On the motion of an oscillator with a periodically time-varying mass. (English) Zbl 1163.34354

Summary: The stability of the motion of an oscillator with a periodically time-varying mass is under consideration. The key idea is that an adequate change of variables leads to a newtonian equation, where classical stability techniques can be applied: Floquet theory for the linear oscillator, KAM method in the nonlinear case. To illustrate this general idea, first we have generalized the results of [W.T. van Horssen, A.K. Abramian, Hartono, On the free vibrations of an oscillator with a periodically time-varying mass, J. Sound Vibration 298 (2006) 1166-1172] to the forced case; second, for a weakly forced Duffing’s oscillator with variable mass, the stability in the nonlinear sense is proved by showing that the first twist coefficient is not zero.


34C25 Periodic solutions to ordinary differential equations
Full Text: DOI


[1] Arnold, V.I., Méthodes mathémathiques de la Mécanique classique, (1976), Mir Moscow
[2] Beletskii, V.V., On the oscillations of a satellite, Iskusst. sputn. zamli, 3, 1-3, (1959)
[3] Broer, H.; Levi, H., Geometrical aspects of stability theory for hill’s equations, Arch. rat. mech. anal., 131, 3, 225-240, (1995) · Zbl 0840.34047
[4] Broer, H.; Simo, C., Resonance tongues in hill’s equations: A geometric approach, J. differential equations, 166, 2, 290-327, (2000) · Zbl 1046.34072
[5] van der Burg, A.H.P.; Hartono; Abramian, A.K., A new model for the study of rain-wind-induced vibrations of a simple oscillator, Int. J. nonlinear mech., 41, 345-358, (2006) · Zbl 1160.74323
[6] Cesari, L., Asymptotic behavior and stability problems in ordinary differential equations, (1971), Springer-Verlag · Zbl 0215.13802
[7] Gan, S.; Zhang, M., Resonance pockets of hill’s equations with two-step potentials, SIAM J. math. anal., 32, 651-664, (2000) · Zbl 0973.34019
[8] Holl, H.J.; Belyaev, A.K.; Irschik, H., Simulation of the Duffing-oscillator with time-varying mass by a BEM in time, Comput. struct., 73, 177-186, (1999) · Zbl 0968.70004
[9] van Horssen, W.T.; Abramian, A.K.; Hartono, On the free vibrations of an oscillator with a periodically time-varying mass, J. sound vibration, 298, 1166-1172, (2006) · Zbl 1243.74061
[10] Irschik, H.; Holl, H.J., Mechanics of variable-mass systems — part 1: balance of mass and linear momentum, Appl. mech. rev., 57, 2, 145-160, (2004)
[11] Lei, J.; Li, X.; Yan, P.; Zhang, M., Twist character of the least amplitude periodic solution of the forced pendulum, SIAM J. math. anal., 35, 844-867, (2003) · Zbl 1189.37064
[12] J. Lei, M. Zhang, Twist property of periodic motion of an atom near a charged wire, Lett. Math. Phys. 60, 9-17 · Zbl 1002.78006
[13] Loud, W., Periodic solutions of perturbed second-order autonomous equations, Mem. amer. math. soc., N 47, (1964) · Zbl 0128.31802
[14] Meyer, K.R.; Hall, G.R, Introduction to Hamiltonian dynamical system and the N-body problem, (1992), Springer-Verlag New York · Zbl 0743.70006
[15] Markus, L.; Meyer, K.R., Periodic orbits and solenoids in generic Hamiltonian dynamical system, Am. J. math., 102, 25-92, (1980) · Zbl 0438.58013
[16] Magnus, W.; Winkler, S., Hill’s equation, (1966), Dover · Zbl 0158.09604
[17] Moser, J., Stability and nonlinear character of ordinary differential equations, (), 139-150
[18] Núñez, D., The method of lower and upper solutions and the stability of periodic oscillations, Nonlinear anal. TMA, 51, 1207-1222, (2002) · Zbl 1043.34044
[19] Núñez, D.; Ortega, R., Parabolics fixed points and stability criteria for nonlinear hill’s equation, Z. angew. math. phys., 51, 890-911, (2000) · Zbl 0973.34046
[20] Núñez, D.; Torres, P.J., Periodic solutions of twist type of an Earth satellite equation, Discr. cont. dyn. syst., 7, 303-306, (2001) · Zbl 1068.70027
[21] Núñez, D.; Torres, P.J., Stable odd solutions of some periodic equations modeling satellite motion, J. math. anal. appl., 279, 700-709, (2003) · Zbl 1034.34051
[22] Núñez, D.; Torres, P.J., KAM dynamics and stabilization of a particle sliding over a periodically driven wire, Appl. math. lett., 20, 610-615, (2007) · Zbl 1130.70320
[23] Ortega, R., The twist coefficient of periodic solutions of a time-dependent newton’s equations, J. dynam. differential equations, 4, 651-665, (1992) · Zbl 0761.34036
[24] Ortega, R., The stability of the equilibrium of a non linear hill’s equation, SIAM J. math. anal., 25, 1393-1401, (1994) · Zbl 0807.34065
[25] Ortega, R., Periodic solutions of a Newtonian equation: stability by the third approximation, J. differential equations., 128, 491-518, (1996) · Zbl 0855.34058
[26] Poincaré, H., LES methodes nouvelles de la mechanique celeste, vol. 1, (1957), Dover · Zbl 0079.23801
[27] Siegel, C.L.; Moser, J., Lectures on celestial mechanics, (1971), Springer-Verlag New York, Berlin · Zbl 0312.70017
[28] Torres, P.J., Twist solutions of a hill’s equations with singular term, Adv. nonlinear stud., 2, 279-287, (2002) · Zbl 1016.34044
[29] Vladimirov, V.S., Generalized functions in mathematical physics, (1979), Mir Moscow · Zbl 0515.46033
[30] Wisdom, J.; Peale, S.J.; Mignard, F., The chaotic rotation of hiperion, Icarus, 58, 137-152, (1984)
[31] Zhang, M., The best bound on the rotations in the stability of periodic solutions of a Newtonian equation, J. London math. soc., 67, 2, 137-148, (2003) · Zbl 1050.34075
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.