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Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials
1. | School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China, China |
References:
[1] |
R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the Twist Theorem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 79-95. |
[2] |
S. Laederich and M. Levi, Invariant curves and time-dependent potentials, Ergo.Th. and Dynam. Syst., 11 (1991), 365-378.
doi: 10.1017/S0143385700006192. |
[3] |
X. Yuan, Invariant tori of Duffing-type equations, Adv. in Math. (China), 24 (1995), 375-376. |
[4] |
X. Yuan, Invariant tori of Duffing-type equations, J. Differential Equations, 142 (1998), 231-162. |
[5] |
M. Kunze, "Non-Smooth Dynamical Systems," in: Lecture Notes in Math., vol.1744, Sringer-Verlag, New York, 2000. |
[6] |
H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator, Physica D, 82 (1995), 117-135.
doi: 10.1016/0167-2789(94)00222-C. |
[7] |
P. Boyland, Dual billiards, twist maps and impact oscillators, Nonlinearity, 9 (1996), 1411-1438.
doi: 10.1088/0951-7715/9/6/002. |
[8] |
M. Corbera and J. Llibre, Periodic orbits of a collinear restricted three body problem, Celestial Mech. Dynam. Astronom., 86 (2003), 163-183.
doi: 10.1023/A:1024183003251. |
[9] |
D. Qian and P. J. Torres, Periodic motions of linear impact oscilltors via successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.
doi: 10.1137/S003614100343771X. |
[10] |
D. Qian and X. Sun, Inariant tori for asymptotically linear impact oscillators, Sci. China: Ser. A Math., 49 (2006), 669-687.
doi: 10.1007/s11425-006-0669-5. |
[11] |
V. Zharnitsky, Invariant tori in Hamiltonian systems with impacts, Comm. Math. Phys., 211 (2000), 289-302.
doi: 10.1007/s002200050813. |
[12] |
Z. Wang and Y. Wang, Existence of quasiperiodic solutions and Littlewood's boundedness problem of super-linear impact oscillators, Applied Mathematics and Computation, 217 (2011), 6417-6425.
doi: 10.1016/j.amc.2011.01.037. |
[13] |
Z. Wang, Q. Liu and D. Qian, Existence of quasiperiodic solutions and Littlewood's boundedness problem of sub-linear impact oscillators, Nonlinear Analysis, 74 (2011), 5606-5617.
doi: 10.1016/j.na.2011.05.046. |
[14] |
D. Qian, Large amplitude periodic bouncing in impact oscillators with damping, Proc. Amer. Math. Soc., 133 (2005), 1797-1804.
doi: 10.1090/S0002-9939-04-07759-7. |
[15] |
D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity, Proc. Roy. Soc. Edinburgh, 134 (2004), 201-213.
doi: 10.1017/S0308210500003164. |
[16] |
Z. Wang, C. Ruan and D. Qian, Existence and multiplicity of subharmonic bouncing solutions for sub-linear impact oscillators, J. Nanjing Univ. Math. Biquart., 27 (2010), 17-30.
doi: 10.3969/j.issn.0469-5097.2010.01.003. |
[17] |
R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London. Math. Soc., 79 (1999), 381-413.
doi: 10.1112/S0024611599012034. |
[18] |
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Commun. Math. Phys., 143 (1991), 43-83.
doi: 10.1007/BF02100285. |
[19] |
B. Liu, Boundedness in nonlinear oscillations at resonance, J. Differential Equations, 153 (1999), 142-174.
doi: 10.1006/jdeq.1998.3553. |
[20] |
L. Jiao, D. Piao and Y. Wang, Boundedness for the general semilinear Duffing equation via the twist theorem, J. Differential Equations, 252 (2012), 91-113.
doi: 10.1016/j.jde.2011.09.019. |
[21] |
J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. wiss, Gottingen Math. -phys., Kl. II (1962), 1-20. |
[22] |
H. Rüssman, On the existence of invariant curves of twist mappings of an annulus, Lecture Notes Math., 1007 (1983), 677-718.
