    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

 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.

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 & Continuous Dynamical Systems - B, 2017, 22 (7) : 2669-2685. doi: 10.3934/dcdsb.2017130
##### References:
  V. V. Beletskii, On the oscillations of satellite, Iskusst. Sputn. Zemli, 3 (1959), 1-3.   Google Scholar  V. V. Beletskii, The satellite Motion About Center of Mass, Nauka, Moscow, 1965.   Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  D. D. Hai, Multiple solutions for a nonlinear second order differential equation, Ann. Polon. Math., 52 (1990), 161-164. Google Scholar  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.  Google Scholar  J. Franks, Generalization of Poincaré-Birkhoff theorem, Ann. of Math., 128 (1988), 139-151.  doi: 10.2307/1971464.  Google Scholar  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  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  A. P. Markeev, B. S. Bardin and A. Planar, Rotational motion of a satellite in an elliptic orbit, Cosmic Res., 32 (1994), 583-589.   Google Scholar  S. Maró, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal., 42 (2013), 51-75. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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  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.  Google Scholar  C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971. Google Scholar  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  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.  Google Scholar  A. A. Zevin, On oscillations of a satellite in the plane of elliptic orbit, Kosmich. Issled., 19 (1981), 674-679.   Google Scholar  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.  Google Scholar  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.  Google Scholar

show all references

##### References:
  V. V. Beletskii, On the oscillations of satellite, Iskusst. Sputn. Zemli, 3 (1959), 1-3.   Google Scholar  V. V. Beletskii, The satellite Motion About Center of Mass, Nauka, Moscow, 1965.   Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  D. D. Hai, Multiple solutions for a nonlinear second order differential equation, Ann. Polon. Math., 52 (1990), 161-164. Google Scholar  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.  Google Scholar  J. Franks, Generalization of Poincaré-Birkhoff theorem, Ann. of Math., 128 (1988), 139-151.  doi: 10.2307/1971464.  Google Scholar  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  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  A. P. Markeev, B. S. Bardin and A. Planar, Rotational motion of a satellite in an elliptic orbit, Cosmic Res., 32 (1994), 583-589.   Google Scholar  S. Maró, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal., 42 (2013), 51-75. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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  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.  Google Scholar  C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971. Google Scholar  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  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.  Google Scholar  A. A. Zevin, On oscillations of a satellite in the plane of elliptic orbit, Kosmich. Issled., 19 (1981), 674-679.   Google Scholar  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.  Google Scholar  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.  Google Scholar
  Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037  Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021026  Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087  Yi Guan, Michal Fečkan, Jinrong Wang. Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1157-1176. doi: 10.3934/dcds.2020313  François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221  Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266  Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021010  Michal Fečkan, Kui Liu, JinRong Wang. $(\omega,\mathbb{T})$-periodic solutions of impulsive evolution equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021006  Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021003  Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160  Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296  Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176  Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136  Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252  Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081  Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364  Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454  Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448  Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262  Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032

2019 Impact Factor: 1.27