×

zbMATH — the first resource for mathematics

Stable odd solutions of some periodic equations modeling satellite motion. (English) Zbl 1034.34051
This paper is concerned with the existence and stability of solutions of the periodic boundary value problem \[ (m(t)x')'+f(t,x)=0,\quad x(0)=x(T), x'(0)=x'(T) , \tag{1} \] where \(f\in C^{0, 4}(\mathbb{R}/TZ\times \mathbb{R}, \mathbb{R})\), \(m\in C(\mathbb{R}/TZ, \mathbb{R}^+)\) satisfy the symmetry condition \[ f(-t, x)=-f(t,x), \qquad m(t)=m(-t). \] By using a result due to the first author [Nonlinear Anal., Theory Methods Appl. 51, 1207–1222 (2002; Zbl 1043.34044)], the authors establish a new stability theorem for equations of the form \[ (m(t)x')'+a(t)x+b(t)x^2+c(t)x^3+R(t,x)=0. \tag{2} \] On the basis of this new theorem, they further prove that equation (1) has an odd solution which is of twist type by using the method of lower and upper solutions. Finally, the parameters’ region of Lyapunov stability for the satellite equation \[ (1+e\cos (t))x''-2e\sin (t)x'+\lambda \sin x=4e\sin(t) \] is studied. For related works, one can see W. V. Petryshyn and Z. S. Yu [ Nonlinear Anal., Theory Methods Appl. 9, 969–975 (1985; Zbl 0581.70024)].

MSC:
34C25 Periodic solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnold, V.I.; Avez, A., Ergodic problems of classical mechanics, (1968), Benjamin New York · Zbl 0167.22901
[2] Bhardwaj, R.; Bhatnagar, K.B., Chaos in nonlinear planar oscillation of a satellite in an elliptical orbit under the influence of a third body torque, Indian J. pure appl. math., 28, 391-422, (1997) · Zbl 0879.70019
[3] Beletskii, V.V., On the oscillations of a satellite, Iskusst. sputn. zemli, 3, 1-3, (1959)
[4] Beletskii, V.V., The satellite motion about center of mass, (1965), Nauka Moscow · Zbl 0138.20301
[5] Dancer, E.N.; Ortega, R., The index of Lyapunov stable fixed points in two dimensions, J. dynam. differential equations, 6, 631-637, (1994) · Zbl 0811.34018
[6] De Coster, C.; Habets, P., Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, () · Zbl 0889.34018
[7] Hai, D.D., Note on a differential equation describing the periodic motion of a satellite in its elliptic orbit, Nonlinear anal., 12, 1337-1338, (1980) · Zbl 0669.70028
[8] Hai, D.D., Multiple solutions for a nonlinear second order differential equation, Ann. polon. math., 52, 161-164, (1990) · Zbl 0724.34021
[9] Kill, I.D., On periodic solutions of a certain nonlinear equation, Pmm, 27, 1107-1110, (1963) · Zbl 0134.07204
[10] Moser, J., On invariant curves of area preserving mappings of an annulus, Nachr. akad. wiss. Göttingen math.-phys. kl. II, 1-20, (1962) · Zbl 0107.29301
[11] D. Núñez, The method of lower and upper solutions and the stability of periodic oscillations, Nonlinear Anal., to appear
[12] Núñez, D.; Torres, P.J., Periodic solutions of twist type of an Earth satellite equation, Discrete contin. dynam. systems, 7, 303-306, (2001) · Zbl 1068.70027
[13] Ortega, R., The twist coefficient of periodic solutions of a time-dependent Newton’s equation, J. dynam. differential equations, 4, 651-665, (1992) · Zbl 0761.34036
[14] Ortega, R., Periodic solutions of a Newtonian equation: stability by the third approximation, J. differential equations, 128, 491-518, (1996) · Zbl 0855.34058
[15] Ortega, R., The number of stable periodic solutions of time-dependent Hamiltonian systems with one degree of freedom, Ergodic theory dynam. systems, 18, 1007-1018, (1998) · Zbl 0946.34036
[16] Petryshyn, W.V.; Yu, Z.S., On the solvability of an equation describing the periodic motions of a satellite in its elliptic orbit, Nonlinear anal., 9, 969-975, (1985) · Zbl 0581.70024
[17] Shlapak, Y.D., Periodic solutions of of nonlinear second-order equations which are not solvable for the highest derivative, Ukrainian math. J., 26, 850-854, (1974)
[18] Siegel, C.L.; Moser, J.K., Lectures on celestial mechanics, (1971), Springer-Verlag · Zbl 0312.70017
[19] Torzhevskii, A.P., Periodic solutions of the equation for two-dimensional vibrations of a satellite with elliptical orbit, Kosmich. issled., 2, 667-678, (1964)
[20] Wisdom, J.; Peale, S.J.; Mignard, F., The chaotic rotation of hyperion, Icarus, 58, 137-152, (1984)
[21] Zevin, A.A., On oscillations of a satellite in the plane of elliptic orbit, Kosmich. issled., XIX, 674-679, (1981)
[22] Zevin, A.A.; Pinsky, M.A., Qualitative analysis of periodic oscillations of an Earth satellite with magnetic attitude stabilization, Discrete contin. dynam. systems, 6, 293-297, (2000) · Zbl 1109.70308
[23] Zlatoustov, V.A.; Markeev, A.P., Stability of planar oscillations of a satellite in an elliptic orbit, Celestial mech., 7, 31-45, (1973) · Zbl 0263.70031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.