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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 & 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

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