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September  2017, 22(7): 2669-2685. doi: 10.3934/dcdsb.2017130

Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass

Jifeng Chu 1,, , Zaitao Liang 2, , Pedro J. Torres 3, and  Zhe Zhou 4,

1. 

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2. 

Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China
3. 

Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain
4. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Jifeng Chu

Received  July 2016 Revised  November 2016 Published  April 2017

Fund Project: Jifeng Chu was supported by the National Natural Science Foundation of China (Grant No. 11171090 and No. 11671118). Zaitao Liang was supported by the Fundamental Research Funds for the Central Universities (Grant No. KYZZ15−0155). Pedro Torres was partially supported by Spanish MICINN Grant with FEDER funds MTM2014-52232-P. Zhe Zhou was supported by the National Natural Science Foundation of China (Grant No. 11301512) and the Key Lab of Random Complex Structures and Data Science, Chinese Academy of Sciences (Grant No. 2008DP173182).

We study the existence and stability of periodic solutions of a differential equation that models the planar oscillations of a satellite in an elliptic orbit around its center of mass. The proof is based on a suitable version of Poincaré-Birkhoff theorem and the third order approximation method.

Keywords: Satellite equation, twist periodic solutions, unstable periodic solutions, Poincaré-Birkhoff theorem, third order approximation.
Mathematics Subject Classification: Primary:34C25.
Citation: Jifeng Chu, Zaitao Liang, Pedro J. Torres, Zhe Zhou. Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2669-2685. doi: 10.3934/dcdsb.2017130
References:
[1]

V. V. Beletskii, On the oscillations of satellite, Iskusst. Sputn. Zemli, 3 (1959), 1-3. 

[2] V. V. Beletskii, The satellite Motion About Center of Mass, Nauka, Moscow, 1965. 
[3]

V. V. Beletskii and A. N. Shlyakhtin, Resonsnce Rotations of a Satellite with Interactions Between Magnetic and Gravitational Fields Preprint No. 46, Moscow: Institute of Applied Mathematics, Academy of Sciences of the USSR, 1980.

[4]

B. S. Bardin, E. A. Chekina and A. M. Chekin, On the stability of a planar resonant rotation of a satellite in an elliptic orbit, Regul. Chaotic Dyn., 20 (2015), 63-73.  doi: 10.1134/S1560354715010050.

[5]

J. Chu and M. Zhang, Rotation number and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst., 21 (2008), 1071-1094.  doi: 10.3934/dcds.2008.21.1071.

[6]

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.

[7]

J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, J. Differential Equations, 247 (2009), 530-542.  doi: 10.1016/j.jde.2008.11.013.

[8]

J. Chu and T. Xia, The Lyapunov stability for the linear and nonlinear damped oscillator with time-periodic parameters Abstr. Appl. Anal. 2010, Art. ID 286040, 12 pp. doi: 10.1155/2010/286040.

[9]

J. Chu, N. Fan and P. J. Torres, Periodic solutions for second order singular damped differential equations, J. Math. Anal. Appl., 388 (2012), 665-675.  doi: 10.1016/j.jmaa.2011.09.061.

[10]

J. Chu, J. Ding and Y. Jiang, Lyapunov stability of elliptic periodic solutions of nonlinear damped equations, J. Math. Anal. Appl., 396 (2012), 294-301.  doi: 10.1016/j.jmaa.2012.06.024.

[11]

J. Chu, P. J. Torres and F. Wang, Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem, Discrete Contin. Dyn. Syst., 35 (2015), 1921-1932.  doi: 10.3934/dcds.2015.35.1921.

[12]

D. D. Hai, Note on a differential equation describing the periodic motion of a satellite in its elliptic orbits, Nonlinear Anal., 12 (1980), 1337-1338.  doi: 10.1016/0362-546X(88)90081-8.

[13]

D. D. Hai, Multiple solutions for a nonlinear second order differential equation, Ann. Polon. Math., 52 (1990), 161-164. 

[14]

A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane, Adv. Nonlinear Stud., 12 (2012), 395-408.  doi: 10.1515/ans-2012-0210.

[15]

J. Franks, Generalization of Poincaré-Birkhoff theorem, Ann. of Math., 128 (1988), 139-151.  doi: 10.2307/1971464.

[16]

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.

[17]

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.

[18]

A. P. Markeev, B. S. Bardin and A. Planar, Rotational motion of a satellite in an elliptic orbit, Cosmic Res., 32 (1994), 583-589. 

[19]

S. Maró, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal., 42 (2013), 51-75. 

[20]

D. Núñez, The method of lower and upper solutions and the stability of periodic oscillations, Nonlinear Anal., 51 (2002), 1207-1222.  doi: 10.1016/S0362-546X(01)00888-4.

[21]

D. Nuñez and P. J. Torres, Periodic solutions of twist type of an earth satellite equation, Discrete Contin. Dyn. Syst., 7 (2001), 303-306.  doi: 10.3934/dcds.2001.7.303.

[22]

D. Nuñez and P. J. Torres, Stable odd solutions of some periodic equations modeling satellite motion, J. Math. Anal. Appl., 279 (2003), 700-709.  doi: 10.1016/S0022-247X(03)00057-X.

[23]

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.

[24]

W. V. Petryshyn and Z. S. Yu, On the solvability of an equation describing the periodic motions of a satellite in its elliptic orbit, Nonlinear Anal., 9 (1985), 969-975.  doi: 10.1016/0362-546X(85)90079-3.

[25] C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971. 
[26]

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.

[27]

M. Zhang and W. Li, A Lyapunov-type stability criterion using $L^α$ norms, Proc. Amer. Math. Soc., 130 (2002), 3325-3333.  doi: 10.1090/S0002-9939-02-06462-6.

