Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials

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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

Daxiong Piao ^1, and Xiang Sun ^1,

1.	School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China, China

Received January 2013 Revised July 2013 Published October 2013

Abstract
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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 $p_i(t)\in C^5(R/Z).$

Keywords: Canonical transformation, Time-dependent polynomial potentials, Moser's small twist theorem., Boundedness of solutions, Impactor oscillators.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

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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