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

Previous Article
The Fueter primitive of biaxially monogenic functions
CPAA Home
This Issue
Next Article
A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities

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
Related Papers

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.

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.

[1]	Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017
[2]	Huiping Jin. Boundedness in a class of duffing equations with oscillating potentials via the twist theorem. Communications on Pure and Applied Analysis, 2011, 10 (1) : 179-192. doi: 10.3934/cpaa.2011.10.179
[3]	Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
[4]	Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911
[5]	Božzidar Jovanović. Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems. Journal of Geometric Mechanics, 2018, 10 (2) : 173-187. doi: 10.3934/jgm.2018006
[6]	Alessandro Fortunati, Stephen Wiggins. Normal forms à la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1109-1118. doi: 10.3934/dcdss.2016044
[7]	Florian Wagener. A parametrised version of Moser's modifying terms theorem. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 719-768. doi: 10.3934/dcdss.2010.3.719
[8]	Yavar Kian, Alexander Tetlow. Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation. Inverse Problems and Imaging, 2020, 14 (5) : 819-839. doi: 10.3934/ipi.2020038
[9]	Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations and Control Theory, 2022, 11 (2) : 515-536. doi: 10.3934/eect.2021011
[10]	Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations and Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015
[11]	Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1833-1849. doi: 10.3934/cpaa.2021044
[12]	Haixia Li. Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity. Evolution Equations and Control Theory, 2021, 10 (4) : 723-732. doi: 10.3934/eect.2020088
[13]	Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure and Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003
[14]	Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240
[15]	Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 865-900. doi: 10.3934/dcdss.2018053
[16]	Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141
[17]	Morteza Fotouhi, Mohsen Yousefnezhad. Homogenization of a locally periodic time-dependent domain. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1669-1695. doi: 10.3934/cpaa.2020061
[18]	G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks and Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1
[19]	Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure and Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969
[20]	Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Daxiong Piao Xiang Sun
on Google Scholar
Daxiong Piao Xiang Sun

[Back to Top]