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A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities
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.
|
[2] |
S. Laederich and M. Levi, Invariant curves and time-dependent potentials,, Ergo.Th. and Dynam. Syst., 11 (1991), 365.
doi: 10.1017/S0143385700006192. |
[3] |
X. Yuan, Invariant tori of Duffing-type equations,, Adv. in Math. (China), 24 (1995), 375. Google Scholar |
[4] |
X. Yuan, Invariant tori of Duffing-type equations,, J. Differential Equations, 142 (1998), 231. Google Scholar |
[5] |
M. Kunze, "Non-Smooth Dynamical Systems,", in: Lecture Notes in Math., (2000).
|
[6] |
H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator,, Physica D, 82 (1995), 117.
doi: 10.1016/0167-2789(94)00222-C. |
[7] |
P. Boyland, Dual billiards, twist maps and impact oscillators,, Nonlinearity, 9 (1996), 1411.
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.
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.
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.
doi: 10.1007/s11425-006-0669-5. |
[11] |
V. Zharnitsky, Invariant tori in Hamiltonian systems with impacts,, Comm. Math. Phys., 211 (2000), 289.
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.
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.
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.
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.
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.
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.
doi: 10.1112/S0024611599012034. |
[18] |
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials,, Commun. Math. Phys., 143 (1991), 43.
doi: 10.1007/BF02100285. |
[19] |
B. Liu, Boundedness in nonlinear oscillations at resonance,, J. Differential Equations, 153 (1999), 142.
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.
doi: 10.1016/j.jde.2011.09.019. |
[21] |
J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. wiss, Kl. (1962), 1.
|
[22] |
H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, Lecture Notes Math., 1007 (1983), 677.
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.
|
[2] |
S. Laederich and M. Levi, Invariant curves and time-dependent potentials,, Ergo.Th. and Dynam. Syst., 11 (1991), 365.
doi: 10.1017/S0143385700006192. |
[3] |
X. Yuan, Invariant tori of Duffing-type equations,, Adv. in Math. (China), 24 (1995), 375. Google Scholar |
[4] |
X. Yuan, Invariant tori of Duffing-type equations,, J. Differential Equations, 142 (1998), 231. Google Scholar |
[5] |
M. Kunze, "Non-Smooth Dynamical Systems,", in: Lecture Notes in Math., (2000).
|
[6] |
H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator,, Physica D, 82 (1995), 117.
doi: 10.1016/0167-2789(94)00222-C. |
[7] |
P. Boyland, Dual billiards, twist maps and impact oscillators,, Nonlinearity, 9 (1996), 1411.
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.
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.
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.
doi: 10.1007/s11425-006-0669-5. |
[11] |
V. Zharnitsky, Invariant tori in Hamiltonian systems with impacts,, Comm. Math. Phys., 211 (2000), 289.
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.
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.
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.
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.
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.
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.
doi: 10.1112/S0024611599012034. |
[18] |
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials,, Commun. Math. Phys., 143 (1991), 43.
doi: 10.1007/BF02100285. |
[19] |
B. Liu, Boundedness in nonlinear oscillations at resonance,, J. Differential Equations, 153 (1999), 142.
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.
doi: 10.1016/j.jde.2011.09.019. |
[21] |
J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. wiss, Kl. (1962), 1.
|
[22] |
H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, Lecture Notes Math., 1007 (1983), 677.
doi: 10.1007/BFb0061441. |
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