# American Institute of Mathematical Sciences

March  2014, 13(2): 645-655. doi: 10.3934/cpaa.2014.13.645

## 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

Received  January 2013 Revised  July 2013 Published  October 2013

In this paper, we consider the boundedness of solutions for a class of impact oscillators with time dependent polynomial potentials, \begin{eqnarray} \ddot{x}+x^{2n+1}+\sum_{i=0}^{2n}p_{i}(t)x^{i}=0, \quad for\ x(t)> 0,\\ x(t)\geq 0,\\ \dot{x}(t_{0}^{+})=-\dot{x}(t_{0}^{-}), \quad if\ x(t_{0})=0, \end{eqnarray} where $n\in N^+$, $p_i(t+1)=p_i(t)$ and $p_i(t)\in C^5(R/Z).$
Citation: Daxiong Piao, Xiang Sun. Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials. Communications on Pure and Applied Analysis, 2014, 13 (2) : 645-655. doi: 10.3934/cpaa.2014.13.645
##### 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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##### 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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