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Extinction in stochastic predator-prey population model with Allee effect on prey
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 |
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.
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. |

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