doi: 10.1007/BFb0061441. |
show all references
References:
[1] |
R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the Twist Theorem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 79-95. |
[2] |
S. Laederich and M. Levi, Invariant curves and time-dependent potentials, Ergo.Th. and Dynam. Syst., 11 (1991), 365-378.
doi: 10.1017/S0143385700006192. |
[3] |
X. Yuan, Invariant tori of Duffing-type equations, Adv. in Math. (China), 24 (1995), 375-376. |
[4] |
X. Yuan, Invariant tori of Duffing-type equations, J. Differential Equations, 142 (1998), 231-162. |
[5] |
M. Kunze, "Non-Smooth Dynamical Systems," in: Lecture Notes in Math., vol.1744, Sringer-Verlag, New York, 2000. |
[6] |
H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator, Physica D, 82 (1995), 117-135.
doi: 10.1016/0167-2789(94)00222-C. |
[7] |
P. Boyland, Dual billiards, twist maps and impact oscillators, Nonlinearity, 9 (1996), 1411-1438.
doi: 10.1088/0951-7715/9/6/002. |
[8] |
M. Corbera and J. Llibre, Periodic orbits of a collinear restricted three body problem, Celestial Mech. Dynam. Astronom., 86 (2003), 163-183.
doi: 10.1023/A:1024183003251. |
[9] |
D. Qian and P. J. Torres, Periodic motions of linear impact oscilltors via successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.
doi: 10.1137/S003614100343771X. |
[10] |
D. Qian and X. Sun, Inariant tori for asymptotically linear impact oscillators, Sci. China: Ser. A Math., 49 (2006), 669-687.
doi: 10.1007/s11425-006-0669-5. |
[11] |
V. Zharnitsky, Invariant tori in Hamiltonian systems with impacts, Comm. Math. Phys., 211 (2000), 289-302.
doi: 10.1007/s002200050813. |
[12] |
Z. Wang and Y. Wang, Existence of quasiperiodic solutions and Littlewood's boundedness problem of super-linear impact oscillators, Applied Mathematics and Computation, 217 (2011), 6417-6425.
doi: 10.1016/j.amc.2011.01.037. |
[13] |
Z. Wang, Q. Liu and D. Qian, Existence of quasiperiodic solutions and Littlewood's boundedness problem of sub-linear impact oscillators, Nonlinear Analysis, 74 (2011), 5606-5617.
doi: 10.1016/j.na.2011.05.046. |
[14] |
D. Qian, Large amplitude periodic bouncing in impact oscillators with damping, Proc. Amer. Math. Soc., 133 (2005), 1797-1804.
doi: 10.1090/S0002-9939-04-07759-7. |
[15] |
D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity, Proc. Roy. Soc. Edinburgh, 134 (2004), 201-213.
doi: 10.1017/S0308210500003164. |
[16] |
Z. Wang, C. Ruan and D. Qian, Existence and multiplicity of subharmonic bouncing solutions for sub-linear impact oscillators, J. Nanjing Univ. Math. Biquart., 27 (2010), 17-30.
doi: 10.3969/j.issn.0469-5097.2010.01.003. |
[17] |
R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London. Math. Soc., 79 (1999), 381-413.
doi: 10.1112/S0024611599012034. |
[18] |
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Commun. Math. Phys., 143 (1991), 43-83.
doi: 10.1007/BF02100285. |
[19] |
B. Liu, Boundedness in nonlinear oscillations at resonance, J. Differential Equations, 153 (1999), 142-174.
doi: 10.1006/jdeq.1998.3553. |
[20] |
L. Jiao, D. Piao and Y. Wang, Boundedness for the general semilinear Duffing equation via the twist theorem, J. Differential Equations, 252 (2012), 91-113.
doi: 10.1016/j.jde.2011.09.019. |
[21] |
J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. wiss, Gottingen Math. -phys., Kl. II (1962), 1-20. |
[22] |
H. Rüssman, On the existence of invariant curves of twist mappings of an annulus, Lecture Notes Math., 1007 (1983), 677-718.
doi: 10.1007/BFb0061441. |
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