]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[4]

Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021017

[5]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[6]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475

[7]

Lipeng Duan, Jun Yang. On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021056

[8]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[9]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055

[10]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104

[11]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[12]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[13]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027

[14]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

[15]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[16]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[17]

Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211

[18]

Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021060

[19]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241

[20]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]