[28]

A. A. Zevin, On oscillations of a satellite in the plane of elliptic orbit, Kosmich. Issled., 19 (1981), 674-679. 

[29]

A. A. Zevin and M. A. Pinsky, Qualitative analysis of periodic oscillations of an earth satellite with magnetic attitude stabilization, Discrete Contin. Dyn. Syst., 6 (2000), 193-297.  doi: 10.3934/dcds.2000.6.293.

[30]

V. A. Zlatoustov and A. P. Markeev, Stability of planar oscillations of a satellite in an elliptic orbit, Celestial Mech., 7 (1973), 31-45.  doi: 10.1007/BF01243507.

show all references

References:
[1]

V. V. Beletskii, On the oscillations of satellite, Iskusst. Sputn. Zemli, 3 (1959), 1-3. 

[2] V. V. Beletskii, The satellite Motion About Center of Mass, Nauka, Moscow, 1965. 
[3]

V. V. Beletskii and A. N. Shlyakhtin, Resonsnce Rotations of a Satellite with Interactions Between Magnetic and Gravitational Fields Preprint No. 46, Moscow: Institute of Applied Mathematics, Academy of Sciences of the USSR, 1980.

[4]

B. S. Bardin, E. A. Chekina and A. M. Chekin, On the stability of a planar resonant rotation of a satellite in an elliptic orbit, Regul. Chaotic Dyn., 20 (2015), 63-73.  doi: 10.1134/S1560354715010050.

[5]

J. Chu and M. Zhang, Rotation number and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst., 21 (2008), 1071-1094.  doi: 10.3934/dcds.2008.21.1071.

[6]

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.

[7]

J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, J. Differential Equations, 247 (2009), 530-542.  doi: 10.1016/j.jde.2008.11.013.

[8]

J. Chu and T. Xia, The Lyapunov stability for the linear and nonlinear damped oscillator with time-periodic parameters Abstr. Appl. Anal. 2010, Art. ID 286040, 12 pp. doi: 10.1155/2010/286040.

[9]

J. Chu, N. Fan and P. J. Torres, Periodic solutions for second order singular damped differential equations, J. Math. Anal. Appl., 388 (2012), 665-675.  doi: 10.1016/j.jmaa.2011.09.061.

[10]

J. Chu, J. Ding and Y. Jiang, Lyapunov stability of elliptic periodic solutions of nonlinear damped equations, J. Math. Anal. Appl., 396 (2012), 294-301.  doi: 10.1016/j.jmaa.2012.06.024.

[11]

J. Chu, P. J. Torres and F. Wang, Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem, Discrete Contin. Dyn. Syst., 35 (2015), 1921-1932.  doi: 10.3934/dcds.2015.35.1921.

[12]

D. D. Hai, Note on a differential equation describing the periodic motion of a satellite in its elliptic orbits, Nonlinear Anal., 12 (1980), 1337-1338.  doi: 10.1016/0362-546X(88)90081-8.

[13]

D. D. Hai, Multiple solutions for a nonlinear second order differential equation, Ann. Polon. Math., 52 (1990), 161-164. 

[14]

A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane, Adv. Nonlinear Stud., 12 (2012), 395-408.  doi: 10.1515/ans-2012-0210.

[15]

J. Franks, Generalization of Poincaré-Birkhoff theorem, Ann. of Math., 128 (1988), 139-151.  doi: 10.2307/1971464.

[16]

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.

[17]

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.

[18]

A. P. Markeev, B. S. Bardin and A. Planar, Rotational motion of a satellite in an elliptic orbit, Cosmic Res., 32 (1994), 583-589. 

[19]

S. Maró, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal., 42 (2013), 51-75. 

[20]

D. Núñez, The method of lower and upper solutions and the stability of periodic oscillations, Nonlinear Anal., 51 (2002), 1207-1222.  doi: 10.1016/S0362-546X(01)00888-4.

[21]

D. Nuñez and P. J. Torres, Periodic solutions of twist type of an earth satellite equation, Discrete Contin. Dyn. Syst., 7 (2001), 303-306.  doi: 10.3934/dcds.2001.7.303.

[22]

D. Nuñez and P. J. Torres, Stable odd solutions of some periodic equations modeling satellite motion, J. Math. Anal. Appl., 279 (2003), 700-709.  doi: 10.1016/S0022-247X(03)00057-X.

[23]

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.

[24]

W. V. Petryshyn and Z. S. Yu, On the solvability of an equation describing the periodic motions of a satellite in its elliptic orbit, Nonlinear Anal., 9 (1985), 969-975.  doi: 10.1016/0362-546X(85)90079-3.

[25] C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971. 
[26]

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.

[27]

M. Zhang and W. Li, A Lyapunov-type stability criterion using $L^α$ norms, Proc. Amer. Math. Soc., 130 (2002), 3325-3333.  doi: 10.1090/S0002-9939-02-06462-6.

[28]

A. A. Zevin, On oscillations of a satellite in the plane of elliptic orbit, Kosmich. Issled., 19 (1981), 674-679. 

[29]

A. A. Zevin and M. A. Pinsky, Qualitative analysis of periodic oscillations of an earth satellite with magnetic attitude stabilization, Discrete Contin. Dyn. Syst., 6 (2000), 193-297.  doi: 10.3934/dcds.2000.6.293.

[30]

V. A. Zlatoustov and A. P. Markeev, Stability of planar oscillations of a satellite in an elliptic orbit, Celestial Mech., 7 (1973), 31-45.  doi: 10.1007/BF01243507.

Figure 1.  The region of stability $\Delta$
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